Test Bank for Calculus with Applications, 12th Edition.
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1|Pa ge
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Perform the indicated operation. 1) (8x2 - 3x - 9) + (-5x2 - 7x + 12)
A) 13x2 - 10x + 21
1)
B) 3x2 - 10x + 3
C) -3x2 + 10x + 3
D) 13x2 - 10x - 21
2) (9n5 + 20n3 + 5) - (5n5 - 8n3 - 2) A) 4n5 + 28n3 + 3 B) 4n5 + 28n3 + 7
C) 4n5 + 25n3 + 3
D) 39n8
2)
3) (2m 5 + 7m 2 - 3m) + (6m 5 + 5m 2 + 6m) A) 8m 5 + 12m 2 + 3m C) 23m 8
3) B) 3m 5 + 7m 2 + 13m D) 8m + 12m 5 + 3m 2
4) (4q2 - 6q - q3 + 2) - (6q2 - 6q - q3 + 7) A) -3q2 - 2q3 + 5 B) -2q2 - 5
4) C) -2q2 - 12q - 9
5) (5p2 + 4p - p3 + 7) - (8p2 - 2p - p3 + 4) A) -3p2 - 6p - 11 C) -3p2 + 6p + 3
D) -4p2 + 2p - 11
6) (7r4 + 9r2 - 7r ) - (-8r3 + 3r2 - 7r - 1) A) 11r7 + 12r3 C) 8r4 + 9r3 + 12r2 + 1
B) 7r4 + 8r3 + 6r2 + 1 D) 7r4 - 8r3 + 12r2 - 1
D) -3q2 - 9 5)
B) -4p2 - 2p3 + 2p + 3
6)
7) (6k4 - 8k3 + 6k2 + 2) + (5k4 + 2k3 + 3k2 + 5) A) 11k4 - 6k3 + 9k2 + 7 C) 11k8 - 6k6 + 9k4 + 7
B) 14k18 + 7 D) 4k4 + 4k3 + 11k2 - 3
8) (9y7 - 9y6 - 8y5 + 1) - (4y7 - 7y6 + 6y5 + 2) A) 13y7 - 16y6 - 2y5 - 1 C) 5y7 - 16y6 - 2y5 + 3
B) 5y7 - 2y6 - 14y5 - 1 D) 13y7 - 16y6 - 2y5 + 3
9) 3(q2 - 3q + 3) + 2(1q2 + q - 5) A) 4q2 + 11q + 1 B) 5q2 + 11q + 1
C) 5q2 - 7q - 1
7)
8)
9)
1
D) 5q2 + 7q - 1
10) 0.36(2x2 + 2x - 5) - (3.23x2 - 6x + 6.865) A) -2.51x2 + 6.72x - 8.665 C) -2.51x2 - 4x - 8.665
10) B) -1.23x2 + 6.72x - 8.665 D) -1.23x2 + 8x - 11.865
11) -5p(4p2 - 12p + 6) A) -20p3 + 60p2 + 6p C) -20p3 + 60p2 - 30p
D) -20p3 - 12p2 - 30p
12) 4m(-9m 2 - 2m + 12) A) -36m 2 - 8m + 48 C) 36m 3 + 8m 2 - 48m
B) -36m 3 - 8m + 48 D) -36m 3 - 8m 2 + 48m
11) B) -20p3 + 30p2
12)
13) 5y2 (12y2 - 6y - 1) A) 60y4 - 30y2 - 5y C) 60y4 - 30y3 - 5y2 14) (4x + 10)(x + 2) A) 4x2 + 18x + 20 15) (3k + 2)(k + 11) A) 3k2 + 33k + 22
13) B) 60y3 - 30y2 - 5y D) - 60y4 + 30y3 + 5y2 14) B) x2 + 18x + 17
C) x2 + 20x + 18
15) B) 3k2 + 35k + 35
C) 3k2 + 35k + 22
16) (m + 12)(-4m - 7) A) -4m 2 - 55m - 55 C) -4m 2 - 84m - 55
B) -4m 2 - 57m - 84 D) -4m 2 - 55m - 84
17) (-4p - 10)(-4p - 11) A) -8p2 + 84p + 110 C) 16p2 + 84p + 84
B) -8p2 + 84p + 84 D) 16p2 + 84p + 110
18) (x - 11y)(x + 3y) A) x2 - 8xy - 8y2 C) x - 8xy - 33y
B) x2 - 11xy - 33y2 D) x2 - 8xy - 33y2
19) (a - 3b)(-3a + 7b) A) a2 + 16ab - 21b2 C) -3a 2 + 16ab + 16b2
B) -3a 2 + 16ab - 21b2 D) a2 + 16ab + 16b2
20) (3x + 2)(3x - 2) A) 9x2 + 12x - 4
D) 4x2 + 17x + 20
D) 3k2 + 22k + 35 16)
17)
18)
19)
20) B) 3x2 - 12x - 4
C) 9x2 - 4
2
D) 9x2 - 12x - 4
21) (2r + 5)(2r + 5) A) 2r2 + 20r + 19
21) B) 2r2 + 19r + 25
C) 4r2 + 20r + 25
D) 4r2 + 25r + 20
22) q + 4 q + 3 3
22)
4
B) q2 + 25 q + 1
A) q2 + 1
C) q2 + 25 q + 1
12
6
D) 2q2 + 25 q + 1 12
23) 1 m - 1 5 m - 2 3
23)
3
A) 5 m 2 - 7 m + 2 3
3
B) 5 m 2 - 7 m + 2 9
C) 5 m 2 - 7 m + 2
3
6
24) (x + 5)(x2 - x + 7) A) x3 + 6x2 + 12x + 35 C) x3 + 4x2 + 35
3
D) 5 m 2 - 7 m + 4 3
3
24) B) x3 + 4x2 + 2x + 35 D) x3 + 35
25) (m - 5)(9m 2 + m + 8) A) 9m 3 - 44m 2 + 13m - 40 C) 9m 3 - 46m 2 + 3m - 40
25) B) 9m 3 - 44m 2 + 3m - 40 D) 9m 3 + 44m 2 + 3m - 40
26) (3m - 5)(2m 3 + 4m 2 + 5m + 5) A) 6m 4 - 8m 3 - 5m 2 - 10m - 25 C) 6m 4 + 2m 3 - 5m 2 + 15m - 25
26) B) 6m 4 + 2m 3 - 10m 2 - 10m - 25 D) 6m 4 + 2m 3 - 5m 2 - 10m - 25
27) (-4x + 6y)(2x - 2y + 1) A) -8x2 + 20xy + 20y2 C) -8x2 + 8xy - 4x - 12y2
B) -8x2 + 20xy - 4x - 12y2 + 6y D) -8x2 + 12xy - 4x - 12y2 + 6y
28) (-4a 2 + 4b)(3a 2 + 2b + c) A) -12a2 + 4ab - 4a2 c + 8b2 + 4c C) -12a4 + 4a 2 b - 4a 2 c + 8b2 + 4bc
B) -12a4 + 4a 2 b2 + 8b2 D) -12a4 + 4a 2 b + 8b4 - 4a 2bc
27)
28)
29) (3x2 - 2x - 4)(x2 + 4x + 2) A) 3x4 + 12x3 - 6x2 - 20x - 8 C) 3x4 + 10x3 - 2x2 - 20x - 8
29) B) 3x4 + 10x3 - 6x2 - 20x - 8 D) 3x4 + 12x3 - 2x2 - 20x - 8
30) (x - 5)(x - 2)(x + 2) A) x3 - 5x2 - 4x + 20 C) x3 - 9x2 + 24x - 20
30) B) x3 + 20 D) x3 + 4x2 - 5x - 20
3
31) (n + 2)2 A) n2 + 4
31) C) 4n2 + 4n + 4
B) n + 4
32) (7x - 11y)2 A) 49x2 + 121y2 C) 49x2 - 154xy + 121y2
32) B) 7x2 + 121y2 D) 7x2 - 154xy + 121y2
33) (3a - b)3 A) 27a 3 - b3 C) 27a 3 - 9a 2 b + 3ab2 - b3 Factor out the greatest common factor. 34) 9a 6 + 36a 4
A) 9a 4 (a 2 + 4)
D) n2 + 4n + 4
33) B) 27a 3 -27a 2 b + 9ab2 - b3 D) 27a 3 - 27a 2b - 9ab2 - b3
34) B) 36(a 2 + 4a)
C) a4 (9a 2 + 36)
35) 10x3 - 20x2 + 15x A) 5x(2x2 - 4x + 3) C) 10x(x2 - 2x + 3)
D) 9(a 6 + 4a 4) 35)
B) 5(2x3 - 4x2 + 3x) D) 5x(2x2 - 20x2 + 15x)
36) 27m 9 + 6m 6 + 18m 2 A) m 2(27m 7 + 6m 4 + 18) C) no common factor
36) B) 3m 2 (9m 7 + 2m 4 + 6) D) 3(9m 9 + 2m 6 + 6m 2)
37) 84m 9 + 72m 6 + 84m 3 A) 12m 3 (7m 6 + 6m 3 + 7) C) 12(7m 9 + 6m 6 + 7m 3 )
37) B) no common factor D) m 3(84m 6 + 72m 3 + 84)
38) 18x4y3 - 6x3y2 + 10x2 y A) 2x2 y2(9x2 y - 3x + 5) C) 2(9x4 y3 - 3x3 y2 + 5x2 y)
38) B) 2xy(9x3 y2 - 3x2 y + 5x) D) 2x2 y(9x2 y2 - 3xy + 5)
39) 24x9y8 + 36x6 y6 + 20x3y2 A) 4(6x9 y8 + 9x6 y6 + 5x3 y2 ) C) 4x3 y2(6x6 y6 + 9x3 y4 + 5)
39) B) no common factor D) 4x3 (6x6 y8 + 9x3 y6 + 5y2 )
40) 72x8y7 + 45x3 y5 + 63x6y2 A) 9x3 y2(8x5 y5 + 5y3 + 7x3 ) C) 9x3 (8x5 y7 + 5y5 + 7x3 y2 )
40) B) 9(8x8 y7 + 5x3 y5 + 7x6 y2 ) D) no common factor
4
41) 13m 2 - 25r3 A) 2(6m 2 + 12r3) C) no common factor
41) B) 3(4m 2 - 8r3 ) D) m 2(13 - 25m)
Factor completely. State that the polynomial is prime if it cannot be factored. 42) x2 + 3x - 108
A) (x - 12)(x + 1)
42)
B) prime
C) (x - 12)(x + 9)
D) (x + 12)(x - 9)
43) x2 - 4x - 45 A) (x + 5)(x - 9)
B) (x - 5)(x + 1)
C) prime
D) (x - 5)(x + 9)
44) x2 + 7x + 10 A) (x - 5)(x - 2)
B) (x + 10)(x + 7)
C) (x + 10)(x - 1)
D) (x + 5)(x + 2)
45) x2 - 10x + 24 A) (x - 4)(x - 6) 46) x2 - x - 45 A) (x - 45)(x + 1) 47) 4x2 - 25x - 21 A) (x + 4)(x - 7)
43)
44)
45) B) (x + 4)(x - 6)
C) (x - 4)(x + 6)
46) B) (x + 5)(x - 9)
C) (x - 5)(x + 9)
B) (4x + 3)(x - 7)
C) (x - 3)(x - 7)
52) 6x2 - 5xt - 4t2 A) prime
D) (3x + 4)(x - 7) 48)
B) (9x + 14)(9x - 2) D) (9x - 2)(x - 8) 49) B) prime
C) (3z + 2)(5z - 4)
50) u2 - 4uv - 12v2 A) (u - 2v)(u + v) C) prime 51) x2 - 9xy + 14y2 A) (x + 7y)(x - 2y)
D) prime 47)
48) 9x2 - 74x + 16 A) (2 - 9x)(9x - 2) C) (9x - 2)(x + 8) 49) 15z 2 - 2z - 8 A) (15z + 2)(z - 4)
D) (x + 4)(x + 6)
D) (3z - 2)(5z + 4) 50)
B) (u - 2v)(u + 6v) D) (u + 2v)(u - 6v) 51) B) (x - 14y)(x - y)
C) (x - 7y)(x - 2y)
D) prime 52)
B) (2x + t)(3x - 4t)
C) (6x + t)(x - 4t)
5
D) (2x - t)(3x + 4t)
53) 12m 2 - 31mn + 20n2 A) (4m - 5n)(3m + 4n) C) (4mn - 5)(3mn - 4) 54) 8x2 - 8x - 48 A) prime
53) B) (4mn + 5)(3mn + 4) D) (4m - 5n)(3m - 4n) 54) B) 8(x - 2)(x + 3)
C) 8(x + 2)(x - 3)
55) 24x2 - 104x - 80 A) prime C) 8(3x + 2)(x - 5) 56) 6y2 + 27y - 15 A) 3(2y - 1)(y + 5)
D) (8x + 16)(x - 3) 55)
B) (24x + 16)(x - 5) D) 8(3x - 2)(x + 5) 56) B) (6y - 3)(y + 5)
C) prime
57) 2x2 - 6xy - 8y2 A) (2x - 2y)(x + 4y) C) 2(x + y)(x - 4y)
D) 3(2y + 1)(y - 5) 57)
B) prime D) 2(x - y)(x + 4y)
58) 5y3 - 5y2 - 30y A) (y - 2)(5y2 + 15) C) 5y(y + 2)(y - 3)
58) B) 5y(y - 2)(y + 3) D) (5y2 + 10y)(y - 3)
59) 18x3 - 78x2 - 60x A) x(3x + 2)(6x - 30) C) 6(3x - 2)(x + 5)
59) B) (x2 - 5)(18x + 12) D) 6x(3x + 2)(x - 5)
60) 24x2 + 14xy + 2y2 A) (6x + 2y)(4x + y) C) prime
60) B) 2(3x - y)(4x - y) D) 2(3x + y)(4x + y)
61) x3 - 11x2 + 30x A) x(x + 5)(x - 6) C) x(x - 5)(x + 6)
61) B) x(x - 5)(x - 6) D) x(x2 - 11x + 30)
62) a3 b - 9a 2b2 + 20ab3 A) ab(a - 20b)(a - b) C) a(ab - 5)(ab - 4)
62) B) ab(a - 5b)(a - 4b) D) ab(a - 5b)(a + 4b)
63) 4x3 - 28x2 y - 120xy2 A) 4x(x - 10y)(x + 3y) C) 4x(x + 10y)(x - 3y)
63) B) 4(x - 10y)(x + 3y) D) 4xy(x - 10y)(x + 3y)
6
64) 32x4 + 72x3y + 36x2 y2 A) 4xy(2x + 1)(4x + 9) C) 4x2 (2x + 1)(4x + 9) 65) 81x2 - 16 A) (9x - 4)2
64) B) 4x2 (2x + 3y)(4x + 3y) D) 4x2 y2(2x + 3)(4x + 3) 65) B) prime
C) (9x + 4)(9x - 4)
66) 9k2 - 49m 2 A) prime C) (3k + 7m)2
D) (9x + 4)2 66)
B) (3k + 7m)(3k - 7m) D) (3k - 7m)2
67) 81y4 - 64 A) (9y2 + 8)2 C) prime
B) (9y2 - 8)2 D) (9y2 + 8)(9y2 - 8)
68) 81s2 - 25t4 A) (9s + 5t2 )(9s - 5t2 ) C) (9s + 5t2 )2
B) prime D) (9s - 5t2 )2
67)
68)
69) 147a4 - 48b2 A) 3(7a 2 + 4b)2 C) 3(7a 2 + 4b)(7a 2 - 4b)
69) B) prime D) 3(7a 2 - 4b)2
70) 8a 4 b - 18b3 A) 2b(2a + 3b)2 C) 2b(2a 2 + 3b)(2a 2 - 3b)
D) prime
71) 36x2 + 25 A) (6x + 5)2
C) (6x - 5)2
70) B) 2b(2a - 3b)2
71) B) (6x + 5)(6x - 5)
72) 25x4 - 49y4 A) prime C) (5x2 - 7y2)(5x2 + 7y2)
D) prime 72)
B) (5x - 7y)(5x + 7y)(5x2 + 7y2) D) -24x4
73) 100pm 4 - 100pn 4
73)
A) 100(m 2p + n2)(mp + n)(mp - n)
B) 100p(m 2 + n2)(m + n)(m - n)
C) 100p(m 2 - n2)2
D) p(10m 2 + n2)(10m + n)(10m - n)
7
74) x4 - 625 A) (x + 5)2 (x - 5)2 C) prime 75) x2 + 50x + 625 A) (x + 25)(x - 25) 76) 16x2 + 24x + 9 A) (4x + 3)(4x - 3) 77) x2 - 6xy + 9y2 A) prime 78) x2 - 8x + 64 A) (x - 8)2
74) B) (x2 - 25)(x + 5)(x - 5) D) (x2 + 25)(x + 5)(x - 5) 75) B) (x + 25)2
C) (x - 25)2
D) prime 76)
B) (4x - 3)2
C) (4x + 3)2
D) Prime
B) (x - 3y)(x + 3y)
C) (x + 3y)2
D) (x - 3y)2
C) prime
D) (x + 8)2
77)
78) B) (x + 8)(x - 8)
79) 27x2 + 90x + 75 A) (9x + 15)(3x + 5) C) 3(3x - 5)(3x + 5)
79) B) 3(9x2 + 30x + 25) D) 3(3x + 5)2
80) 196x2 - 336xy + 144y2 A) 4(7x + 6y)(7x - 6y) C) 4(49x2 - 84xy + 36y)
80) B) (28x - 24y)(7x - 6y) D) 4(7x - 6y)2
Factor completely. 81) 729p3 - 1
81)
A) (729p - 1)(p2 + 9p + 1) C) (9p + 1)(81p2 - 9p + 1)
B) (9p - 1)(81p2 + 1) D) (9p - 1)(81p2 + 9p + 1)
82) x3 - 343 A) (x + 7)(x2 - 7x + 49) C) (x + 343)(x2 - 1)
82) B) (x - 7)(x2 + 49) D) (x - 7)(x2 + 7x + 49)
83) 1,000y3 - 343 A) (10y - 7)(100y2 + 70y + 49) C) (10y + 7)(100y2 - 70y + 49)
B) (10y - 7)(100y2 + 49) D) (1,000y - 7)(y2 + 70y + 49)
84) 27a 3 - 64b3 A) (3a - 4b)(9a 2 + 12ab + 16b2) C) (3a - 4b)(9a 2 + 16b2)
B) (27a - 4b)(a2 + 12ab + 16b2 ) D) (3a + 4b2 )(9a2 - 12ab + 16b2 )
83)
84)
8
85) 250k3 m - 16m 4 A) 2m(125k - 2m)(k2 + 10km + 4m 2 ) C) 2m(5k - 2m)(25k2 + 10km + 4m 2 )
85) B) (10km - 4m 2)(25k2 + 4m 2 ) D) 2m(5k + 2m 2 )(25k2 - 10km + 4km2)
86) 1,000s3 + 1 A) (10s - 1)(100s2 + 10s + 1) C) (1,000s + 1)(s2 - 10s + 1)
B) (10s + 1)(100s2 - 10s + 1) D) (10s + 1)(100s2 + 1)
87) t3 + 64 A) (t + 4)(t2 - 4t + 16) C) (t - 64)(t2 - 1)
B) (t - 4)(t2 + 4t + 16) D) (t + 4)(t2 + 16)
88) 343c3 + 1,000 A) (7c - 10)(49c2 + 70c + 100) C) (7c + 10)(49c2 + 100)
B) (343c + 10)(c2 - 70c + 100) D) (7c + 10)(49c2 - 70c + 100)
86)
87)
88)
Write the expression in lowest terms. 89) 8k 10
A) 5
4k
89) B) 4k
C) 4
5
5
D) - 8k 10
3
90) 10k
90)
5k
A) 2k2
B) 5k2
C) 2k
D) 5
91) (8x - 3)
91)
-(3 - 8x)
A) -1
C) 8x - 3
B) 1
-3x + 8
D) 3 - 8x
-3 + 8x
92) (y + 7)(y - 5)
92)
(y - 5)(y + 8)
A) 2y - 5 2y + 3
93)
B) y + 7
C) y + 5
y+ 8
y+ 3
D) y - 7 y- 8
a 2 - 5a (a + 6)(a - 5)
A)
a a+6
93) 2 C) a
B) a - 5 a+6
a+6
9
D)
1 a+6
94)
3x + 3
94)
12x2 + 21x + 9
A)
3x + 3 12x2 + 21x + 9
B)
1 4x + 3
C)
3x 4x + 3
D) 3x + 4
4x + 21
2
95) y - 2y - 15
95)
y2 + 3y - 40
A) y + 3
B) -2y - 3
y+ 8
2
D) - y - 2y - 15
C) -2y - 15
3y - 8
y2 + 3y - 40
3y - 40
2
96) y + 5y - 24
96)
y2 + 2y - 48 2
A) - y + 5y - 24 y2 + 2y - 48
97)
C) y - 3
2y - 2
D) 5y - 24
y- 6
2y - 48
a 2 - 49
97)
a 2 + 11a + 28
A) a - 7
B) a - 7
a-4
98)
B) 5y - 1
C) a + 7
a+4
D) a + 7
a-4
a+4
36 - k2
98)
k2 - 11k + 30
A) - k - 6 k- 5
B) - k + 6
C) - k + 6
k+ 5
D) k + 6
k- 5
k- 5
Perform the indicated operation and simplify. 2 99) 3x · 28 4 x3
B) 84x
A) 21 x
2
C) 21x
4x3
100) 3p - 3 · 2p p
99) 2
D) x
21
x3
2
100)
8p - 8 3
B) 6p - 6p
A) 3p
2
8p2 - 8p
4
2
C) 24p + 48p + 24
D) 4
2p3
3p
10
2
101) 2x ÷ x 5
3
101)
40
A) x
C) 16x
B) 16
16
2
D) 80x
x3
x
2
5x3
102) 4p - 4 ÷ 8p - 8 p
102)
3p2
3
B) 12p - 12p
A) 2
2
8p2 - 8p
3p 2
C) 32p + 64p + 32
D) 3p
3p3
2
2
k2 + 5k k2 + 9k + 20 k2 + 8k + 16
103) k + 8k + 16 · A)
1 k+ 4
103) B)
2
2
k2 + 15k + 54
k2 + 7k + 12
2
C) k + 5k
k
k2 + 9k + 20
k+ 4
D)
k k+ 4
104) k + 10k + 24 · k + 12k + 27 B) k + 9
A) 1 2
104) C) k + 4
k+ 3
k+ 9
D)
1 k+ 3
2
105) x - y · x + y (x + y)2
105)
x-y
A) x2 - y
B) 1
2
106) z + 10z + 24 ÷ z 2 + 13z + 36
C)
1 x+y
z 2 + 6z
106)
z 2 + 12z + 27
A) z + 3
B)
z 2 + 9z
z
C) z + 3
z 2 + 13z + 36
2
107) 16x - 9 ÷ 4x + 3 x2 - 64
D) x - y
D) z + 3 z
107)
x-8
A) 4x + 3
B) 4x - 3
x-8
x+8
2
C) (4x + 3)(16x - 9)
D) x + 8
4x - 3
(x2 - 8)(x -8)
11
108)
x2 - 25 10x - 50 ÷ 2 x - 8x + 16 x2 - x - 12
A) (x - 5)(x + 3)
108) B)
10(x - 4)
10(x + 5) (x + 3)(x - 4)
C) x + 3
D) (x + 5)(x + 3) 10(x - 4)
Perform the indicated operations and simplify. 109) 2 + 8 13x 13x
A) 13x
109)
B) 10
10
D) 10
C) 1
26x
13x
110) 2 + 6 r
110)
r- 7
A) 14r - 8
B) 14r - 8
r(7 - r)
C) 8r - 14
r(r - 7)
r(7 - r)
D) 8r - 14
r(r - 7)
111) 5 + 6 r
111)
r+ 3
A) 11r + 15
B) -15r - 11
r(-3 - r)
112)
m-1
m 2 + 5m - 24
C) 11r + 15
r(-3 - r)
+
r(r + 3)
5m - 3
113)
114)
6m - 4 2 2m - 3m - 9
B)
6m 2 + 31m - 19 (m + 3)(m - 8)(m + 5)
D)
6m 2 + 31m - 19 (m - 3)(m + 8)(m - 5)
2 5 + 2 2 y - 3y + 2 y - 1
113)
A)
20y - 8 (y - 1)(y + 1)(y - 2)
B)
8y - 7 (y - 1)(y + 1)(y - 2)
C)
7y - 8 (y - 1)(y - 2)
D)
7y - 8 (y - 1)(y + 1)(y - 2)
x
x2 - 16
-
r(r + 3)
112)
m 2 - 8m + 15
A) 6m - 4 C)
D) -15r - 11
6
114)
x 2 + 5x + 4
A)
x2 + 5x + 24 (x - 4)(x + 4)(x + 1)
B) x - 5x + 24
2
C)
x2 - 5x + 24 (x - 4)(x + 4)(x + 1)
D)
(x - 4)(x + 4)
12
x2 - 5 (x - 4)(x + 4)(x + 1)
115) 6 - 8 z2
115)
z
B) 2(4z - 3)
A) 2(3z + 4) z2
116)
2ab
a 2 - b2
-
C) 2(3 + 4z) z2
z
116)
A) (a - b)(2a + 3b)
B) 2a + 3b
C) 2ab - b + 2
D) 2a + 3b
a 2 - b2
a+b+1
118)
z2
b 4 + a-b 2 a 2 - b2
117)
D) 2(3 - 4z)
1
3m 2 - 8mp - 3p2
a+b
+
1
18m 2 + 3mp - p2
-
7
117)
6m 2 - 19mp + 3p2
A)
28m + 3p (3m + p)(m - 3p)(6m - p)
B)
28m - 11p (3m + p)(m - 3p)(6m - p)
C)
-14m - 11p (3m + p)(m - 3p)(6m - p)
D)
-14m + 3p (3m + p)(m - 3p)(6m - p)
49x 1 3 + 5(7x + 1) 5x(7x + 1) x
118) 2
B) 49x + 105x + 14
A) 7(x + 2) 5x
5x
2
C) 49x + 105x + 14
D) 7(x + 2)
35x2 + 5x
35x2 + 5x
Solve the equation. 119) 2x + 3 = 5 - 7x A) 2 9
119) B) 8 9
C) - 8 5
B) 5
C) 15 4
120) 7y - 3 = 27 + y A) 4 121) 8x + 6 = 4x - 18 A) 6 122) 0.5x - 0.2 = 0.4x + 0.5 A) 3
D) - 2 5 120) D) 3
121) B) - 9 2
C) -24
D) -6
122) B) 7
C) 0.3
13
D) 0.7
123) -4.8q + 1.2 = -34.8 - 1.8q A) 7.5
123) B) 12
C) 7.9
D) -39
124) p - 3p = 2 4
124)
8
A) 14
C) 16
B) -16
D) -14
125) 7 z - 4z + 1 = 2 10
5
125)
5
A) - 1
B) - 2
33
C) - 2
11
126) 36(x - 144) = 72 A) 72 127) 4x - (2x - 1) = 2 A) - 1 6
128) 6(8x - 1) = 24
33
D) 2 3
126) B) 146
C) 142
D) 144
B) 1 6
C) 1 2
D) - 1 2
127)
128)
A) 5
B) 25
8
C) 3
48
129) (y - 4) - (y + 4) = 5y A) - 4
D) 23
8
48
129) B) - 8
C) - 1
5
5
D) - 2
130) 1 (20x - 25) = 1 (20x - 16) 5
130)
4
A) 1
20
131) (y - 7) - (y + 6) = 10y A) - 13 8
B) -1
C) -20
D) 1
B) - 13 10
C) - 1 10
D) - 1 8
131)
132) 1 (12x - 20) = 1 (15x - 9) 4
132)
3
A) -8 133) -8b + 5 + 6b = -3b + 10 A) -10
B) 1
C) 1
D) -1
B) 10
C) 5
D) -5
8
133)
14
134) 6[7m - (6m + 3) + 4] = 5m + 7 A) 6
134)
B) - 35
D) 13 11
C) 1
Solve the equation. If the solutions involve square roots, give both the exact solutions and the approximate solutions to three decimal places. 135) 4d2 + 16d + 15 = 0 135)
A) - 2 , - 5 3
2
136) 4b2 + 12b = -9 A) - 2 , - 3 3
2
B) 2 , 2
C) 3 , 5
3 5
2 2
138) 2m 2 - 6m = 0 A) 3, - 3
2
B) - 3 , - 3
C) 2 , 2
D) 3 , 3
B) -3, 7
C) 1 , - 1 20 3
D) - 1 , 3 3
2
2
3 3
2 2
137)
138) B) - 3, 0
C) 3, 0
D) 0
B) 0
C) 4 , - 4 7 7
D) - 4 , 0 7
139) 21n2 + 12n = 0 A) 4 , 0 7
2
136)
137) 3k2 - 20k - 7 = 0 A) - 1 , 7 3
D) - 3 , - 5
139)
140) 3m 2 + 8m + 2 = 0 A) -4 + 22 3
140) -4 - 22 0.230, 3
B) -4 + 10 6
-2.897
C) -4 + 10 -0.279, -4 - 10 -2.387 3
-4 - 10 -0.140, 6
-1.194
D) -8 + 10 -1.613, -8 - 10 -3.721
3
3
3
141) 5n2 = -8n - 2
141)
A) -8 + 6 -1.110, -8 - 6 -2.090
B) -4 + 26 0.220, -4 - 26 -1.820
C) -4 + 6 -0.155, -4 - 6 -0.645
D) -4 + 6 -0.310, -4 - 6 -1.290
5
10
5
5
10
5
5
5
142) 2x2 + 10x = -5
142)
A) -5 + 15 -0.282, -5 - 15 -2.218
B) -5 + 35 0.458, -5 - 35 -5.458
C) -10 + 15 -3.064, -5 - 15 -2.218
D) -5 + 15 -0.564, -5 - 15 -4.436
4
2
4
2
4
2
15
2
2
143) x2 - x = 12 A) -3, 4
143) B) 3, 4
144) x2 + 2x - 63 = 0 A) 9, 7
D) 1, 12
C) -3, -4
144) B) -9, 7
C) 9, -7
D) -9, 1
145) 1 + 1 = 20 x
145)
x2
B) - 1 , 1
A) -5, 4
146)
C) 4, 5
5 4
D) -4, 5
1 7 2 = y + 5 y - 5 y2 - 25
A) 30 5.477 147) 2 =
146) B) 7
D) 42
C) -7
t -5t - 12
t
147) B) 0, 36
A) -4, -6
C) No solution
D) 0, 12 5
148) 12 = 1 + 14 x-4
148)
x+4
A) No solution
B) 10, -12
C) -10, 12
D) -14, 12
149) 2y + 3 = 3 y
149)
2
B) 0
A) -6
D) 2 1.414
C) 6
150) 1 - 3 = 7 2x
150)
4
B) - 1
A) 2
D) 1
C) -2
2
2
151) 5 - a + 3 = 7 a
4
A) -4
152)
151)
a
B) -8
C)
29 20
1.204
D) 8
x 2x - 3 -2x = + 2x + 2 4x + 4 x+1
A) -3
152) B) 3
C) - 12
2
5
16
D) 3
153)
6 1 3 + = 2 2 2 x - 3x + 2 x + 4x - 5 x + 3x - 10
A) 31 4
154)
153)
B) - 31
C) - 1
4
D) - 31 10
6 5 -36 = m + 3 m - 3 m2 - 9
A) 6
154) C) No solution
B) -3
Write the expression in interval notation. Graph the interval. 155) x > 20
D) 3
155)
A) [20, ]
B) (20, ]
C) (20, )
D) [20, )
156) x < 6
156)
A) (6, )
B) (- , 6)
C) [- , 6)
D) (- , 6]
157) x 1
157)
A) [1, ]
B) (1, )
C) (1, ]
D) [1, )
17
158) x 7
158)
A) (- , 7]
B) [- , 7]
C) [7, )
D) (- , 7)
159) 1 x
159)
A) [1, )
B) [- , 1)
C) (1, )
D) (- , 1]
160) -8 < x
160)
A) (-8, ]
B) [- , -8)
C) (-8, )
D) (- , -8)
161) -9 < x < -5
161)
A) (-9, -5)
B) (-9, -5]
C) [-9, -5)
D) [-9, -5]
18
162) -9 x -6
162)
A) (-9, -6]
B) (-9, -6)
C) [-9, -6)
D) [-9, -6]
163) -9 < x -3
163)
A) (-9, -3)
B) [-9, -3]
C) [-9, -3)
D) (-9, -3]
Using the variable x, write the interval as an inequality. 164) (-4, 9] A) -4 < x < 9 B) x 9
165) [-2, 6) A) x < 6 166) [1, ) A) x 1
164) C) -4 x 9
D) -4 < x 9 165)
B) -2 < x 6
C) -2 x 6
D) -2 x < 6 166)
B) x 1
C) x > 1
D) x < 1
167) - , 2
167)
9
A) 9 x 2
B) x > 2
C) x 2
9
9
D) x < 2 9
168) - , 1
168)
6
A) 6 x 1
B) x > 1
C) x 1
6
6
19
D) x < 1 6
169)
169)
A) -4 x < 5
B) -4 < x 5
C) -4 < x < 5
D) -4 x 5
170)
170)
A) 2 < x < 6
B) 2 < x 6
C) 2 x < 6
D) 2 x 6
171)
171)
A) x > 4
B) x 4
C) x < 4
D) x 4
172)
172)
A) x < 2 or x > 6
B) 2 < x < 6
C) x 2 or x 6
173)
D) 2 x 6 173)
A) x < -6 or x -2 C) -6 < x -2
B) -6 x -2 D) x < -6 or x > -2
20
Solve the inequality and graph the solution. 174) 2x - 8 12
174)
A) [2, )
B) (- , 2]
C) [10, )
D) (- , 10]
175) 9x - 9 4x - 11
A)
,-
2 5
B)
,-
2 5
175)
C) (- , 5]
D) [-2, )
21
176) -12 + 7t + 1 6t - 15
176)
A) (- , -4]
B) [ -4, )
C) (- , 7)
D) (7, )
177) 35a + 35 > 5(6a + 13)
177)
A) (6, )
B) (35, )
C) (- , 35)
D) (- , 6)
22
178) -5(3x + 6) < -20x - 15
178)
A) (- , 3)
B) (-20, )
C) (- , -20)
D) (3, )
179) m - 3(m - 7) < 5m
179)
A) (- , 3)
B) (- , -1)
C) (-1, )
D) (3, )
23
180) 21 - (2x + 3) 2(x - 5) + 3x
180)
A) ( , 4]
B)
,
34 7
C) 34 , 7
D) [4, )
181) -2(x + 5) + 15x < -5(-3x + 8) - 3x
181)
A) (-30, )
B) (30, )
C) (- , 30)
D) (- , -30)
24
182) 1 (x + 3) > 1 (7x + 3) 2
182)
6
A) 3 , 2
B) - , 3 2
C) ( , -1)
D) (-1, )
183) 2 (4 + 5k) > 1 (5k + 3) 3
183)
2
A) - 7 , 5
B) - , - 1 5
C) - 1 , 5
D) - , - 7 5
25
Solve the inequality, then graph the solution. 184) 0 < -4y 16
184)
A) [-4, 0]
B) (0, 4]
C) (-4, 0)
D) [-4, 0)
185) 2 < 5y + 2 22
185)
A) [0, 4]
B) [0, 4)
C) (0, 4]
D) (0, 4)
186) -1 < -3a + 2 11
186)
A) [-3, 1)
B) (-3, 1]
C) (-1, 3]
D) [-1, 3)
26
187) 10 < 10x - 7 < 14
187)
3
A) - 37 , 49
B) 37 , 49
C) - 49 , 37
D) - , 37
10 10
10 10
10 10
10
49 , 10
188) -8 < 4 - 8x 8
188)
3
A) - 5 , 7
B) - , - 5
C) - 5 , 7
D) 5 , 7
2 2
2
2 2
7 , 2
2 2
Solve the quadratic inequality. Graph the solution. 189) (x - 3)(x + 5) > 0
189)
A) (-5, )
B) (-5, 3)
C) (- , -5) (3, )
D) (- , -3) (5, )
27
190) p2 + 8p + 15 > 0 A) (-5, -3)
190) B) (- , -5) (-3, )
C) (-3, )
D) (- , -5)
191) s2 - 3s - 28 < 0 A) (-4, 7)
191) B) (- , -4) (7, )
C) (- , -4)
D) (7, )
192) t2 - 3t - 18 0 A) (- , -3] [6, )
B) [6, )
C) [-3, 6]
D) (- , -3]
192)
193) v2 - 9v + 18 0 A) (- , 3] [6, )
B) [6, )
C) (- , 3]
D) [3, 6]
193)
194) 5x2 + 2x + 3 0
194)
A) -3, - 1 5
B) (- , )
C) -5, - 3
D) no solution
5
28
195) -3x2 + 7x - 5 0
195)
A) -3, - 1
B) 1 , 5
C) (- , )
D) no solution
5
3
196) x2 16 A) [-4, 4]
196)
B) [4, )
C) (- , -4] [4, )
D) (- , -4) (4, )
197) y2 - 3y < 0 A) (-3, 3)
197)
B) (0, 3)
C) [0, 3]
D) (- , 0) (3, )
29
Solve the inequality. 198) -7 > 0 -3x - 5
198) B) - , 5
A) (0, )
C) - , - 3
3
D) - 5 ,
5
3
199) x + 24 < 4
199)
x+9
A) - 4, 9 C) - , - 4 200)
B) -9, - 4 D) - , -9
9,
6 1 > x+9 7
200)
A) no solution C) - , -9 33, 201)
4
(x + 3)2
B) - 33, 9 D) -9, 33
<0
A) no solution 202)
- 4,
201) B) (-1, )
C) (- , )
D) (1, )
7 2 > x+7 x+7
A) (-7, )
202) B) no solution
C) (- , )
D) (- , -7)
203) 2x + 9 1
203)
x-4
A) [-13, 4) C) (- , -13] [4, )
B) (- , -13] (4, ) D) [-13, )
2
204) x + x 2
204)
x2 - 1
A) (- , -1) (-1, 1) [2, ) C) (- , 1) [2, )
B) (1, 2] D) (- , -1) [2, )
Evaluate the expression. Write your answer without exponents. 205) 3-4
A) 81
B) -81
205) C) 1 -81
D) 1 81
206) 1
206)
2 -3
A) 6
B) 8
C) 4
30
D) 16
207) 5
-2
207)
2 -3
A) 25 8
208)
B) 16
C) 8
125
25
D) 125 16
1
208)
-10-2
A) 10 209) (-4)-1 A) -4
B) 100
C) -10
D) -100
B) 1 4
C) 4
D) - 1 4
B) 25
C) 1 -25
209)
210) (-5)-2
A) 1 25
211) -(-5 -2)
210) D) -25
211)
A) -25
B) - 1
C) 25
D) 1
A) -1
B) 0
C) 1
D) 5
213) (-5)0 A) 1
B) -5
C) 0
D) -1
25
25
212) 50
214) 1
212)
213)
-2
214)
2
A) 2
C) 1
B) 4
4
D) 1
Simplify the expression. If the expression contains any variables, assume that they represent positive real numbers. Write your answer with only positive exponents. 8 7 m · 8 -2 m
215)
215)
8 -9 m
A) 814m
C) m
B) 8-4 m
8 14
31
D) m
8 -4
-8 -5 216) 5 p · 5 p
216)
5 7 p3
A) 520p
7
217) 243 · 3
B)
C) 5
1 5 20p
6
p
D) 1
56p
-3
217)
97
B) 2718
A) 1 218) (5x)
C) 318
D) 37
12
218)
x12
A) 5
B) 1
x
C) 512
5
D) 5
-8 219) x
219)
(6x)-8
A) 6
C) 1
B) 6x
68
D) 68
2 -9 -9 220) x (x )
220)
(x-5 )-5
A) x104
B) x58
C) x26
D)
1 104 x
-2 -8 4 221) x (x )
221)
(x-5 )-6
A) 1
x4
B) x5
C) x64
D) 1
x64
-4 5 -2 222) x y
222)
y-2
14
A) y
x8
B)
C) x
1 8 x y14
8
y12
D) x
8
y14
-6 -2 1/2 223) x y
223)
x8 y-6
A)
1 x7 y2
2
2
B) y
C) y
x7
x14
32
D) x
7
y4
Simplify the expression, writing the answer as a single term without negative exponents. m -1 + z-1
224)
A) z + m
B) z - m
m
225) a-2 - b-2 2
A) b - a
C) z + m
z
D) z + m
z
z-m
225) 2
ab2
226) (2k)-1 + 3m -1 A) m + 6k 3m
227) (a-1 + b-1)-1 A) ab a
228)
224)
m -1 - z-1
2
B) b + a
2
a 2 b2
2
2
C) b - a
D) b - a
C) m - 6k 3km
D) m + 6k 2k
a 2 b2
ab
226) B) m + 6k 2km
227) B) ab
C) ab
a+b
D) a + b
a-b
ab
x-2 x-2 - y-2
A)
229) -4a
y2 2 y + x2
228) B)
2
C) y - x
y 2 y - x2
2
y2
D)
y2 2 y - x2
-1 - 6b-1
229)
(-6ab)-1
A) 36b + 24a
B) 5
C) 24b + 36a
3
D) -4b - 6a -6
-1 + 9yx-1
230) -6xy
230)
-6x2 + 9y2
A)
1 -54xy
231) (2a -1 - 7b-1 )-1 A) ab -7b + 2a
B) 1
C) 1
D) xy
B) -7a + 2b ab
C) ab 2b - 7a
D) -14ab -7a + 2b
xy
231)
33
-1 232) (5m + 2n)
232)
4m -2 - 25n -2
A) C)
-3
233) x
2
100m 2 n 2
B)
(2n + 5m)(2n - 5m)2
7
D)
4m -1 - 25n -1
+
m 2n2 (2n + 5m)2 (2n - 5m) m 2n2
(2n + 5m)(2n - 5m)2
y -1 5
A) 8y + 5x 40
3
233) B) 8y + 5x
3
D) 40 + x y
xy
x3 y
3
C) 8y + 5x
8y
Evaluate the expression. 234) 5761/2
234)
A) 12
B) 96
C) 24
D) 48
235) 81/3 A) 6
B) 2
C) 48
D) 16
236) 4,0961/4 A) 256
B) 32,768
C) 8
D) 32
237) -321/5 A) 32
B) -2
C) -8
D) 16
238) 49
235)
236)
237)
1/2
238)
81
A) 7 9
239) 274/3 A) 2,187 240) 165/4 A) 512 241) 324/5 A) 512
B) 6
C) 7
10
10
9
239) B) 81
C) 729
D) 243 240)
B) 128
C) 256
D) 32 241)
B) 256
C) 16
D) 128
B) -4
C) 1 4
D) - 1 4
242) 8-2/3
A) 1 2
D) 6
242)
34
243) 27
-2/3
243)
64
A) 9
B) - 9
16
C) 16
16
D) 16
27
9
Simplify the expression. Write the answer with only positive exponents. Assume that all variables represent positive real numbers. 244) 7a 3/4 · 7a-23/4 244)
A) 7a -20
B) a
5
C) 7
D) 49
a6
49
a5
7/5 3/2 245) x · y
245)
x3/5 · y-11/2
A) x4 y14
246) 5
B) x
4/5
C) x
y4
7/3
y3/11
D) x4/5y7
-2/3 55 x- 5
246)
5 1/3x-2
A) 5 4 x6
B) 5
4
C) 5
x3
6
D) 5 x7
x7
-6 -2 -1 247) 3 k (3 k )
247)
6k3/2
A)
1
6 k11/2
B)
3 k15/2
1 6k12
C) -
D) - 3k13
1/5 -1 248) 2 m
248)
2 -7/5m - 4
A) 48/25m 4
C) 2
B) 28/5m 3
8/5
m3
D) -2 6/5m 3
-2 1/4 249) - 5b b
249)
b-7
A) - 5
b21/4
21/4
C) b
B) -5b23/4
35
5
D) -5b21/4
250) 2
1/5 x-8
250)
2 -9 x- 9
A) 246/5x
251) x
B) x
C) 244/5x
29
D) 2
48/5 x
-4/5 y-1/3 z 7/6
251)
x-1/5y7/3 z -5/6
A)
252) m
z xy1/3
B)
z2 x3/5y4
C) xy1/3
D)
z2 x3/5y8/3
-2/3 n -2/5 p29/8
252)
m 7/3n 13/5p-3/8
A)
p4 m3n3
B)
p4 m 8/3n 3
C)
p7 m 11/3n 3
3 D) m
n 3 p4
-1/5 · m -1/9 · n 3/7
253) k
253)
k7/5 · m -8/9 · n -2/7
A) m
7/9 · n5/7 k8/5
B)
n5/7 k6/5 · m 7/9
C) m
7/9 · n1/7 k8/5
Factor the expression. 254) (y - 7)(m - 2) + (y - 7)(m - 5)
A) (y - 7)(m + 7)
D) m
7/9 · n5/7 k6/5
254)
B) (y - 7)(m - 7)
C) (y - 7)(2m - 7)
255) (8q + 11)(6q + 8) + (8q + 11)(q - 8) A) (16q + 22)(7q) C) (8q + 11)(7q - 16)
D) (y - 7)(m 2 - 7) 255)
B) (8q + 11)(7q) D) (8q + 11)(7q + 16)
256) 3(x + y)3 - 6(x + y)2 + 9(x + y)4 A) (x + y)2[3x + 3y - 6 + 9(x + y)2] C) 3(x + y)3 (-2 + 3x + 3y)
B) 3(x + y)2 [x + y - 2 + 3(x + y)2 ] D) 3(x + y)[(x + y)2 - 2(x + y) + 3(x + y)3 ]
257) 20(p + 3)2 + 17(p + 3) + 3 A) (5p + 6)(4p + 4) C) (5p + 18)(4p + 13)
B) (5p + 3)(4p + 1) D) (5p + 16)(4p + 15)
256)
257)
258) 8(m - 4)2 - 6(m - 4) - 5 A) (4m + 17)(2m + 13) C) (4m + 5)(2m + 1)
258) B) (4m + 9)(2m + 5) D) (4m - 21)(2m - 7)
36
259) a2 (a + b)2 - ab(a + b)2 - 12b2(a + b)2 A) (a - 4b)(a + 3b)(a + b) C) (a + b)2(a - 4)(a + 3)
259) B) (a + b)2(a - 3b)(a + 4b) D) (a + b)2(a - 4b)(a + 3b)
260) 8x2 (x2 + 2)2 - 4x(5x3 + 1)(x2 + 2)
260)
A) 4x(x2 + 2)2(-3x3 + 4x - 1)
B) 4x(x2 + 2)(-5x3 + 2x2 + 3)
C) 4x(x2 + 2)(-3x3 + 4x - 1)
D) 4x(x2 + 2)(-3x3 + 4x + 1)
261) 9m 1/2 + 3m -1/2 A) m -1/2 (9m - 3) C) m -1/2 (9m + 3)
261) B) m -1/2 (9m 1/2 + 3) D) m 1/2 (9m + 3)
262) x(2x + 5)2 (x2 - 6)-1/2 + 4(x2 - 6)1/2(2x + 5)
262)
A) (2x + 5)(x2 - 6)-1/2(6x2 + 5x - 24)
B) (2x + 5)2 (x2 - 6)-1/2(4x2 + x - 24)
C) (2x + 5)(x2 - 6)1/2(6x2 + 5x - 24)
D) (2x + 5)(x2 - 6)-1/2(4x2 + 2x - 19)
263) (x - 7)-3/2 - (x - 7)-1/2 + (x - 7)1/2 A) (x - 7)-3/2(x2 + 15x - 57) C) (x - 7)-3/2(x2 - 15x - 50)
263) B) (x - 7)-3/2(x2 - 15x + 57) D) (x - 7)3/2(x2 + 14x - 57)
Simplify. Assume that all variables represent positive real numbers. 264) 96 A) 4 B) 4 6 C) 9
265) - 125 A) -5 5 266)
3
D) 11
3
C) 5 3
B) 11
D) 5
2,592
267) 4
A) 6 2 268) -
C) 5
266) 3
4
265) B) -25 5
135
A) 3 5
267)
264) D) 16 6
4
B) 50
4
D) 2 6
C) 7
1,280
A) 5
268) 4
C) -4 5
B) -5
37
4
D) -5 4
269)
3
-125
269)
A) -5 270) 343x2 A) 7x 7 271) 48k7 q8 A) (4q4 ) 3k7 272)
273)
3
3
C) 5
B) -25
1,000x4 y5 3 A) 10xy( xy2)
D) 25 270)
B) 7x2 7
C) 7 7x
D) 343x 271)
B) (4k7 q8) 3k
C) (4k3 q4) 3
D) (4k3 q4) 3k
272) 3
3
C) 3xy( xy2)
B) 10xy( xy)
2
D) 10xy( xy2)
-8a 8 b5
273) 3
A) 2ab( a 2 b2)
3
B) 2ab( a 3 b3)
C) 2( a 2 b2)
3
D) -2a 2 b( a 2b2)
Simplify the expression by removing as many factors as possible from under the radical. Assume that all variables represent positive real numbers. 274) 12 · 12 274) A) 12 B) 24 C) 144 D) 144
275) 2 · 18 A) 12
B) 36
C) 6
D) 72
276) 28 · 7 A) 2 7
B) 196
C) 14
D) 7 2
277) 5 · 10 A) 5 2
B) 25 2
C) 10
D) 2 5
275)
278) 15 · 75 A) -25 3 279) 11x3 · 11x5 A) x4 22
280)
7
u· u 14
A)
276)
277)
278) B) 15 5
C) -15 5
D) 25 3 279)
C) 11x4
B) 121x8
D) 11x4 280)
u9
B)
9
2u
C)
38
9
u2
D)
7
u2
281)
7
4
x3 11 A) x4
x·
281) B)
28
x4
C)
11
x3
D)
28
x25
4
282) m · m 13 4
A) m 6 m
282) 4
B) m 3 m 3
C)
4
D) m 6 m
m
Perform the indicated operations and simplify. Assume all variables represent positive real numbers. 283) -8 98 + 4 18 - 5 32 A) -416 2 B) 416 2 C) -8 2 D) -64 2
284) 18 - 6 32 - 4 8 A) -32 2 285) 6 + 5 24 - 4 54 A) 1 84 286) 3a - 3 12a - 6 27a A) -23 42a
284) B) 22 2
C) 29 2
D) -29 2 285)
B) -1 84
C) 1 6
D) -1 6 286)
B) -9 42a
C) -9 3a
D) -23 3a
287) 5x2 + 4 125x2 + 7 125x2 A) 56x 186 B) 11x 186
C) 11x 5
D) 56x 5
288) 2 - 4 128 - 5 98 A) -66 2
C) 66 2
D) -3 2
3
287)
288) B) 3 2
3
289) 6 3 + 14 3 3
A) 20 9 3
289) 3
3
B) 20 6
C) 20 3
3
D) 8 3
3
290) 9 2 - 4 128 3
A) 5 2
291) 11
283)
290) 3
B) 7 2
C) -7
4 x7 - 5x x3 4 A) 6x x3 4 4 C) 11 x7 - 5x x3
3
2
4
3
3
D) 9 2 - 4 128
291) B) 16 D) 6x
39
4 4
x3 x7
3
3
292) 3 a +
64a
3
A) 7 a
292) 3
B) 3 a +
3
3
64a
C) 4 64a
3
D) 12 a
Simplify the root, if possible. 293) 9x2 + 24x + 16
293)
A) (3x + 4)2
B) 3x + 4 D) cannot be simplified
C) 3x + 4 294) 4m 2 + 49n 2 A) 2m + 7n C) cannot be simplified 295) z 2 + 2z + 1 A) z + 1
294) B) 2m + 7n D) (2m + 7n)2 295) B) -z - 1
C) z + 1
D) z + 1
Rationalize the denominator. Assume that all radicands represent positive real numbers. 296) - 25 24
A) -5 6
297)
C) - 5 6
6
D) -24
12
5 9- 5
297)
A) 45 - 5 5 76
298)
B) - 5 6
296)
B) 45 + 5 5
C) 5 - 5 9
-4
5
D) 45 + 5 5 76
7 2+5
298)
A) 3 14 + 27 10
B)
14 - 5 7 7
14 - 5 7 -23
C)
D)
14 + 5 7 -23
299) 1 - 10 1+
299)
10
A) 11 + 2 10 -9
300)
B) -9 - 2 10
C) 1
11
D) 11 - 2 10 -9
7 7 5- 7
A) 1 ( 35 - 1) 34
300) B) 1 ( 5 + 1)
C) 1 ( 35 + 1)
34
34
40
D) 1 ( 35 + 1) 36
301)
4 x+4
A)
301)
16 x + 16
B)
C) 4 x - 16
16 x - 16
D) 4 x + 16
x - 16
x + 16
302) 4 x + 1
302)
3 x-5
A) 16x + 1
B) 12x - 5
C) 12x + 23 x + 5
D) 4x + 23 x + 5
9x - 25
9x - 25
9x - 25
303)
9x - 25
4x 5x - 7
303)
A) 16x
2
5x + 7 5x + 7
C) 16x
B) 4x 5x - 7
2
D) 4x 5x - 7
5x - 7
5x + 7
5x - 7
304) -7
304)
3
305)
yy+
C) -7 3
B) - 7
A) -7 3
3
3
3
y+ 3 y+ 3
305)
A) -2y - 3 + 2 y(y + 3)
B) -2y - 3 3
3
C)
D) -7 7
D) 2y + 3 - 2 y(y + 3)
-3 2y + 3
3
Rationalize the numerator. Assume that all radicands represent positive real numbers. 306) 5 + 2 2
A) -5
B)
5
5 5-1
C)
2 2 5-1
306) D)
307) 8 - 3 10 -
A) C)
307)
10
61 80 + 2 -7 - 30 80 - 10
1 2 5-4
61 3-8
B) 10 -
D)
30
41
80 + 10
61 3-8
10 - 30
80 + 10
61 3-8
10 -
30
308) 8 + 6 4+
A) C)
309)
310)
308)
7 32 - 4
58 6+8
32 + 4
58 6-8
77-
B)
42
D)
42
58 32 - 12 13 - 42 32 - 4
58 6-8
7 - 42
x+ y 8x
309)
A)
8x - 8y x x- x y
B)
x+y - 8x x + 8x y
C)
x-y 8x x - 8x y
D) -
y 8 x-8 y
m- n 5+ n
310)
A)
m-n 5 m +n
B)
m+n 5 m - 5 n - mn - n
C)
m-n 5 m + 5 n + mn + n
D)
m-1 5 m+5 n+
42
mn
Answer Key Testname: UNTITLED1
1) B 2) B 3) A 4) B 5) C 6) B 7) A 8) B 9) C 10) A 11) C 12) D 13) C 14) A 15) C 16) D 17) D 18) D 19) B 20) C 21) C 22) B 23) B 24) B 25) B 26) D 27) B 28) C 29) B 30) A 31) D 32) C 33) B 34) A 35) A 36) B 37) A 38) D 39) C 40) A 41) C 42) D 43
Answer Key Testname: UNTITLED1
43) A 44) D 45) A 46) D 47) B 48) D 49) C 50) D 51) C 52) B 53) D 54) C 55) C 56) A 57) C 58) C 59) D 60) D 61) B 62) B 63) A 64) B 65) C 66) B 67) D 68) A 69) C 70) C 71) D 72) C 73) B 74) D 75) B 76) C 77) D 78) C 79) D 80) D 81) D 82) D 83) A 84) A 44
Answer Key Testname: UNTITLED1
85) C 86) B 87) A 88) D 89) B 90) A 91) B 92) B 93) A 94) B 95) A 96) C 97) B 98) C 99) A 100) A 101) B 102) D 103) D 104) A 105) B 106) D 107) B 108) D 109) D 110) D 111) C 112) D 113) D 114) C 115) D 116) D 117) C 118) A 119) A 120) B 121) D 122) B 123) B 124) B 125) C 126) B 45
Answer Key Testname: UNTITLED1
127) C 128) A 129) B 130) B 131) B 132) D 133) C 134) C 135) D 136) B 137) A 138) C 139) D 140) C 141) D 142) D 143) A 144) B 145) A 146) C 147) A 148) B 149) A 150) C 151) B 152) D 153) B 154) C 155) C 156) B 157) D 158) A 159) D 160) C 161) A 162) D 163) D 164) D 165) D 166) A 167) D 168) C 46
Answer Key Testname: UNTITLED1
169) A 170) B 171) A 172) C 173) A 174) C 175) A 176) B 177) A 178) A 179) D 180) D 181) D 182) B 183) A 184) D 185) C 186) A 187) B 188) A 189) C 190) B 191) A 192) C 193) A 194) B 195) C 196) C 197) B 198) D 199) D 200) D 201) A 202) A 203) B 204) A 205) D 206) B 207) C 208) D 209) D 210) A 47
Answer Key Testname: UNTITLED1
211) D 212) C 213) A 214) B 215) A 216) B 217) C 218) C 219) D 220) B 221) D 222) D 223) B 224) D 225) D 226) B 227) B 228) D 229) C 230) C 231) C 232) B 233) B 234) C 235) B 236) C 237) B 238) A 239) B 240) D 241) C 242) C 243) D 244) D 245) D 246) B 247) A 248) B 249) D 250) A 251) D 252) A 48
Answer Key Testname: UNTITLED1
253) A 254) C 255) B 256) B 257) C 258) D 259) D 260) C 261) C 262) A 263) B 264) B 265) A 266) A 267) A 268) C 269) A 270) A 271) D 272) A 273) D 274) A 275) C 276) C 277) A 278) B 279) C 280) A 281) D 282) B 283) D 284) D 285) D 286) D 287) D 288) A 289) C 290) C 291) A 292) A 293) C 294) C 49
Answer Key Testname: UNTITLED1
295) D 296) C 297) D 298) C 299) D 300) C 301) C 302) C 303) D 304) C 305) A 306) D 307) D 308) A 309) C 310) C
50
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the slope of the line passing through the given pair of points. 1) (5, 6) and (3, 1) A) 2 B) 5 C) - 5 5 2 2
2) (8, -4) and (3, -2) A) 2 5
3) (5, 6) and (5, -7) A) - 1 10
4) (-3, 6) and (-4, 6) A) - 12 5) (11, -18) and (-6, 2) A) - 20 17
1) D) 7 8 2)
B) - 5
C) - 6
2
11
D) - 2 5
3) B) 13 10
C) Not defined
D) 0
4) B) - 12 7
C) 0
D) Not defined
B) - 16 5
C) 20 17
D) - 17 20
5)
Find the slope of the line. 6) y = 2 x 9
A) 0 7) y = 4x - 8 A) -4
6) C) 2
B) 1
9
9) 3x - 4y = -27 A) 27 4
2
7) B) 1
C) 4
D) 0
B) 4
C) 1 2
D) - 1 2
8) 2x + 4y = 16 A) 2
D) 9
8)
9) B) - 3
C) - 4
4
3
1
D) 3 4
10) The x-axis A) 1
B) 0
C) Not defined
D) -1
11) x = 10 A) 10
B) Not defined
C) 0
D) 1
C) 2 5
D) - 6 5
C) 11 2
D) 3 2
10)
11)
12) A line parallel to -2y + 5x = -6 A) 5 2
12)
B) - 5 2
13) A line parallel to 2x = 3y + 11 A) - 2 3
13)
B) 2 3
14) A line perpendicular to 8x + 5y = -102 A) - 5
14) C) 5
B) 8
8
D) 8
8
5
15) A line perpendicular to 7x = 2y + 8 A) - 7
15)
B) - 2
2
D) 2
C) 4
7
7
Find an equation in slope-intercept form (where possible) for the line. 16) Through (0, 2), m = 1 2
16)
A) y = 1 x - 2
B) y = - 1 x - 2
C) y = - 1 x + 2
D) y = 1 x + 2
17) Through (-13, 5), m = -4 A) y = -4x - 47
B) y = 4x + 7
C) y = -4x + 5
D) y = 4x + 47
18) Through (3, 0), m = -1 A) y = x - 3
B) y = -3x
C) y = -x + 3
D) y = 3x
2
19) Through (4, 6), m = -4 A) y = 4x - 10
2
2
17)
18)
19) B) y = -4x - 10
C) y = -4x + 22
20) Through (-9, -6), m = 0 A) y = - 3 x 2
D) y = 4x + 22 20)
B) x = -9
C) y = - 2 x
D) y = -6
C) x = 3
D) 3 x - 4y = 0 4
3
21) Through (3, -4), with undefined slope A) 4 x + 3y = 0 3
2
21)
B) y = -4
2
22) Through (5, 5), m = - 7
22)
8
A) y = 7 x + 35 8
B) y = - 7 x + 75
8
8
C) y = 7 x - 75
8
8
D) y = - 7 x + 35
8
8
8
23) Through (0, -5), m = 9
23)
4
A) y = - 9 x + 5
B) y = 9 x + 5
4
C) y = - 9 x - 5
4
D) y = 9 x - 5
4
4
24) Through (3, 5), m = - 3
24)
4
A) y = - 3 x + 9 4
B) y = - 3 x + 29
4
4
25) Through (-4, 3), m = 2.5 A) y = 2.5x - 7
C) y = 3 x - 29
4
4
D) y = 3 x + 9
4
4
25) B) y = -2.5x + 13
C) y = 2.5x + 13
26) Through (-3, 8) and (-9, 19)
B) y = 11 x + 27
C) y = - 6 x + 70
D) y = - 11 x + 5
11
11
A) y = 7 x + 8 8
28) Through (-5, 0) and (-7, -3) A) y = - 3 x + 15 2 2
29) Through (9, 7) and (1, -2) A) y = 2 x - 8 3
6
11
27) Through (2, -5) and (0, 8)
3
30) Through (-1, 1) and (10, 3) A) y = 2 x + 41
D) y = -2.5x - 7 26)
A) y = - 11 x + 106 6
4
2
6
2
27) B) y = - 7 x + 8 8
C) y = - 13 x + 8 2
D) y = 13 x + 8 2
B) y = - 5 x + 23 4 4
C) y = 5 x + 23 4 4
D) y = 3 x + 15 2 2
28)
29) B) y = - 9 x - 25 8
C) y = - 2 x - 8
8
3
3
D) y = 9 x - 25 8
8
30) B) y = - 2 x + 13
C) y = - 2 x + 41
D) y = 2 x + 13
31) Through (-7, 1.5) and (-5, 5.5) A) y = 0.5x + 5 B) y = -2x - 12.5
C) y = -0.5x - 2
D) y = 2x + 15.5
7
7
11
11
7
7
11
11
31)
3
32) Through (-3, 6) and (-3, 4) A) x = -3
32) C) - 2 x + 6y = 0 3
B) y = 6
33) Through (-9, 10) and (-8, 10)
33)
B) - 9 x - 8y = 0
A) y = 10
8
34) y-intercept -6, x-intercept -9 A) y = 2 x - 6 3
C) x = -9
D) - 8 x - 9y = 0
C) y = - 2 x - 6 3
D) y = 3 x - 9 2
C) y = - 3 x 2
D) y = 3 x - 3 2 2
9
34)
B) y = - 3 x - 9 2
35) Through (-3, 6), perpendicular to -2x + 3y = 24 A) y = - 2 x + 3 3
35)
B) y = - 3 x + 3 2 2
36) Through (-4, 11), parallel to -6x + 7y = 66
36)
A) y = 6 x + 101 7 7
B) y = - 6 x - 101 7 7
C) y = 4 x + 66
D) y = 7 x - 11
7
7
6
37) Through (-1, -3), parallel to -8x - 3y = -10 A) y = - 1 x + 10 3
3
B) y = - 3 x + 3 8
8
C) y = 8 x + 17
D) y = - 8 x - 17
C) y = - 5 x - 64 7 7
D) y = - 3 x - 56 5 5
C) y = 9 x - 95 2
D) y = 2 x 9
3
3
40) C) y = 2
41) The line with y-intercept 3 and perpendicular to x + 7y = -5 B) y = 7x + 3
C) y = -7x + 3
42) The line with x-intercept -10 and perpendicular to 5x - y = -9 A) y = - 1 x - 2 5
3
39)
B) y = - 2 x + 95 9 9
40) Through (1, -2), perpendicular to x = -5 A) x = -5 B) y = -2
A) y = 1 x + 3 7
3
38)
B) y = 5 x - 64 7 7
39) Through (7, -9), perpendicular to -9x - 2y = -45 A) y = 2 x - 95 9 9
6
37)
38) Through (3, -7), perpendicular to -7x - 5y = -56 A) y = 7 x + 7 5 5
D) - 3 x + 4y = 0 2
B) y = - 1 x - 10
C) y = -5x - 50
5
4
D) y = -5 41) D) y = - 1 x + 1 7 42) D) y = 1 x - 2 5
Find the slope of the line.
43)
43)
A) 2 5
B) - 2
C) 5
5
2
D) - 5 2
44)
44)
A) -1
C) 1
B) -5
D) 5
45)
45)
A) 2
C) 1
B) -2 5
D) -1
46)
46)
A) 1 2
C) - 1
B) 2
2
D) - 2
47)
47)
A) 3 2
B) 4
C) undefined
D) 0
48)
48)
A) 0
B) undefined
C) 2 6
D) -2
Graph the equation. 49) y = 3x - 4
49)
A)
B)
C)
D)
7
50) y = - 1 x + 1
50)
3
A)
B)
C)
D)
8
51) 4y + 9x = 4
51)
A)
B)
C)
D)
9
52) 3y - 5x = -15
52)
A)
B)
C)
D)
10
53) 5x + 5y = 12
53)
A)
B)
C)
D)
11
54) 5x + y = -1
54)
A)
B)
C)
D)
12
55) 5x - 25y = 0
55)
A)
B)
C)
D)
13
56) x = -1
56)
A)
B)
C)
D)
14
57) y = -1
57)
A)
B)
C)
D)
15
58) x + 1 = 0
58)
A)
B)
C)
D)
16
59) y - 2 = 0
59)
A)
B)
C)
D)
Solve the problem. 60) In a certain city, the cost of a taxi ride is computed as follows: There is a fixed charge of $2.70 as soon as you get in the taxi, to which a charge of $1.95 per mile is added. Find a linear equation that can be used to determine the cost, C, of an x-mile taxi ride. A) C = 4.65x B) C = 3.15x
C) C = 1.95x + 2.70
D) C = 2.70x + 1.95
17
60)
61) After two years on the job, an engineer's salary was $55,000. After seven years on the job, her
61)
salary was $61,000. Let y represent her salary after x years on the job. Assuming that the change in her salary over time can be approximated by a straight line, give an equation for this line in the form y = mx + b. A) y = 6,000x + 43,000 B) y = 1200x + 55,000
C) y = 6,000x + 55,000
D) y = 1200x + 52,600
62) Suppose that the population of a certain town, in thousands, was 105 in 1990 and 141 in 2002.
62)
Assume that the population growth can be approximated by a straight line. Find the equation of a line which will estimate the population of the town, in thousands, in any given year since 1990. A) y = -3x + 177 where x is the number of years since 1990
B) y = 4.25x + 90 where x is the number of years since 1990 C) y = 2.5x + 105 where x is the number of years since 1990 D) y = 3x + 105 where x is the number of years since 1990 63) Assume that the sales of a certain appliance dealer can be approximated by a straight line.
63)
Suppose that sales were $11,500 in 1982 and $85,000 in 1987. Let x = 0 represent 1982. Find the equation giving yearly sales S. A) S = 14,700x + 85,000 B) S = 73,500x + 11,500
C) S = 14,700x + 11,500
D) S = 73,500x + 85,000
64) The cost of owning a home includes both fixed costs and variable utility costs. Assume that it costs
64)
$3,516 per month for mortgage and insurance payments and it costs an average of $4.65 per unit for natural gas, electricity, and water usage. Determine a linear equation that computes the annual cost of owning this home if x utility units are used. A) y = -4.65x + 3,516 B) y = -4.65x + 42,192
C) y = 4.65x + 3,516
D) y = 4.65x + 42,192
65) In a lab experiment 14 grams of acid were produced in 10 minutes and 19 grams in 39 minutes. Let
65)
y be the grams produced in x minutes. Write a linear equation for grams produced. A) y = 5 x - 356 B) y = - 5 x - 356 29 29 29 29
C) y = 29 x - 356 5
D) y = 5 x + 356
29
29
29
66) A biologist recorded 2 snakes on 25 acres in one area and 12 snakes on 39 acres in another area.
Let y be the number of snakes in x acres. Write a linear equation for the number of snakes. A) y = 7 x + 111 B) y = 5 x + 111 C) y = 5 x - 111 D) y = - 5 x + 111 5 7 7 7 7 7 7 7
18
66)
67) The following data show the list price, x, in thousands of dollars, and the dealer invoice price, y,
67)
also in thousands of dollars, for a variety of sport utility vehicles. Find a linear equation that approximates the data, using the points (16.5, 16.1) and (20.0, 18.3). List Price Dealer Invoice Price 16.5 16.1 17.6 17.0 20.7 18.2 23.1 19.3 20.0 18.3 24.6 21.0
A) y = 0.629x + 5.73 C) y = 1.59x - 10.2
B) y = 1.59x - 9.11 D) y = 0.629x + 6.38
68) The information in the chart gives the salary of a person for the stated years. Model the data with
68)
a linear function using the points (1, 24,300) and (3, 26,200). Year, x Salary, y 1990, 0 $23,500 1991, 1 $24,300 1992, 2 $25,200 1993, 3 $26,200 1994, 4 $27,200
A) y = 950x + 23,350 C) y = 28.1x + 23,350
B) y = 950x D) y = -1,193x + 23,350
69) The change in a certain engineer's salary over time can be approximated by the linear equation
69)
70) The relationship between the list price, x, in thousands of dollars, and the dealer invoice price, y,
70)
y = 1500x + 47,500 where y represents salary in dollars and x represents number of years on the job. According to this equation, after how many years on the job was the engineer's salary $64,000? A) 10 years B) 11 years C) 12 years D) 13 years
also in thousands of dollars, for pickup trucks can be approximated by the linear equation y = 0.715x + 2.82. Use this equation to predict the dealer invoice price for a pickup truck with a list price of 22.0 thousand dollars. A) 15.730 thousand dollars B) 23.595 thousand dollars
C) 26.825 thousand dollars
D) 18.550 thousand dollars
71) Suppose the sales of a particular brand of appliance satisfy the relationship S = 100x + 1,200,
where S represents the number of sales in year x, with x = 0 corresponding to 1982. Find the number of sales in 1999. A) 2,800 sales B) 2,900 sales C) 5,800 sales D) 5,700 sales
19
71)
72) The mathematical model C = 900x + 60,000 represents the cost in dollars a company has in
72)
73) Suppose the function y = 4.5t - 0.4 determines the actual time that has elapsed, in minutes, for t
73)
74) A car rental company charges $28 per day to rent a particular type of car and $0.12 per mile. Juan
74)
manufacturing x items during a month. Based on this, how much does it cost to produce 200 items? A) $66.67 B) $240,000 C) $0.33 D) $180,000
minutes of a person's estimate of the elapsed time. Find the actual time that has elapsed for an estimate of t = 60 minutes. A) 61.8 min B) 269.6 min C) 270.4 min D) 58.2 min
is charged $40.84 for a one-day rental. How many miles did he drive? A) 340 mi B) 122 mi C) 107 mi
D) 312 mi
75) If an object is dropped from a tower, then the velocity, V (in feet per second), of the object after t
75)
seconds can be obtained by multiplying t by 32 and adding 10 to the result. Write an equation expressing the velocity, V, in terms of the number of seconds, t. Use this function to predict the velocity of the object at time t = 7.9 seconds. A) 260.8 feet per second B) 262.8 feet per second
C) 262.1 feet per second
D) 264.1 feet per second
76) The information in the chart below gives the salary of a person for the stated years. Model the
76)
data with a linear function using the points (1, 24,200) and (3, 26,200). Then use this function to predict the salary for the year 2002. Year, x Salary, y 1990, 0 $23,500 1991, 1 $24,200 1992, 2 $25,200 1993, 3 $26,200 1994, 4 $27,200
A) $35,540
B) $35,480
C) $35,500
D) $35,520
77) In order to receive a B in a course, it is necessary to get an average of 80% correct on two one-hour
77)
exams of 100 points each, on one midterm exam of 200 points, and on one final exam of 500 points. If a student scores 91, and 82 on the one-hour exams, and 140 on the midterm exam, what is the minimum score on the final exam that the person can get and still earn a B? A) 407 B) 452 C) 587 D) 317
Evaluate the function as indicated. 78) Find f(13) when f(x) = -16x - 17. A) -209.7 B) -225
78) C) 191
79) Find f(3) when f(x) = 2x + 9. A) -3 B) 15
D) -191 79)
C) 11
20
D) 33
80) Find f(0) when f(x) = 13x + 8. A) 21 B) 8
C) 0
D) 13
81) Find f(1.2) when f(x) = -1.6x - 17. A) 15.08 B) -15.08
C) -3.62
D) -18.92
82) Find f(-1.5) when f(x) = -6x - 0.9. A) 9.9 B) 8.91
C) 8.1
D) -9.9
80)
81)
82)
83) Find g - 5 when g(x) = 5 - 5x.
83)
3
A) 10 3
B) - 40
C) 40
3
3
84) Find f(8.6) when f(x) = -11. A) 11 B) -94.6
D) - 10 3
84) C) 8.6
85) Find f(-r) when f(x) = 3 - 5x. A) r - 5x B) 3 + 5r
D) -11 85)
C) 3 + rx
86) Find g(k2) when g(x) = -4 - 5x. A) -4 + -5k2 B) -4 - 5x2
D) 3 - 5r 86)
C) -4 + k2
D) -4 + 5k2
87) Find g(a + 1) when g(x) = 2x + 1. A) 2a + 1
87)
B) 1 a + 1 2
C) 2a + 3
D) 2a - 1
Write a cost function for the problem. Assume that the relationship is linear. 88) A moving firm charges a flat fee of $35 plus $30 per hour. Let C(x) be the cost in dollars of using the moving firm for x hours. A) C(x) = 35x - 30 B) C(x) = 35x + 30 C) C(x) = 30x - 35 D) C(x) = 30x + 35
89) A cab company charges a base rate of $1.00 plus 10 cents per minute. Let C(x) be the cost in dollars of using the cab for x minutes. A) C(x) = 1.00x - 0.10
89)
B) C(x) = 0.10x - 1.00 D) C(x) = 1.00x + 0.10
C) C(x) = 0.10x + 1.00
90) An electrician charges a fee of $45 plus $30 per hour. Let C(x) be the cost in dollars of using the electrician for x hours. A) C(x) = 45x + 30
88)
B) C(x) = 45x - 30
C) C(x) = 30x - 45
21
D) C(x) = 30x + 45
90)
91) A cable TV company charges $23 for the basic service plus $5 for each movie channel. Let C(x) be the total cost in dollars of subscribing to cable TV, using x movie channels. A) C(x) = 5x - 23 B) C(x) = 23x - 5 C) C(x) = 23x + 5
92) Fixed cost, $450; 10 items cost $1,810 to produce A) C(x) = 136x + 450 C) C(x) = 272x + 1,810
B) C(x) = 136x + 1,810 D) C(x) = 272x + 450
93) Marginal cost, $20; 50 items cost $1,400 to produce A) C(x) = 20x + 1,400 C) C(x) = 8x + 1,400
B) C(x) = 20x + 400 D) C(x) = 8x + 400
91)
D) C(x) = 5x + 23 92)
93)
Solve the problem. 94) Let the supply and demand functions for a certain model of electric pencil sharpener be given by 2 2 p = S(q) = q and p = D(q) = 16 - q , 3 3 where p is the price in dollars and q is the quantity of pencil sharpeners (in hundreds). Graph these functions on the same axes (graph the supply function as a dashed line and the demand function as a solid line). Also, find the equilibrium quantity and the equilibrium price.
A)
B)
Equilibrium quantity: 1,200 Equilibrium price: $8
Equilibrium quantity: 960 Equilibrium price: $6.40
22
94)
C)
D)
Equilibrium quantity: 950 Equilibrium price: $7
Equilibrium quantity: 640 Equilibrium price: $9.60
95) Let the supply and demand functions for raspberry-flavored licorice be given by 5 p = S(q) = q 4
3 and p = D(q) = 70 - q , 4
where p is the price in dollars and q is the number of batches. Graph these functions on the same axes (graph the supply function as a dashed line and the demand function as a solid line). Also, find the equilibrium quantity and the equilibrium price.
A)
B)
Equilibrium quantity: 28 Equilibrium price: $35.00
Equilibrium quantity: 43.75 Equilibrium price: $35
23
95)
C)
D)
Equilibrium quantity: 35 Equilibrium price: $43.75
Equilibrium quantity: 35.00 Equilibrium price: $28
96) Given the supply and demand functions below, find the price when the demand is 145. S(p) = 9p + 12 D(p) = 280 - 9p A) $15
B) $47
C) $1317
96)
D) $292
97) Suppose that the demand and price for a certain model of graphing calculator are related by
97)
98) Given the supply and demand functions below, find the demand when p = $12.
98)
p = D(q) = 99 - 1.5q, where p is the price (in dollars) and q is the demand (in hundreds). Find the price if the demand is 500 calculators. A) $24.00 B) $91.50 C) $106.50 D) $174.00
S(p) = 5p D(p) = 120 - 4p A) 72
B) 60
C) 132
D) 48
99) Suppose that the demand and price for a certain model of graphing calculator are related by
99)
p = D(q) = 109 - 3.5q, where p is the price (in dollars) and q is the demand (in hundreds). Find the demand for the calculator if the price is $26. Round to the nearest whole number if necessary. A) 33,200 calculators B) 24 calculators
C) 2,371 calculators
D) 593 calculators
100) Suppose that the price and supply for a certain model of graphing calculator are related by
p = S(q) = 3.5q, where p is the price (in dollars) and q is the supply (in hundreds) of calculators. Find the supply if the price is $91. Round to the nearest whole number if necessary. A) 260 calculators B) 2,600 calculators
C) 1,300 calculators
D) 650 calculators
24
100)
101) Let the demand and supply functions be represented by D(p) and S(p), where p is the price in
101)
102) Let the demand and supply functions be represented by D(p) and S(p), where p is the price in
102)
103) Let the demand and supply functions be represented by D(p) and S(p), where p is the price in
103)
104) A book publisher found that the cost to produce 1000 calculus textbooks is $25,600, while the cost
104)
105) In deciding whether or not to set up a new manufacturing plant, analysts for a popcorn company
105)
106) A toilet manufacturer has decided to come out with a new and improved toilet. The fixed cost for
106)
dollars. Find the equilibrium price and equilibrium quantity for the given functions. D(p) = 4,750 - 50p S(p) = 140p A) $33; 3,100 B) $90; 250 C) $90; 3,500 D) $25; 3,500
dollars. Find the equilibrium price and equilibrium quantity for the given functions. D(p) = 201,400 - 250p S(p) = 810p A) $560; 153,900 B) $190; 153,900 C) $560; 61,400 D) $248; 139,400
dollars. Find the equilibrium price and equilibrium quantity for the given functions. D(p) = 2,784 - 60p S(p) = 230p - 696 A) $15; 2,064 B) $15; 1,884 C) $12; 2,064 D) $20; 1,584
to produce 2000 calculus textbooks is $52,200. Assume that the cost C(x) is a linear function of x, the number of textbooks produced. What is the marginal cost of a calculus textbook? A) $2.66 B) $0.03 C) $26.60 D) $26,600.00
have decided that a linear function is a reasonable estimation for the total cost C(x) in dollars to produce x bags of microwave popcorn. They estimate the cost to produce 10,000 bags as $5,100 and the cost to produce 15,000 bags as $7,560. Find the marginal cost of the bags of microwave popcorn to be produced in this plant. A) $4.92 B) $0.49 C) $49.20 D) $2,460.00
the production of this new toilet line is $16,600 and the variable costs are $68 per toilet. The company expects to sell the toilets for $150. Formulate a function C(x) for the total cost of producing x new toilets and a function R(x) for the total revenue generated from the sales of x toilets. A) C(x) = 16,668; R(x) = 150 B) C(x) = 16600 + 150x; R(x) = 68x
C) C(x) = 68x; R(x) = 150x
D) C(x) = 16600 + 68x; R(x) = 150x
107) A toilet manufacturer has decided to come out with a new and improved toilet. The fixed cost for the production of this new toilet line is $16,600 and the variable costs are $70 per toilet. The company expects to sell the toilets for $152. Formulate a function P(x) for the total profit from the production and sale of x toilets. A) P(x) = 82x - 16600 B) P(x) = 82x + 16600
C) P(x) = 82x
D) P(x) = 152x - 16600
25
107)
108) A shoe company will make a new type of shoe. The fixed cost for the production will be $24,000.
108)
109) Midtown Delivery Service delivers packages which cost $1.20 per package to deliver. The fixed
109)
110) Regrind, Inc. regrinds used typewriter platens. The cost per platen is $1.20. The cost to regrind 90
110)
111) Regrind, Inc. regrinds used typewriter platens. The cost per platen is $2.70. The fixed cost to run
111)
112) Northwest Molded molds plastic handles which cost $1.00 per handle to mold. The fixed cost to
112)
113) Midtown Delivery Service delivers packages which cost $1.20 per package to deliver. The fixed
113)
114) A lumber yard has fixed costs of $5,662.00 a day and variable costs of $1.00 per board-foot
114)
The variable cost will be $33 per pair of shoes. The shoes will sell for $103 for each pair. What is the profit if 600 pairs are sold? A) $42,000 B) $57,600 C) $66,000 D) $18,000
cost to run the delivery truck is $144 per day. If the company charges $5.20 per package, how many packages must be delivered daily to make a profit of $80? A) 120 packages B) 22 packages C) 56 packages D) 36 packages
platens is $600. Find the linear cost function to regrind platens. If reground platens sell for $8.30 each, how many must be reground and sold to break even? A) C(x) = 1.20x + 492 B) C(x) = 1.20x + 600 break-even = 69 break-even = 64 C) C(x) = 1.20x + 492 D) C(x) = 1.20x + 600 break-even = 500 break-even = 86
the grinding machine is $315 per day. If the company sells the reground platens for $7.70, how many must be reground daily to break even? A) 30 platens B) 42 platens C) 116 platens D) 63 platens
run the molding machine is $4,000 per week. If the company sells the handles for $3.00 each, how many handles must be molded weekly to break even? A) 1,333 handles B) 4,000 handles C) 1,000 handles D) 2,000 handles
cost to run the delivery truck is $56 per day. If the company charges $5.20 per package, how many packages must be delivered daily to break even? A) 9 packages B) 46 packages C) 8 packages D) 14 packages
produced. The company gets $2.90 per board-foot sold. How many board-feet must be produced daily to break even? A) 5,662 board-feet B) 1,986 board-feet
C) 1,451 board-feet
D) 2,980 board-feet
115) A shoe company will make a new type of shoe. The fixed cost for the production will be $24,000.
The variable cost will be $32 per pair of shoes. The shoes will sell for $100 for each pair. How many pairs of shoes will have to be sold for the company to break even on this new line of shoes? A) 353 pairs B) 751 pairs C) 68 pairs D) 241 pairs
26
115)
116) When going more than 38 miles per hour, the gas mileage of a certain car fits the model
116)
y = 43.81 - 0.395x where x is the speed of the car in miles per hour and y is the miles per gallon of gasoline. Based on this model, at what speed will the car average 15 miles per gallon? (Round to nearest whole number.) A) 98 miles per hour B) 149 miles per hour
C) 48 miles per hour
D) 73 miles per hour
117) The temperature of water in a certain lake on a day in October can be determined by using the
117)
118) The bank's temperature display shows that it is 20° Celsius. What is the temperature in
118)
model y = 15.2 - 0.537x where x is the number of feet down from the surface of the lake and y is the Celsius temperature of the water at that depth. Based on this model, how deep in the lake is the water 10 degrees? (Round to the nearest foot.) A) 66 feet B) 28 feet C) 10 feet D) 47 feet
Fahrenheit? A) 68.0°
B) 28.9°
C) -6.7°
D) 93.6°
119) On a summer day, the surface water of a lake is at a temperature of 29° Celsius. What is this temperature in Fahrenheit? A) 29° B) 84.2°
C) 52.2°
D) 61°
120) On a summer day, the bottom water of a lake is at a temperature of 10° Celsius. What is this temperature in Fahrenheit? A) 50° B) 18°
C) 10°
119)
120)
D) 42°
121) The outdoor temperature rises to 10° Fahrenheit. What is this temperature in Celsius? A) -22° B) 5.6° C) -12.2° D) 10°
121)
122) A meteorologist in the Upper Peninsula of Michigan predicts an overnight low of -8° Fahrenheit.
122)
123) Find the temperature at which the Celsius and Fahrenheit scales coincide. A) 42° B) -22° C) -40°
123)
What would a Canadian meteorologist predict for the same location in Celsius? A) -22.2° B) -8° C) -4.4° D) -40°
124) For the following table of data,
124)
a. Draw a scatterplot. b. Calculate the correlation coefficient. c. Calculate the least squares line and graph it on the scatterplot. d. Predict the y-value when x is 16. x y
1 5
2 4.5
3 4
4 4
5 3
6 3.5
D) 0°
7 2.5
8 2
9 1
27
A) a.
B) a.
b. -0.965 c. Y = -0.45x + 5.53
b. 0.965 c. Y = -0.45x + 5.53
d. -1.67
d. -1.67
C) a.
D) a.
b. 0.965 c. Y = 0.45x - 5.53
b. -0.965 c. Y = -0.45x - 5.53
d. 1.67
d. -12.73
28
125) For the following table of data,
125)
a. Draw a scatterplot. b. Calculate the correlation coefficient. c. Calculate the least squares line and graph it on the scatterplot. d. Predict the y-value when x is -23. x -4 y -3.5
-3 -2
-2 -1
-1 -1.5
0 1
1 0.5
2 1.5
3 2
A) a.
4 4
B) a.
b. -0.966 c. Y = -0.82x - 0.11
b. 0.966 c. Y = 0.82x + 0.11
d. 18.75
d. -18.75
29
C) a.
D) a.
b. -0.966 c. Y = 0.82x + 0.11
b. 0.966 c. Y = 0.82x - 0.11
d. -18.97
d. -18.97
126) For the following table of data,
126)
a. Draw a scatterplot. b. Calculate the correlation coefficient. c. Calculate the least squares line and graph it on the scatterplot. d. Predict the y-value when x is 22. x y
1 10
2 12
3 11
4 14
5 15
6 17
7 22
8 19
9 24
30
A) a.
B) a.
b. 0.950 c. Y = 1.7x + 7.5
b. 0.950 c. Y = 1.7x
d. 44.9 C) a.
d. 37.4 D) a.
b. 0.950 c. Y = 1.7x + 8.5
b. 0.903 c. Y = 2x + 6.5
d. 45.9
d. 50.5
31
Find the correlation coefficient. 127) Consider the data points with the following coordinates: x 29.6 27.9 14.1 42.5 37.2 y 3 5 5 9 3 A) 0.3,445 B) 0 C) 0.3,066
127) D) -0.3,445
128) The test scores of 6 randomly picked students and the number of hours they prepared are as follows: Hours Score
5 64
10 86
4 69
6 86
A) -0.2242
10 59
9 87
C) 0.6781
B) -0.6781
D) 0.2242
129) The test scores of 6 randomly picked students and the number of hours they prepared are as follows: Hours 4 Score 54 A) -0.2241
10 99
5 56
3 3 70 72 B) -0.6781
59 163
C) 0.2015
D) 0.6039
61 177
53 56 159 175 B) -0.0783
130)
60 151
C) 0.2145
D) 0.1085
131) Consider the data points with the following coordinates: x 62 y 158 A) 0.7537
53 176
64 151
52 164
52 54 164 174 B) -0.7749
131)
58 162
C) -0.0810
D) 0
132) Consider the data points with the following coordinates: x 121 y 171 A) 0.0537
101 152
128 168
160 157
154 126 164 169 B) 0.2245
132)
134 160
C) -0.0781
D) 0.5370
133) The following are costs of advertising (in thousands of dollars) and the number of products sold (in thousands): Cost 9 Number 85 A) -0.0707
2 52
3 55
4 2 68 67 B) 0.2353
5 86
9 83
3 75
7 91
6 10 57 96 B) 0.6756
C) 0.2456
4 52
7 92
133)
10 73
D) 0.7077
134) The following are costs of advertising (in thousands of dollars) and the number of products sold (in thousands): Cost 6 Number 54 A) -0.3707
129)
5 99
130) Consider the data points with the following coordinates: x 57 53 y 156 164 A) -0.0537
128)
7 100
C) 0.6112
32
D) 0.2635
134)
135) The following are the temperatures on randomly chosen days and the amount a certain kind of plant grew (in millimeters): Temp 62 76 50 51 71 46 Growth 36 39 50 13 33 33 A) 0.1955 B) -0.2105
51 17
44 6
79 16 C) 0.2563
D) 0
136) The following are the temperatures on randomly chosen days and the amount a certain kind of plant grew (in millimeters): Temp 77 88 85 61 64 72 Growth 39 17 12 22 15 29 A) -0.3105 B) -0.0953
73 14
74 43 C) 0.0396
136)
63 25
D) 0
Find the equation of the least squares line. 137) Ten students in a graduate program were randomly selected. Their grade point averages (GPAs) when they entered the program were between 3.5 and 4.0. The following data were obtained regarding their GPAs on entering the program versus their current GPAs. Entering GPA (x) 3.5 3.8 3.6 3.6 3.5 3.9 4.0 3.9 3.5 3.7
135)
137)
Current GPA (y) 3.6 3.7 3.9 3.6 3.9 3.8 3.7 3.9 3.8 4.0
A) y = 3.67 + 0.0313x C) y = 4.91 + 0.0212x
B) y = 2.51 + 0.329x D) y = 5.81 + 0.497x
138) The paired data below consist of the test scores of 6 randomly selected students and the number of hours they studied for the test.
Hours (x) 5 10 4 6 10 9 Score (y) 64 86 69 86 59 87
A) y = -67.3 + 1.07x C) y = 67.3 + 1.07x
B) y = 33.7 - 2.14x D) y = 33.7 + 2.14x
33
138)
139) The paired data below consist of the costs of advertising (in thousands of dollars) and the number
139)
of products sold (in thousands).
Cost (x) 9 2 3 4 2 5 9 10 Number (y) 85 52 55 68 67 86 83 73
A) y = -26.4 - 1.42x C) y = 26.4 + 1.42x
B) y = 55.8 + 2.79x D) y = 55.8 - 2.79x
140) The paired data below consist of the temperatures on randomly chosen days and the amount a
140)
certain kind of plant grew (in millimeters).
Temp (x) 62 76 50 51 71 46 51 44 79 Growth (y) 36 39 50 13 33 33 17 6 16
A) y = 14.6 + 0.211x C) y = 7.30 + 0.122x
B) y = -14.6 - 0.211x D) y = 7.30 - 0.112x
141) A study was conducted to compare the average time spent in the lab each week versus course
141)
grade for computer students. The results are recorded in the table below. Number of hours spent in lab (x) 10 11 16 9 7 15 16 10
Grade (percent)(y) 96 51 62 58 89 81 46 51
A) y = 1.86 + 88.6x C) y = 88.6 - 1.86x
B) y = 0.930 + 44.3x D) y = 44.3 + 0.930x
142) Two separate tests are designed to measure a student's ability to solve problems. Several students are randomly selected to take both tests and the results are shown below. Test A (x) 48 52 58 44 43 43 40 51 59 Test B (y) 73 67 73 59 58 56 58 64 74
A) y = 19.4 + 0.930x C) y = -0.930 + 19.4x
B) y = -19.4 - 0.930x D) y = 0.930 - 19.4x
34
142)
143) Managers rate employees according to job performance and attitude. The results for several
143)
randomly selected employees are given below.
Attitude (x) 59 63 65 69 58 77 76 69 70 64 Performance (y) 72 67 78 82 75 87 92 83 87 78
A) y = -47.3 + 2.02x C) y = 92.3 - 0.669x
B) y = 11.7 + 1.02x D) y = 2.81 + 1.35x
144) Two different tests are designed to measure employee productivity and dexterity. Several
144)
employees of a company are randomly selected and asked to complete the tests. The results are below. Dexterity (x) 23 25 28 21 21 25 26 30 34 36 Productivity (y) 49 53 59 42 47 53 55 63 67 75
A) y = 75.3 - 0.329x C) y = 5.05 + 1.91x
B) y = 10.7 + 1.53x D) y = 2.36 + 2.03x
145) In the table below, x represents the number of years since 2000 and y represents annual sales (in
145)
thousands of dollars) for a clothing company. Year x 0 1 3 5 Sales y 21 30 35 39
A) y = 5.18x + 20.6 C) y = 3.31x + 23.8
B) y = 2.61x + 25.9 D) y = 4.37x + 21.7
146) In the table below, x represents the number of years since 2000 and y represents the population (in
146)
thousands) of the town Boomville. Year x 1 2 3 4 5 Sales y 30 40 60 90 130
A) y = 28x - 10
B) y = 25x - 5
C) y = 12x + 20
D) y = 18x + 8
Solve the problem. 147) Find an equation for the least squares line representing weight, in pounds, as a function of height, in inches, of men. Then, predict the weight of a man who is 68 inches tall to the nearest tenth of a pound. The following data are the (height, weight) pairs for 8 men: (66, 150), (68, 160), (69, 166), (70, 175), (71, 181), (72, 191), (73, 198), (74, 206). A) 165.1 pounds B) 151.4 pounds C) 160.0 pounds D) 161.2 pounds
148) Find an equation for the least squares line representing weight, in pounds, as a function of height, in inches, of men. Then, predict the height of a man who is 145 pounds to the nearest tenth of an inch. The following data are the (height, weight) pairs for 8 men: (66, 150), (68, 160), (69, 166), (70, 175), (71, 181), (72, 191), (73, 198), (74, 206). A) 63.2 inches B) 64.6 inches C) 68.2 inches D) 65.7 inches
35
147)
148)
149) For some reason the quality of production decreases as the year progresses at a light bulb
149)
manufacturing plant. The following data represent the percentage of defective light bulbs produced at a light bulb manufacturing plant in the corresponding month of the year. month (x) 2 3 5 7 8 9 12 % defective (y) 1.3 1.6 2.0 2.4 2.6 2.8 3.1 Use the equation of the least squares line to predict the percentage of defective bulbs in June. A) 2.3% B) 2.0% C) 2.20% D) 2.15%
150) For some reason the quality of production decreases as the year progresses at a light bulb
150)
manufacturing plant. The following data represent the percentage of defective light bulbs produced at a light bulb manufacturing plant in the corresponding month of the year. month (x) 2 3 5 7 8 9 12 % defective (y) 1.3 1.6 2.0 2.4 2.6 2.8 3.1 Use the equation of the least squares line to predict in which month the percentage of defective light bulbs would be 1.83%. A) April B) May C) March D) February
151) Ten students in a graduate program were randomly selected. Their grade point averages (GPAs)
151)
when they entered the program were between 3.5 and 4.0. The following data were obtained regarding their GPAs on entering the program versus their current GPAs. Use the equation of the least squares line to predict the current GPA of a student whose entering GPA is 3.5. Entering GPA (x) 3.5 3.8 3.6 3.6 3.5 3.9 4.0 3.9 3.5 3.7
A) 3.78
Current GPA(y) 3.6 3.7 3.9 3.6 3.9 3.8 3.7 3.9 3.8 4.0
B) 3.29
C) 3.58
D) 3.40
152) The paired data below consist of the test scores of 6 randomly selected students and the number of hours they studied for the test. Use the equation of the least squares line to predict the score on the test of a student who studies 11 hours. Hours (x) 5 10 4 6 10 9 Score (y) 64 86 69 86 59 87
A) 74.1
B) 79.1
C) 83.8
36
D) 84.1
152)
153) The paired data below consist of the costs of advertising (in thousands of dollars) and the number
153)
of products sold (in thousands). Use the equation of the least squares line to predict the number of products sold if the cost of advertising is $15,000. Cost (x) 9 2 3 4 2 5 9 10 Number (y) 85 52 55 68 67 86 83 73
A) 97.65 products sold C) 94.65 products sold
B) 41,905.8 products sold D) 104.35 products sold
154) The paired data below consist of the temperatures on randomly chosen days and the amount a
154)
certain kind of plant grew (in millimeters). Use the equation of the least squares line to predict the growth of a plant if the temperature is 71. Temp (x) 62 76 50 51 71 46 51 44 79 Growth (y) 36 39 50 13 33 33 17 6 16
A) 28.09 mm
B) 30.15 mm
C) 30.72 mm
D) 29.58 mm
155) In the table below, x represents the number of years since 2000 and y represents annual sales (in
155)
thousands of dollars) for a clothing company. Use the least squares regression equation to estimate sales in the year 2006. Round to the nearest thousand dollars. Year (x) 1 2 3 4 5 Sales (y) 30 40 60 90 130 A) $140,000 B) $145,000
C) $142,000
D) $147,000
156) A study was conducted to compare the average time spent in the lab each week versus course
156)
grade for computer students. The results are recorded in the table below. Use the equation of the least squares line to predict the grade of a student who spends 16 hours in the lab. Number of hours spent in lab (x) 10 11 16 9 7 15 16 10
A) 62.0%
Grade (percent) (y) 96 51 62 58 89 81 46 51
B) 54.8%
C) 58.8%
D) 72.6%
Provide an appropriate response. 157) Find k so that the line through (3, k) and (1, -2) is parallel to 3x - 3y = -6. Find k so that the line is perpendicular to 4x + 2y = 7. A) 4; - 1 B) 0; - 3 C) 4; - 3 D) 0; - 1
37
157)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
158) John has been a teacher at West Side High School for the past 12 years. His salary during
158)
159) If a company decides to make a new product, there are fixed costs and variable costs
159)
160) Give a definition or an example of the word or phrase: Perpendicular lines
160)
161) Why is the slope of a horizontal line equal to zero? Give an example.
161)
162) Explain what is wrong with the statement "The line has no slope."
162)
163) Why is the slope of a vertical line undefined?
163)
164) Can an equation of a vertical line be written in slope-intercept form? Explain.
164)
165) The total number of reported cases of AIDS in the United States has risen from 372 in 1981
165)
166) Show that the points P1 (2,4), P2 (5,2), and P3 (7,5) are the vertices of a right triangle.
166)
that time can be modeled by the linear equation y = 800x + 33,000 where x is the number of years since he began teaching at West Side and y is his salary in dollars. Explain what the slope, 800, represents in this context.
associated with this new product. Explain the differences of the two types of costs and why they occur. Use an example to illustrate your point.
to 100,000 in 1989 and 200,000 in 1992. Does a linear equation fit this data? Explain.
38
Answer Key Testname: UNTITLED1
1) B 2) D 3) C 4) C 5) A 6) C 7) C 8) D 9) D 10) B 11) B 12) A 13) B 14) C 15) B 16) D 17) A 18) C 19) C 20) D 21) C 22) B 23) D 24) B 25) C 26) D 27) C 28) D 29) D 30) D 31) D 32) A 33) A 34) C 35) B 36) A 37) D 38) B 39) A 40) B 41) B 42) A 39
Answer Key Testname: UNTITLED1
43) B 44) A 45) C 46) B 47) C 48) A 49) A 50) B 51) D 52) D 53) A 54) B 55) D 56) B 57) D 58) D 59) D 60) C 61) D 62) D 63) C 64) D 65) D 66) C 67) A 68) A 69) B 70) D 71) B 72) B 73) B 74) C 75) B 76) C 77) A 78) B 79) B 80) B 81) D 82) C 83) C 84) D 40
Answer Key Testname: UNTITLED1
85) B 86) A 87) C 88) D 89) C 90) D 91) D 92) A 93) B 94) A 95) C 96) A 97) B 98) A 99) C 100) B 101) D 102) B 103) C 104) C 105) B 106) D 107) A 108) D 109) C 110) A 111) D 112) D 113) D 114) D 115) A 116) D 117) C 118) A 119) B 120) A 121) C 122) A 123) C 124) A 125) B 126) A 41
Answer Key Testname: UNTITLED1
127) A 128) D 129) D 130) D 131) B 132) A 133) D 134) C 135) A 136) B 137) A 138) C 139) B 140) A 141) C 142) A 143) B 144) C 145) C 146) B 147) D 148) D 149) D 150) A 151) A 152) B 153) A 154) D 155) B 156) C 157) D 158) The slope of 800 indicates that during his 12 years at the school, John's salary has increased by approximately $800
per year. 159) Fixed costs occur only once. These costs may be startup costs related to the production of the new product. Variable costs depend on how much product is made. These costs may consist of labor, material, and maintenance. For example, a company decided to make oak filing cabinets. Fixed costs would include the costs of purchasing and renovating plant space and the cost of manufacturing equipment. Variable costs would include the cost labor and the cost of materials. 160) Two lines which intersect at right angles. (Answers may vary.) 161) Answers may vary. One possibility: The slope of a horizontal line is equal to zero because the y-values do not change as the x-values change. For example, the points (3, 4) and (7, 4) are two points on a horizontal line. The slope of this 4-4 0 = = 0. line is zero because m = 7-3 4
42
Answer Key Testname: UNTITLED1
162) Answers may vary. One possibility: It is not specific enough. The slope of a horizontal line is 0, while the slope of a
vertical line is undefined. 163) Answers may vary. One possibility: Let (a, b) and (a, c), b c, be any two different points on a vertical line. The slope y1 - y2 b - c b - c = = of the line = . Division by zero is undefined. x 1 - x2 a - a 0
164) No. In the slope-intercept form of the equation of a line, x is multiplied by slope; however, the slope of a vertical line is undefined. (Explanations will vary.)
165) No, the data cannot be modeled by a linear equation because the reported cases are not increasing at a constant rate. Assume a linear equation, and examine the slope of the two line segments. The slope of the segment from (0, 372) to
(8, 100,000) is 12,453.5 while the slope of the segment from (8, 100,000) to (11, 200,000) is 33,333.3.(Explanations will vary.) 166) Answers will vary. One possibility: The slope of the line through P1 and P2 is -2/3. The slope of the line through P2
and P3 is 3/2. Therefore, since the product of these slopes is -1, the lines are perpendicular and constitute a right angle in the triangle, making the triangle formed by these points a right triangle.
43
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the rule defines y as a function of x. 1) X Y
4
24
7
42
10
60
13
78
1)
A) Function
B) Not a function
2)
2) X
Y
7
3
16
95
70
10
35
30
A) Function
B) Not a function
1
3)
3) X
7 17 75 39 107
Y
95 15 30 9
A) Function
B) Not a function
4)
4) x -6 -6 -1 4 7
y 1 -4 7 -1 7 A) Function
B) Not a function
5)
5) x y -3 2 2 1 5 8 7 6 12 -5 A) Function
B) Not a function
6) y = x4 + 6 A) Function
6) B) Not a function
7) x = y2 + 7 A) Function
7) B) Not a function
Give the range for the function if the domain is {-2, -1, 0, 1, 2}. 8) y = x + 7 A) {-5, -3, -1, 1, 3}
C) {-2, -1, 0, 1, 2}
2
8) B) {5, 7, 9, 11, 13} D) {5, 6, 7, 8, 9}
9) y = 2x - 1 A) {-5, -3, -1, 1, 3} C) {-2, -1, 0, 1, 2}
B) {-3, -1, 1, 3, 5} D) {-4, -3, -2, -1, 0}
10) 3x + y = 11 A) {13, 11, 9, 7, 5} C) {-5, -7, -9, -11, -13}
B) {-5, -8, -11, -14, -17} D) {17, 14, 11, 8, 5}
9)
10)
11) 5x - y = 2 A) {-10, 0, 10} C) {-12, -7, -2, 3, 8}
11) B) {-10, -5, 0, 5, 10} D) {-12, 0, 12}
12) y = x(x - 1) A) {0, 2, 6}
B) {0, 4, 8}
C) {-6, -2, 0, 2, 6}
D) {-8, -4, 0, 4, 8}
13) y = x2 A) {-2, -1, 0, 1, 2}
B) {0, 1, 4}
C) {0, 1, 2}
D) {-4, -1, 0, 1, 4}
12)
13)
14) y = - 4x2 A) {-16, -4, 0}
14) B) {-16, 0, 16}
C) {0, 4, 16}
D) {-4, 0, 4}
15) y = x
15)
x+3
A) -1, - 1 , 0, 3 , 7
B) -2, - 1 , 0, 1 , 2
C) -1, 1 , 0, 3 , 7
D) -2, 1 , 0, 1 , 2
2
4 5
2
2
4 5
4 5
2
4 5
16) y = -3
16)
x+7
A) - 3 , - 1 , - 3 , - 3 , -1
B) - 3 , - 1 , - 3 , - 1 , -1
C) - 3 , - 1 , - 3 , - 3 , - 1
D) - 3 , - 1 , - 3 , - 3 , - 1
8 5
4 2
5
5
7
8
7
3
11
2
2
8
3
7
8
3
17) y = x - 5
17)
x+5
A) - 7 , - 3 , 1, - 2 , - 3 6
4
5
B) - 7 , - 3 , -1, - 2 , - 3
8
5
C) - 7 , - 3 , -1, - 2 , - 3 3
2
3
4
3
7
D) - 7 , - 3 , 1, - 2 , - 3
7
4
3
2
5
8
Give the domain of the function. 18) f(x) = -9x - 1 A) (0, )
18) B) (- , 0) (0, )
C) [1, )
D) (- , )
19) f(x) = 7x + 8
19)
A) [0, ) C)
,-
B) (- , ) 8 7
-
8 , 7
20) f(x) = 2x2 + 4x - 2 A) (- , 0) 21) f(x) =
D) - 8 , 7
20) B) (- , 0) (0, )
C) (0, )
x4 + 5 x2 + 2x - 8
21)
A) ( , -2) (-2, 4) (4, ) C) ( , -4) (-4, -2) (-2, ) 22) f(x) = (-x - 5)1/2 A) [5, )
B) ( , 2) (2, 4) (4, ) D) ( , -4) (-4, 2) (2, ) 22)
B) (- , -5]
C) [-5, )
23) f(x) = 10 - x A) (- , ) C) [0, 10] 24) f(x) =
D) (- , 5] 23)
B) (- , 10) (10, ) D) (- , 10]
x+3 x-6
A) (- , -3) (6, ) 25) g(z) = 16 - z 2 A) [0, ) 26) f(x) =
D) (- , )
24) B) (- , -3] [6, )
C) (-3, 6)
D) (- , -3] (6, ) 25)
B) [-4, 4]
C) (-4, 4)
D) (- , )
1 2 x + 5x - 36
A) (9, 4)
26) B) (- , -9) (4, )
C) (- , )
4
D) (- , 4) (9, )
Give the domain and range of the function.
27)
27)
A) Domain [-5, 4] ; Range [-4, 4] C) Domain (-5, 4) ; Range [-2, 4)
B) Domain [-4, 4) ; Range (-5, 4] D) Domain (-5, 4] ; Range [-4, 4)
28)
28)
A) Domain (- , ) ; Range [-2, 4] C) Domain (- , ) ; Range [-2, )
B) Domain (- , ) ; Range [0, ) D) Domain (-5, 5) ; Range [-2, 8)
5
29)
29)
A) Domain [-1, 6] ; Range [-8, 8] B) Domain {-1, 1, 6} ; Range {-8, -4, -1, 1, 4, 8} C) Domain [-8, 8] ; Range [-1, 6] D) Domain {-8, -4, -1, 1, 4, 8} ; Range {-1, 1, 6} 30)
30)
A) Domain = (-4, 8) ; Range (-4, 2) C) Domain [-4, 8] ; Range [-4, 2]
B) Domain {-4, 8} ; Range {-4, 2} D) Domain (- , ) ; Range (- , )
6
31)
31)
A) Domain (- , 0) ; Range (- , 0) B) Domain (- , ) ; Range [-4, ) C) Domain (- , 0) (0, ) ; Range (- , 0) (0, ) D) Domain (0, ) ; Range [8, ) 32)
32)
A) Domain (- , ) ; Range {-5, 4, 1} C) Domain {-5, 4, 1} ; Range (- , )
B) Domain (- , ) ; Range (- , ) D) Domain (- , ) ; Range [4, )
7
33)
33)
A) Domain [0, ) ; Range (- , -3] B) Domain (- , -3) (-3, ) ; Range (- , 0) (0, ) C) Domain (- , -3] ; Range [0, ) D) Domain (- , ) ; Range [0, ) 34)
34)
A) Domain {-4} ; Range (-7, 7) C) Domain ( , ) ; Range {-4}
B) Domain (-7, 7) ; Range {-4} D) Domain [-7, 7] ; Range {-4}
8
Use the graph to evaluate the function f(x) at the indicated value of x. 35) Find f(1.5).
35)
B) 0 D) None of these are correct.
A) -2 C) 1 36) Find f(2).
36)
A) -3 C) -2
B) -4 D) None of these are correct.
Evaluate the function. 37) f(x) = x2 - 5x - 4; Find f(-3). A) 20 B) -10
37) C) -2
38) f(x) = x2 - 2x + 5; Find f(0). A) 25 B) 5
D) 28 38)
C) -5
39) f(x) = 4x2 + 2x + 4; Find f(-4). A) 12 B) 52
D) 0 39)
C) 76
40) f(x) = (x - 1)(x + 5); Find f(-3). A) 32 B) 16
D) 60 40)
C) -8 9
D) -4
41) f(x) = x - 10 ; Find f(2).
41)
x-4
B) 5
A) 3
D) - 4
C) 4
2
3
42) f(x) = 3x ; Find f(3).
42)
2x - 9
A) 3
B) - 3
2
7
43) f(x) = 4x2 + 3x + 1; Find f(a). A) 7a B) 4a 2 + 3a
C) - 1
D) - 3
C) 7a + 1
D) 4a 2 + 3a + 1
43)
44) f(x) = (x - 1)(x + 4); Find f(a). A) a2 + 4 B) (a - 1)(a - 4)
C) a2 - 4
D) (a - 1)(a + 4)
45) f(x) = 5x2 - 3x + 5; Find f(t - 1). A) 5t2 - 13t + 7 B) 5t2 - 13t + 13
C) -13t2 + 5t + 13
D) 5t2 + 22t + 7
44)
45)
46) f(x) = -3x2 + 2x + 4; Find f(r + h). A) -3r2 - 6rh - 3h2 + 2r + 2h + 4 C) -3r2 - 3h2 + 2r + 2h + 4
46) B) -3r2 - 3h2 - 4r - 4h + 4 D) -3r2 - 3rh - 3h2 + 2r + 2h + 4
Evaluate the function for the given value. x-7 1 if x 2x + 1 2 1 47) f(x) = 1 ; f- 2 7 if x = 2
A) 0
C) - 15
B) 7 x-7
48) f(x) = 2x + 4 11 1 A) 18
49) f(x) =
47)
2x + 4 x-7
if x - 2
2
D) - 7 2
; f(7)
48)
if x = - 2
B) 77
if x 7
C) 11
; f(a)
D) 0
49)
5 if x = 7 (2a + 4) A) if a 7, 5 if a = 7 (a - 7)
B) 0 if a 7, 5 if a = 7
C) (2a + 4) if a = 7, 5 if a 7
D) 2 if a 7, 5 if a = 7
(a - 4)
10
2x + 4 x-7
50) f(x) =
9 2 A) if m m
if x 7
; f
if x = 7 2 2 , 9 if m = 7 7
2 m
50) B) 2 if m 2 , 9 if m = 2 7
C) (4m + 4) if m 2 , 9 if m = 2 (2m- 7)
Find
7
7
D) (4 + 4m) if m 2 , 9 if m = 2
7
(2 - 7m)
7
7
f(x + h) - f(x) . h
51) f(x) = 3x - 10 A) 3
51) C) 10
B) -3h
3
D) 10
52) f(x) = 7x2 + 7x - 14
52)
A) 14x + 7 + 7h
B) 14x + 7
C) 14xh + 7h + 7h 2
D) 7x + 6 + 14h
53) f(x) = A) C)
9 x + 22
53)
-9
(x + 9)2
-9 (x + h + 22)(x + 22)
54) f(x) = 10 - 9x3 A) -9(x2 - xh - h2 ) C) -3x2
B)
-198 (x + h + 22)(x + 22)
D)
9 (x + h + 22)(x + 22)
54) B) -9(3x2 + 3xh + h 2 ) D) -9(3x2 - 3x - h)
55) f(x) = 2
55)
x
A) 0
B) -
2 x(x + h)
C) -
2 (x + h)
56) f(x) = 4
D) -
h x(x + h)
56)
x2
A) -
4 (x + h)
B) -
C) -
8x + 4h 2 2 x (x + 2hx + h 2 )
D)
11
h x(x + h)
h x-h
Decide whether the graph represents a function.
57)
57)
A) Function
B) Not a function
58)
58)
A) Function
B) Not a function
59)
59)
A) Function
B) Not a function
12
60)
60)
A) Function
B) Not a function
61)
61)
A) Function
B) Not a function
62)
62)
A) Function
B) Not a function
13
63)
63)
A) Function
B) Not a function
64)
64)
A) Function Classify the function as even, odd, or neither. 65) f(x) = 5x A) Even
66) f(x) = 7x2 A) Even 67) f(x) = -3x3 A) Even 68) f(x) = 2x4 - x2 A) Even
B) Not a function
65) B) Odd
C) Neither
B) Odd
C) Neither
66)
67) B) Odd
C) Neither 68)
B) Odd
C) Neither
14
69) f(x) = -5x2 + 6 A) Even
B) Odd
C) Neither
70) f(x) = 4x3 + 6 A) Even
B) Odd
C) Neither
69)
70)
71) f(x) = 1
71)
x2
A) Even 72) f(x) =
B) Odd
C) Neither
x x2 - 4
72)
A) Even
B) Odd
73) f(x) = -6x3 + 9x A) Even
C) Neither 73)
B) Odd
74) f(x) = x2 + x A) Even
C) Neither 74)
B) Odd
C) Neither
Solve the problem. 75) The table shows the estimated number of pounds of summer flounder harvested in North Carolina each year from 1992-1998. Let y = f(x) represent the number of flounder (in millions of pounds) and x represent the years. What is the independent variable? Year 1992 1993 1994 1995 1996 1997 1998
75)
Millions of lb of Summer Flounder 2.6 3.1 3.6 4.6 4.2 1.5 3.0
A) None of these are correct. B) Millions of pounds of flounder C) The number of hurricanes striking the N.C. coast in the given year D) Years 76) A state park charges $10 per day or fraction of a day to rent a tent site, plus a fixed $7 park maintenance fee. Let T(x) represent the cost to stay in a tent site for x days. Find T 9
A) $90.00
B) $97.00
C) $99.00 15
1 . 5
D) $107.00
76)
77) A hummingbird adds 13 grams per day to its base body weight of 5 grams during the spring
77)
7 migration. Let T(x) represent the hummingbird's weight after x days. Find T 4 . 8
A) 57 g
B) 70 g
C) 68.38 g
D) 52 g
78) Sue wants to put a rectangular garden on her property using 88 meters of fencing. There is a river
78)
that runs through her property so she decides to increase the size of the garden by using the river as one side of the rectangle. (Fencing is then needed only on the other three sides.) Let x represent the length of the side of the rectangle along the river. Express the garden's area as a function of x. A) A(x) = 45x - 2x2 B) A(x) = 43x - 1 x2 4
C) A(x) = 44x - 1 x2
D) A(x) = 44x2 - x
2
79) A farmer has 1,400 yards of fencing to enclose a rectangular garden. Express the area A of the
79)
rectangle as a function of the width x of the rectangle. What is the domain of A? A) A(x) = -x2 + 700x; {x|0 < x < 700} B) A(x) = -x2 + 700x; {x|0 < x < 1,400}
C) A(x) = -x2 + 1,400x; {x|0 < x < 1,400}
D) A(x) = x2 + 700x; {x|0 < x < 700}
80) Suppose a life insurance policy costs $32 for the first unit of coverage and then $8 for each
80)
81) The graph shows the relationship between the area A of a rectangle and the length L, if the width
81)
additional unit of coverage. Let C(x) be the cost for insurance of x units of coverage. What will 10 units of coverage cost? A) $112 B) $104 C) $80 D) $48
is fixed. Find the area if the length is 9 cm.
A) 165 cm2
B) 135 cm2
C) 90 cm2 16
D) 105 cm2
82) The territorial area of an animal is defined to be its defended region, or exclusive region. For
82)
example, a rhinoceros has a certain region over which it is ruler. The area T of that region, in acres, can be approximated by the function T = W1.78,
where W is the weight of the animal, in tons. Find the approximate territorial area of a rhinoceros who weights 1.1 tons. Round to the nearest hundredth. A) 1.18 acres B) 1.89 acres C) 0.53 acres D) 0.84 acres
83) When pouring water from one five gallon bucket to another, a person tends to pour at a faster rate
83)
at first and then slow down in order not to spill. The amount of water left in the original bucket can be approximated by f(t) = 5 - 0.75t0.61,
where f(t) is measured in gallons and t is the time spent pouring in seconds. Find the approximate amount of water left in the original bucket after 6 seconds of pouring. Round to the nearest hundredth. A) 2.76 gal B) 4.39 gal C) 4.25 gal D) 2.24 gal
Match the correct graph to the given function. 84) y = x2 - 4
84)
A)
B)
C)
D)
17
85) y = x2 - 3 A)
B)
C)
D)
85)
18
86) y = (x + 4)2 A)
B)
C)
D)
86)
19
87) y = (x - 4)2 A)
B)
C)
D)
87)
20
88) y = (x + 3)2 + 2 A)
B)
C)
D)
88)
21
89) y = -(x + 6)2 - 1 A)
89) B)
C)
D)
22
90) y = -(1 - x)2 + 4 A)
B)
C)
D)
90)
Graph the parabola and give its vertex, axis, x-intercepts, and y-intercepts. 91) f(x) = x2 + 10x
23
91)
A) vertex (-5, -25); axis is x = -5;
x-intercepts are 0 and - 10; y-intercept is 0
B) vertex (5, 25); axis is x = 5;
no x-intercepts; y intercept is 50
C) vertex (-5, 25); axis is x = -5;
no x-intercepts; y-intercept is 50
24
D) vertex (5, -25); axis is x = 5;
x-intercepts are 0 and 10; y-intercept is 0
92) f(x) = -x2 - 2x
92)
A) vertex (1, -1); axis is x = 1;
no x-intercepts; y-intercept is -2
25
B) vertex (-1, -1); axis is x = -1;
no x-intercepts; y-intercept is -2
C) vertex (-1, 1); axis is x = -1;
x-intercepts are 0 and -2; y-intercept is 0
D) vertex (1, 1); axis is x = 1;
x-intercepts are 0 and 2; y-intercept is 0
26
93) f(x) = x2 + 6x + 9
93)
A) vertex (3, 0); axis is x = 3;
B) vertex (-3, 9); axis is x = -3;
C) vertex (-3, 0); axis is x = -3;
D) vertex (3, 9); axis is x = 3;
x-intercept is 3; y-intercept is 9
no x-intercepts; y-intercept is 18
x-intercept is -3; y-intercept is 9
no x-intercepts; y-intercept is 18
27
94) f(x) = x2 + 8x + 7
94)
A) vertex (-4, -9); axis is x = -4;
x-intercepts are -1 and - 7; y-intercept is 7
B) vertex (-4, 9); axis is x = -4;
x-intercepts are -1 and - 7; y-intercept is -7
28
C) vertex (4, 9); axis is x = 4;
x-intercepts are 1 and 7; y-intercept is -7
D) vertex (4, -9); axis is x = 4;
x-intercepts are 1 and 7; y-intercept is 7
95) f(x) = -x2 - 2x + 3
95)
29
A) vertex (1, -4); axis is x = 1;
x-intercepts are -1 and 3; y-intercept is -3
B) vertex (-1, 4); axis is x = -1;
x-intercepts are 1 and - 3; y-intercept is 3
C) vertex (1, 4); axis is x = 1;
x-intercepts are -1 and 3; y-intercept is 3
30
D) vertex (-1, -4); axis is x = -1;
x-intercepts are 1 and - 3; y-intercept is -3
96) f(x) = x2 - 2x - 8
96)
A) vertex (1, 9); axis is x = 1;
x-intercepts are 4 and - 2; y-intercept is 8
31
B) vertex (-1, -9); axis is x = -1;
x-intercepts are -4 and 2; y-intercept is -8
C) vertex (1, -9); axis is x = 1;
x-intercepts are 4 and - 2; y-intercept is -8
D) vertex (-1, 9); axis is x = -1;
x-intercepts are -4 and 2; y-intercept is 8
32
97) f(x) = -x2 + 8x - 7
97)
A) vertex (4, 9); axis is x = 4;
x-intercepts are 7 and 1; y-intercept is -7
B) vertex (4, -9); axis is x = 4;
x-intercepts are 7 and 1; y-intercept is 7
33
C) vertex (-4, -9); axis is x = -4;
x-intercepts are -7 and - 1; y-intercept is 7
D) vertex (-4, 9); axis is x = -4;
x-intercepts are -7 and - 1; y-intercept is -7
98) f(x) = 2x2 - 12x + 19
98)
34
B) vertex (3, 1); axis is x = 3;
A) vertex (-3, 1); axis is x = -3;
no x-intercepts; y-intercept is 19
no x-intercepts; y-intercept is
C) vertex (-3, 1); axis is x = -3;
11 2
D) vertex (3, 1); axis is x = 3;
11 no x-intercepts; y-intercept is 2
no x-intercepts; y-intercept is 19
99) f(x) = -8x2 - 2x - 10
99)
35
A) vertex 1 , 79 ; axis is x = 1 ; 8
8
B) vertex - 1 , - 79 ; axis is x = - 1 ;
8
8
no x-intercepts; y-intercept is 10
8
8
no x-intercepts; y-intercept is -10
C) vertex 1 , - 79 ; axis is x = 1 ; 8
8
D) vertex - 1 , 79 ; axis is x = - 1 ;
8
8
no x-intercepts; y-intercept is -10
8
8
no x-intercepts; y-intercept is 10
Match the correct graph to the given function. 100) y = x - 3
100)
A)
36
B)
C)
D)
37
101) y = x + 3 A)
B)
C)
D)
101)
102) y = x - 2 + 6 A)
102)
38
B)
C)
D)
39
103) y = x - 1 + 7 A)
B)
C)
D)
103)
40
104) y = -x + 3 - 4 A)
B)
C)
D)
104)
41
105) y = - x - 2 - 1 A)
B)
C)
D)
105)
Using the graph below, sketch the graph of the given function.
42
106) y = - f(x)
106)
A)
B)
C)
D)
43
107) y = f(-x)
107)
A)
B)
C)
D)
44
108) y = f(x + 2) -1
108)
A)
B)
C)
D)
45
109) y = f(-x - 2) + 1
109)
A)
B)
C)
D)
Graph the function.
46
110) f(x) = x + 2 + 3
110)
A)
B)
C)
D)
47
111) f(x) = 2 - x + 3
111)
A)
B)
C)
D)
48
112) f(x) = - -5 - x + 2
112)
A)
B)
C)
D)
Graph the indicated new function, given the graph for y = f(x).
49
113) y = f(ax), where a satisfies 0 < a < 1
113)
A)
B)
C)
D)
50
114) y = f(ax), where a satisfies 1 < a
114)
A)
B)
C)
D)
51
115) y = f(ax), where a satisfies -1 < a < 0
115)
A)
B)
C)
D)
52
116) y = f(ax), where a satisfies a < -1
116)
A)
B)
C)
D)
53
117) y = af(x), where a satisfies 0 < a < 1
117)
A)
B)
C)
D)
54
118) y = af(x), where a satisfies 1 < a
118)
A)
B)
C)
D)
55
119) y = af(x), where a satisfies -1 < a < 0
119)
A)
B)
C)
D)
56
120) y = af(x), where a satisfies a < -1
120)
A)
B)
C)
D)
Solve the problem. 121) John owns a hotdog stand. He has found that his profit is represented by the equation P(x) = -x2 + 78x + 84, where is x the number of hotdogs. How many hotdogs must he sell to earn the most profit? A) 45 hotdogs
B) 78 hotdogs
C) 39 hotdogs
57
D) 84 hotdogs
121)
122) Bob owns a watch repair shop. He has found that the cost of operating his shop is given by
122)
C(x) = 2x2 - 160x + 66, where x is the number of watches repaired. How many watches should he repair to produce the lowest cost? A) 40 watches B) 66 watches
C) 132 watches
D) 80 watches
123) John owns a hotdog stand. He has found that his profit is represented by the equation
123)
P(x) = -x2 + 12x + 46, where x is the number of hotdogs. What is the most he can earn?
A) $12
B) $23
C) $46
D) $82
124) Bob owns a watch repair shop. He has found that the cost of operating his shop is given by
124)
125) Suppose the cost of producing x items is given by C(x) = 2,400 - x3 and the revenue made on the
125)
126) Let C(x) = 16x + 27 be the cost to produce x units of a product, and let R(x) = - x2 + 28x be the
126)
C(x) = 2x2 - 24x + 294, where x is the number of watches repaired. What is his minimum cost? A) $240 B) $444 C) $480 D) $222
sale of x items is R(x) = 300x - 8x2 . Find the number of items which serves as a break-even point. A) 8 items B) 16 items C) 80 items D) 4 items
revenue. Find the maximum profit. A) $6 B) $12
C) $9
D) $7
127) An advertising agency has discovered that when the Holt Company spends x thousands of
127)
dollars on advertising, it results in a profit increase in thousands of dollars given by the function 1 P(x) = - (x - 8)2 + 20. How much should the Holt Company spend on advertising to maximize 4 the profit? A) $23,000
B) $20,000
C) $8,000
D) $6,000
128) A projectile is thrown upward so that its distance above the ground, in feet, after t seconds is h = -11t2 + 352t. After how many seconds does it reach its maximum height? A) 22 sec B) 11 sec C) 16 sec
128)
D) 32 sec
129) If an object is thrown upward with an initial velocity of 12 feet per second, then its height is given
129)
by
h = -12t2 + 72t. After how many seconds does it hit the ground? A) 12 sec B) 6 sec C) 24 sec
D) 3 sec
130) The length and width of a rectangle have a sum of 172. What dimensions will give the maximum
130)
area?
A) 43 by 129
B) 85 by 87
C) 86 by 86
D) 43 by 43
131) A projectile is thrown upward so that its distance above the ground, in feet, after t seconds is h = -11t2 + 440t. What is its maximum height? A) 6,600 ft B) 2,200 ft
C) 3,300 ft 58
D) 4,400 ft
131)
132) If an object is thrown upward with an initial velocity of 11 feet per second, then its height is given
132)
by
h = -11t2 + 44t. What is its maximum height? A) 44 ft B) 66 ft
C) 22 ft
D) 33 ft
133) The number of mosquitoes M(x), in millions, in a certain area depends on the June rainfall x, in
133)
inches: M(x) = 8x - x2 . What rainfall produces the maximum number of mosquitoes?
A) 64 in.
B) 8 in.
C) 0 in.
D) 4 in.
134) A Community College wants to construct a rectangular parking lot on land bordered on one side
134)
by a highway. It has 1,440 feet of fencing to use along the other three sides. What should be the dimensions of the lot if the enclosed area is to be a maximum? (Hint: Let x represent the width of the lot, and let 1,440 - 2x represent the length.) A) 480 ft by 480 ft B) 360 ft by 1,080 ft
C) 360 ft by 720 ft
D) 480 ft by 960 ft
135) What is the maximum area that can be enclosed by 1,320 feet of fencing? A) 193,600 sq ft B) 96,800 sq ft C) 108,900 sq ft Use the principles of translating and reflecting to graph the function. 136) f(x) = (x - 2)3 - 1
A)
B)
59
135) D) 217,800 sq ft
136)
C)
D)
137) f(x) = -(x + 2)4 - 3
137)
A)
B)
60
C)
D)
Match the function to the correct graph. 138) y = x3 - 3x + 2
138)
A)
B)
C)
D)
61
139) y = -x3 - 4x2 - 16x + 20 A)
139) B)
C)
D)
62
140) y = 2x3 - 3x2 + 10x + 7 A)
140) B)
C)
D)
63
141) y = x4 + x3 - 5x2 - 4x + 4 A)
141) B)
C)
D)
64
142) y = x4 - 5x3 + 15x2 + x - 30 A)
142) B)
C)
D)
65
143) y = 2x4 - x3 + 5x2 + 4x - 3 A)
143) B)
C)
D)
66
144) y = x5 - x4 + x3 + 1 A)
144) B)
C)
D)
67
145) y = 2x5 + 9x4 + 8x3 - 45x2 - 44x + 10 A)
145) B)
C)
D)
Match the graph to the correct function.
146)
146)
A) y =
x 2 x -1
2
2
B) y = x - 3
C) y = x + 3
x2 + 1
x3 - 1
68
2
D) y = x + 3 x2 - 1
147)
147)
2
A) y = -3x - 5 x2 + 1
2
2
B) y = 3x + 5
C) y = 3x - 5
x2 - 1
x2 + 1
2
D) y = -3x + 5 x2 - 1
The following is a graph of a polynomial function. State whether the degree of the polynomial is even or odd, and give the sign (+ or -) for the leading coefficient.
148)
148)
A) Degree is odd; + C) Degree is even; -
B) Can't identify degree; + D) Degree is even; +
69
149)
149)
A) Can't identify degree; + C) Degree is even; -
B) Degree is odd; D) Degree is even; +
150)
150)
A) Degree is even; C) Degree is even; +
B) Can't identify degree; + D) Degree is odd; +
70
151)
151)
A) Can't identify degree; C) Degree is odd; -
B) Degree is even; + D) Degree is even; -
152)
152)
A) Degree is even; + C) Degree is odd; +
B) Can't identify degree; + D) Degree is even; -
71
153)
153)
A) Can't identify degree; + C) Degree is even; -
B) Degree is even; + D) Degree is odd; +
154)
154)
A) Degree is even; C) Can't identify degree; +
B) Degree is odd; + D) Degree is even; +
72
155)
155)
A) Degree is odd; C) Can't identify degree; -
B) Degree is even; + D) Degree is even; -
Find the asymptotes of the function. 156) y = 5 x-3
156)
A) Vertical asymptote at x = -3; horizontal asymptote at y = 0 B) Vertical asymptote at x = 3; horizontal asymptote at y = 0 C) Vertical asymptote at x = -3; no horizontal asymptote D) Vertical asymptote at x = 3; horizontal asymptote at y = 5 157) y = -3
157)
x-6
A) Vertical asymptote at x = 6; horizontal asymptote at y = 0 B) Vertical asymptote at x = -6; horizontal asymptote at y = 0 C) Vertical asymptote at x = -6; horizontal asymptote at y = -3 D) Vertical asymptote at x = 6; horizontal asymptote at y = -3 158) y =
5 2 - 5x
158)
A) Vertical asymptote at x = 5; horizontal asymptote at y = 2 5
B) Vertical asymptote at x = 2 ; horizontal asymptote at y = 5 5
C) Vertical asymptote at x = 2 horizontal asymptote at y = 0 5
D) Vertical asymptote at x = 0; horizontal asymptote at y = 2 5
73
159) y = 4x
159)
x-7
A) Vertical asymptote at x = 7; horizontal asymptote at y = 4 B) Vertical asymptote at x = 7; no horizontal asymptote C) Vertical asymptote at x = 4; horizontal asymptote at y = 7 D) Vertical asymptote at x = -7; horizontal asymptote at y = 4 160) y = x + 5
160)
x-9
A) Vertical asymptote at x = -9; horizontal asymptote at y = 1 B) Vertical asymptote at x = 9; horizontal asymptote at y = x C) Vertical asymptote at x = -9; horizontal asymptote at y = 0 D) Vertical asymptote at x = 9; horizontal asymptote at y = 1 161) y = -4x + 5
161)
x+1
A) Vertical asymptote at x = 1; horizontal asymptote at y = -4 B) Vertical asymptote at x = -1; horizontal asymptote at y = -4 C) Vertical asymptote at x = -1; horizontal asymptote at y = 5 4
D) Vertical asymptote at x = -4; horizontal asymptote y = -1 162) y = -4x + 4
162)
9 - 3x
A) Vertical asymptote at x = 3; horizontal asymptote at y = 4 B) Vertical asymptote at x = 3; horizontal asymptote at y = 4 3
C) Vertical asymptote at x = 3; horizontal asymptote at y = - 4 3
D) Vertical asymptote at x = - 4 ; horizontal asymptote at y = 3 3
2
163) y = x - 1
163)
x-1
A) No vertical asymptote; horizontal asymptote at y = 1 B) Vertical asymptote at x = -1; no horizontal asymptote C) No asymptotes; hole at x = 1 D) Vertical asymptote at x = 1; no horizontal asymptote Graph the rational function.
74
164) y = 1
164)
x+3
A)
B)
C)
D)
75
165) y = -1
165)
x+1
A)
B)
C)
D)
76
166) y =
1 4 - 2x
166)
A)
B)
C)
D)
77
167) y = 2x
167)
x-2
A)
B)
C)
D)
78
168) y = x - 3
168)
x-1
A)
B)
C)
D)
79
169) y = 9 + 5x
169)
4x + 5
A)
B)
C)
D)
80
2
170) f(x) = x - 9
170)
x-3
A)
B)
C)
D)
81
2
171) y = x + 8x + 15
171)
x+5
A)
B)
C)
D)
Solve the problem.
172) If the average cost per unit C(x) to produce x units of plywood is given by C(x) = 900 , what is x + 30
the unit cost for 10 units? A) $3.00
B) $60.00
C) $22.50
82
D) $90.00
172)
173) If the average cost per unit C(x) to produce x units of plywood is given by C(x) = 1,500 , what do x + 50
1,000 units cost? A) $500.00
B) $30,000.00
C) $1,428.57
173)
D) $1,499.95
174) Suppose the cost per ton, y, to build an oil platform of x thousand tons is approximated by y =
174)
212,500 . What is the cost for x = 300? x + 425
A) $293.10
B) $283.33
C) $150,000.00
D) $87,931.03
175) Suppose the cost per ton, y, to build an oil platform of x thousand tons is approximated by y =
175)
112,500 . What is the cost per ton for x = 50? x + 225
A) $2,025.00
B) $2,250.00
C) $10.00
D) $409.09
176) Suppose the cost per ton, y, to build an oil platform of x thousand tons is approximated by
176)
262,500 y= . What is the cost per ton for x = 300? x + 525
A) $350.00
B) $95,454.55
C) $318.18
D) $150,000.00
177) Suppose a cost-benefit model is given by y = 2.2x , where y is the cost in thousands of dollars 100 - x
177)
for removing x percent of a given pollutant. Find the cost of removing 65% to the nearest dollar. A) $4,086 B) $1,857 C) $1,430 D) $2,200
178) A function that might describe the entire Laffer curve is y = 0.5x(100 - x)(10000 - x2 ) where y is
178)
179) The polynomial function I(t) = -0.1t2 + 2t represents the yearly income (or loss) from a real estate
179)
180) In the following formula, y is the minimum number of hours of studying required to attain a test
180)
the government revenue in hundreds of thousands of dollars from a tax of x percent, with the function valid for 0 x 100. Find the revenue from a tax rate of 70%. Round your answer to the nearest billion. A) $506 billion B) $536 billion C) $436 billion D) $561 billion
investment, where t is time in years. After what year does income begin to decline? A) 20 B) 9 C) 13.33 D) 10
score of x: y =
A) 101.09 hr
0.34x . How many hours of study are needed to score 84? 100.5 - x
B) 4.82 hr
C) 1.73 hr
D) 17.30 hr
181) The polynomial function A(x) = -0.015x3 + 1.05x gives the alcohol level in an average person's
blood x hours after drinking 8 oz of 100-proof whiskey. If the level exceeds 1.5, a person is legally drunk. Would a person be drunk after 3 hours? A) Yes B) No
83
181)
182) The polynomial function L(p) = p3 - 5p2 + 20 gives the rate of gas leakage from a tank as pressure
182)
183) The polynomial function G(x) = -0.006x4 + 0.140x3 - 0.53x2 + 1.79x measures the concentration of
183)
increases in p units from its initial setting. Will an increase of 4 units result in a lower rate of leakage compared to the initial setting of p = 0? A) Yes B) No
a dye in the bloodstream x seconds after it is injected. Does the concentration increase between 13 and 14 seconds? A) Yes B) No
Match the graph to the function.
184)
184)
A) f(x) = 4 x
B) f(x) = 4 x - 1
C) f(x) = 4 x + 1
D) f(x) = 4 x + 1
185)
185)
A) f(x) = - 1 6
x
B) f(x) = 1
x
C) f(x) =-6 x
6
84
D) f(x) = 6 x
186)
186)
A) f(x) = -2 -x
C) f(x) = -2 x
B) f(x) = 2 -x
D) f(x) = 2 x
187)
187)
A) f(x) = -5 -x
C) f(x) = -5 x
B) f(x) = 5 -x
D) f(x) = 5 x
188)
188)
A) f(x) = 3 -x
C) f(x) = -3 x
B) f(x) = -3 -x
85
D) f(x) = 3 x
189)
189)
A) f(x) = 2 1
x
7
B) f(x) = -2 1
x
C) f(x) = 2(7)x
7
D) f(x) = -2(7)x
190)
190)
A) f(x) = 2 x
B) f(x) = 2 x - 2
C) f(x) = 2 x - 2
D) f(x) = 2 x + 2
191)
191)
A) f(x) = 5 x + 2
B) f(x) = 5 x - 2
C) f(x) = 5 x - 2 86
D) f(x) = 5 x
192)
192)
B) y = 1
A) y = 5 -x - 1
1-x
D) y = 1
C) y = 5 x + 1
5
x-1
5
193)
193)
A) y = 1 3
x-1
B) y = 3 x - 1 - 3
-3
D) y = 1
C) y = 3 x + 1 - 3
3
x+1
-3
Solve the equation. 194) 4x = 16
A) 1
194) B) 3
C) 2
D) 4
195) 5-x = 1
195)
25
A) 1
B) 1
C) 2
D) -2
196) 4(6 - 2x) = 16 A) 4
B) 3
C) -2
D) 2
5
2
196)
87
197) 3(1 + 2x) = 27 A) -1
197) B) 1
C) 9
D) 3
198) 2(7 - 3x) = 1
198)
4
A) 1
B) 1
2
C) -3
D) 3
199) 4x = 1
199)
256
C) 1
B) 4
A) -4
4
D) 1
64
200) 4(7 + 3x) = 1
200)
16
A) 4
C) 1
B) 3
4
D) -3
201) e-3x = (e6 )2 - x A) 2
201) B) 4
3
C) 0
D) - 4
202) 5-|x| = 1
202)
25
A) 5, -5
B) 1, -1
C) 2, -2
Graph the function. 203) y = 3ex + 2
D) 2
203)
88
A)
B)
C)
D)
204) y = -3ex + 3
204)
89
A)
B)
C)
D)
205) y = 4e-x + 1
205)
90
A)
B)
C)
D)
206) y = 3e-2x - 2
206)
91
A)
B)
C)
D)
207) y = -3e-x/2 + 2
207)
92
A)
B)
C)
D)
Solve the problem. 208) Find the amount of interest earned on the following deposit: $1000 at 5% compounded annually for 12 years A) $1,795.86 B) $885.65 C) $710.34 D) $795.86
209) How long will it take for prices in the economy to double at a 6% annual inflation rate? Round to the nearest hundredth when necessary. A) 11.9 yr B) 10.24 yr
C) 18.85 yr
208)
209)
D) 23.45 yr
210) An economist predicts that the buying power B(x) of a dollar x years from now will decrease
210)
according to the formula B(x) = 0.78x. How much will today's dollar be worth in 3 years? Round to the nearest cent. A) $0.47
B) $0.93
C) $2.36
D) $2.34
211) Find the interest earned on $8,000 invested for 5 years at 8% interest compounded quarterly. Round to the nearest cent. A) $3,887.58
B) $1,751.96
C) $11,887.58
93
D) $1.49
211)
212) Find the interest earned on $8,000 invested for 7 years at 6.6% interest compounded monthly. Round to the nearest cent. A) $4,650.18
B) $4,691.52
C) $4,681.90
D) $4,513.83
213) Suppose that the number of bacteria in a culture after x hours is given by f(x) = 1000 · 3 0.5x. How many bacteria are in the culture after 4 hours? A) 214 bacteria B) 2,088 bacteria
C) 9,000 bacteria
213)
D) 55 bacteria
214) Suppose that the number of bacteria in a culture after x hours is given by f(x) = 500 · 6 0.125x. How many bacteria are in the culture after 10 hours? A) 4,695 bacteria
212)
214)
B) 302,108,999 bacteria D) 59,791 bacteria
C) 2 bacteria
215) The population of a particular city is increasing at a rate proportional to its size. It follows the
215)
function P(t) = 1 + ke0.12t where k is a constant and t is the time in years. If the current population is 38,000, in how many years is the population expected to be 95,000? A) 3 yr B) 58 yr C) 8 yr
D) 5 yr
216) The number of dislocated electric impulses per cubic inch in a transformer increases when
216)
217) The number of bacteria growing in an incubation culture increases with time according to
217)
lightning strikes by D = 3,600(2)x, where x is the time in milliseconds of the lightning strike. Find the number of dislocated impulses at x = 0 and x = 3. A) 3,600; 14,400 B) 7,200; 28,800 C) 3,600; 21,600 D) 3,600; 28,800
B = 4,500(4)x, where x is time in days. Find the number of bacteria when x = 0 and x = 5. A) 4,500 bacteria, 4,608,000 bacteria B) 4,500 bacteria, 90,000 bacteria
C) 4,500 bacteria, 1,152,000 bacteria
D) 18,000 bacteria, 4,608,000 bacteria
218) The number of books in a small library increases according to the function B = 7,600e0.03t, where t is measured in years. How many books will the library have after 6 years? A) 13,032 books B) 11,503 books C) 5,660 books
D) 9,099 books
Write the exponential equation in logarithmic form. 219) 62 = 36
A) log36 6 = 2 220) 32 = 9 A) log9 3 = 2
218)
219)
B) log2 36 = 6
C) log6 2 = 36
D) log6 36 = 2 220)
B) log2 9 = 3
C) log3 2 = 9
94
D) log3 9 = 2
221) 5-3 = 1
221)
125
A) log1/125 5 = -3
B) log5 -3 = 1
C) log-3 1 = 5
D) log5 1 = -3
125
125
125
-2 222) 3 = 49 7
222)
9
A) log 3/7 49 = -2
B) log 3/7 (-2) = 49
9
9
C) log 49/9 (-2) = 3
D) log49/9 3 = -2
7
7
Write the logarithmic equation in exponential form. 223) log3 1 = -3 27 3 A) 1 = 3 27
224) log4 16 = 2 A) 42 = 16
223) C) 33 = 1
B) 327 = 3
27
D) 3-3 = 1
27
224) B) 24 = 16
C) 162 = 4
225) log 0.00001 = -5 A) 10-5 = 0.00001 C) -5 10 = 0.00001
D) 416 = 2 225)
B) 0.00001-5 = 10 D) 100.00001 = -5
226) log2 8 = 3 A) 23 = 3
226) B) 23 = 1
C) 23 = 8 + 1
8
D) 23 = 8
227) log 1,000 = 3 A) 103 = 1,000 228) ln x = 4 A) ex = 4
227) B) 103 = 10,000
C) 103 = 3
D) 103 =
B) e4 = x
C) x4 = e
D) 4e = x
1 1,000
228)
229) ln 1 = -3
229)
e3
-3 A) 1 =e e3
e C) 1 = -3
B) e-3 = 1
e3
e3
95
D) -3 e = 1
e3
230) ln e5 = 5 A) ln 5 = 5
230) B) e5 = e5
C) ln e5 = e5
D) e5 = 5
231) ln e1/8 = 1
231)
8
A) e1/8 = 1 8
B) e1/8 = e1/8
C) e8 = e1/8
D) ln 1 = e1/8 8
Evaluate the logarithm without using a calculator. 232) log4 16
A) 4
232)
B) 8
C) 16
D) 2
233) log4 1
233)
4
A) 4
C) 1
B) -1
D) 0
234) log8 1
234)
64
A) 8
B) 2
235) log10 10 A) 1
C) -8
D) -2 235)
B) 10
C) 0
D) -1
236) log9 1
236)
729
A) -3 237) log8 32
C) 81
B) -81
D) 3 237)
A) 4
B) 5
C) 3
D) 5
238) ln e A) 1
B) e
C) 0
D) -1
3
3
2
4
238)
239) ln l A) 0
239) B) e
C) -1
D) 1
240) log5 5 1
240)
25
A) - 2 5
B) - 5
C) 2
2
5
96
D) 5 2
241) ln e7/3
241)
A) 7
B) 3
3
C) 3 e
7
D) 7 e
7
3
Rewrite the expression as a sum, difference, or product of simpler logarithms. 242) log4 10x
A) log2 10 + log2 x C) log4 10 + log4 x
242)
B) log2 10 - log2 x D) log4 10 - log4 x
243) log8 xy A) log8 x - log8 y
243) B) log8 x + log8 y
C) log4 x + log4 y
D) log4 x - log4 y
244) log6 12
244)
13
A) log3 12 - log3 13 C) log6 13 - log6 12
B) log6 12 - log6 13 D) log6 12 + log6 13
10 13
245) log6
245)
A) log6 13 - 1 log6 10
B) 1 log6 10 - log6 13
C) 1 log6 10 + log6 13
D) 1 log3 10 - log3 13
2
2
2
2
246) log3 5p
246)
3k
A)
log3 5 + log3 p
B)
1 + log3 k
C) log3 5p - log3 3k
log3 5log3 p log3k
D) log3 5 + log3 p - 1 - log3 k
5
247) log7 6 5 4
247)
2
A) log7 6 + 1 log7 5 - 1 log7 2 5
B)
4
log7 6 + 5log7 5 4log7 2 log7 6 +
C) log7 6 + 5log7 5 - 4log7 2
D)
97
1 log7 5 5
1 log7 2 4
Use the properties of logarithms to find the value of the expression. 248) Let logb A = 2 and logb B = -3. Find logb AB.
A) 5
C) 6
B) -1
248) D) -6
249) Let logb A = 4 and logb B = -16. Find logb A .
249)
B
A) -12
B) - 1
C) 1
4
4
250) Let logb A = 5 and logb B = -4. Find logb B2 . A) 10 B) -8
D) 20
250) C) -16
251) Let logb A = 2 and logb B = -4. Find logb 2 AB. A) -1.000 B) 2.828
D) 16 251)
C) 2 -8
252) Let logb A = 3.299 and logb B = 0.346. Find logb AB. A) 3.645 B) 9.535 C) 2.953
D) -2.828 252) D) 1.141
253) Let logb A = 2.963 and logb B = 0.339. Find logb A .
253)
B
A) 1.004
B) 2.963
C) 2.624
254) Let logb 5 = a and logb 2 = c. Find logb 125b3 . A) 3(a + b) B) 3b + a - 3
D) 3.302 254)
C) 3a + 3
D) 3ab
Use natural logarithms to evaluate the logarithm to the nearest thousandth. 255) log9 21
A) 0.722 256) log2 0.792 A) -0.101 257) log2.1 293 A) 2.467 258) log7.2 3.8 A) 0.580 259) log 3 206.8 A) 0.239
B) 1.386
C) 2.333
255) D) 1.322 256)
C) 2.525
B) -2.972
D) -0.336 257)
B) 139.524
C) 0.131
D) 7.656 258)
B) 0.528
C) 1.479
D) 0.676 259)
B) 0.103
C) 4.853
98
D) 9.706
Solve the equation. 260) log 5x = log 4 + log (x + 5)
A) 20
260) B) 20
C) 9
9
261) log (x + 3) = log (5x - 5) A) 2 262) log2 x = 3 A) 9
261) C) - 1
B) -2
2
B) 6
C) 1.58
C) 7
B) 31/7
3
D) 71/3
264) C) 3 2
B) -6
265) log5 (7x - 4) = log5 (4x + 7) A) 3
D) 8 263)
264) log (3 + x) - log (x - 3) = log 3 A) 6
D) 1
262)
263) logy 7 = 3 A) 73
D) -20
4
D) No solution
265) C) 11
B) 1
3
D) No solution
266) log2 (3x + 4) = log2 (3x + 5) A) 4 5
267) log4 x2 = log4 (5x + 36) A) 9
266) B) - 9
C) 0
D) No solution
267) B) 9
C) 9, -4
4
D) No solution
268) 1 log2 x2 = log4 4x
268)
2
A) 8
B) 4
C) 4, 0
D) No solution
Solve the equation. Round decimal answers to the nearest thousandth. 269) 2x = 11
A) 1.705 270) e-0.06x = 0.5 A) 0.693
B) 3.459
C) 5.500
269) D) 0.289 270)
B) -11.552
C) -8.333
99
D) 11.552
271) ey + 1 = 3 A) 1.099
B) -0.523
C) 2.099
D) 0.099
272) 5(5x - 1) = 12 A) 0.375
B) 0.109
C) 0.680
D) 0.509
273) 4e3x - 6 = 8 A) 2.173
B) 2.231
C) 4.667
D) -1.769
274) 5e6x+2 = 2 A) -2.091 275) 50.32x = 40.54x A) -1.706
271)
272)
273)
274) B) 1.424
C) -2.396
275) B) 0.127
C) 0.000
D) 0.3
B) 10ex + 4
C) (x + 4)e10
D) e10(x + 4)
B) x6 e10
C) e10x6
D) 10ex6
Write the expression using base e rather than base 10. 276) 10x + 4
A) e(ln 10)(x + 4)
276)
277) 10x6
A) e(ln 10)x6
D) -0.486
277)
Approximate the expression in the form a x without using e. Round to the nearest thousandth when necessary. 278) e8x 278)
A) 2.079x
279) e-5x A) 0.007x Find the domain of the function. 280) f(x) = log (x - 6) A) x > 6
281) f(x) = ln (4 - x) A) x < 4
B) 2,980.958x
C) 21.746x
D) 285.005x
B) 0.544x
C) -1.609x
D) -13.591x
279)
280) B) x > 0
C) x > 1
D) x > -6 281)
B) x < -4
C) x > 4
282) f(x) = log2 (9 - x2 ) A) -3 x 3 C) x < -3 and x > 3
D) x > -4 282)
B) -3 < x < 3 D) -9 < x < 9
100
283) f(x) = ln (5x - x2 ) A) 0 < x < 5
283) B) -5 x < 0
C) x 5
D) -5 < x < 5
Solve the problem. 284) Sonja and Chris both accept new jobs on March 1, 2001. Sonja starts at $48,000 with a raise each March 1 of 3%. Chris starts at $27,000 with a raise on March 1 of each year of 7%. In what year will Chris' salary exceed Sonja's? A) 2,015 B) 2,016 C) 2,017 D) 2,014
284)
285) A college student invests $13,000 in an account paying 7% per year compounded annually. In
285)
286) How long will it take for prices in the economy to double at a 5% annual inflation rate? Round to
286)
how many years will the amount at least double? Round to the nearest tenth when necessary. A) 13.5 yr B) 18.5 yr C) 10.2 yr D) 16.2 yr
the nearest hundredth when necessary. A) 23.45 yr B) 22.52 yr
C) 14.21 yr
D) 11.9 yr
287) Assume the cost of a car is $32,000. With continuous compounding in effect, find the number of
287)
288) Suppose the consumption of electricity grows at 4% per year, compounded continuously. Find the
288)
289) The purchasing power of a dollar is decreasing at the rate of 4% annually, compounded
289)
290) At what interest rate must $4,500 be compounded annually to equal $8,485.42 after 13 years?
290)
years it would take to double the cost of the car at an annual inflation rate of 7%. Round to the nearest hundredth. A) 148.19 yr B) 158.09 yr C) 1.48 yr D) 9.90 yr
number of years before the use of electricity has tripled. Round to the nearest hundredth. A) 27.47 yr B) 75.00 yr C) 0.27 yr D) 2.75 yr
continuously. How long will it take for the purchasing power of $1.00 to be worth $0.62? Round to the nearest hundredth. A) 0.12 yr B) 1.20 yr C) 11.95 yr D) 15.50 yr
Round to the nearest percent. A) 5% B) 4%
C) 6%
D) 7%
291) Kimberly invested $7,000 in her savings account for 5 years. When she withdrew it, she had
$9,449.01. Interest was compounded continuously. What was the interest rate on the account? Round to the nearest tenth of a percent when necessary. A) 6.15% B) 6% C) 6.1% D) 5.9%
101
291)
292) The magnitude of an earthquake, measured on the Richter scale, is given by R(I) = log I , where I I0
292)
is the amplitude registered on a seismograph located 100 km from the epicenter of the earthquake, and I0 is the amplitude of a certain small size earthquake. Find the Richter scale rating of an earthquake with an amplitude of 251,189 I0 .
A) 12.4
B) 4.4
C) 5.4
D) 0.54
293) The magnitude of an earthquake, measured on the Richter scale, is given by R(I) = log I , where I I0
293)
is the amplitude registered on a seismograph located 100 km from the epicenter of the earthquake, and I0 is the amplitude of a certain small size earthquake. An earthquake measured 5 on the Richter scale. Express this reading in terms of I0 .
A) 100,000 I0
B) 148 I0
C) 79,433 I0
D) 10,000 I0
294) The magnitude of an earthquake, measured on the Richter scale, is given by R(I) = log I , where I I0
294)
is the amplitude registered on a seismograph located 100 km from the epicenter of the earthquake, and I0 is the amplitude of a certain small size earthquake. Find the Richter scale rating of an earthquake with an amplitude of 107.3 I0 .
A) 16.8
B) 17.3
C) 2.7
D) 7.3
295) A certain noise has intensity 6.1 × 108 I0 . What is the decibel rating of this sound? Use the formula
295)
D = 10 log I0 , where I0 is a faint threshold sound, and I is the intensity of the sound.”
A) 88 decibels
B) 202 decibels
C) 78 decibels
D) 9 decibels
296) The pH of a solution is defined as pH = -log[H+], where [H+] is the concentration of hydrogen
296)
ions in the solution. The pH of pure water is 7, while the pH of stomach acid is about 1. How much greater is the concentration of hydrogen ions in stomach acid than in pure water? A) 100 times greater B) 100,000 times greater
C) 1,000,000 times greater
D) 6 times greater
297) An RC circuit is a simple electronic circuit consisting of a resistor, a capacitor, and a battery. The
V current i in the circuit at some time t after the battery is connected is i = e-t/(RC), where V is the R battery's voltage, R is the resistance, and C is the capacitance. Solve this equation for C. -t t A) C = B) C = Ve C) C = -R D) C = V e-t/(iR) V iR 2 R R C R ln t ln iR V
102
297)
298) One hundred rats are being trained to run through a maze and are rewarded when they run
298)
through it correctly. Once a rat successfully runs the maze, it continues to run the maze correctly in all subsequent trials. The number of rats that run the maze incorrectly after t attempts is given approximately by N(t) = 100e-.12t. Find the number of trials required such that only 35% of the rats are running the maze incorrectly. Round to the nearest trial. A) 30 trials B) 8 trials C) 9 trials
D) 28 trials
299) The population growth of an animal species is described by F(t) = 200 + 60 log3 (2t + 1) where t is
299)
300) Coyotes are one of the few species of North American animals with an expanding range. The
300)
301) Find the effective rate corresponding to the nominal rate. 8% compounded monthly. Round to the
301)
measured in months. Find the population of this species in an area 40 month(s) after the species is introduced. A) 230 B) 2,550 C) 440 D) 5,060
future population of coyotes in a region of Mississippi can be modeled by the equation P = 49 + 19 ln(12t + 1), where t is time in years. Use the equation to determine when the population will reach 150. (Round to the nearest tenth of a year.) A) 17 yr B) 16.9 yr C) 17.2 yr D) 17,242.7 yr
nearest hundredth. A) 8.41%
B) 8.25%
C) 8.38%
D) 8.30%
302) Find the effective rate corresponding to the nominal rate. 8% compounded quarterly. Round to the nearest hundredth. A) 8.24%
B) 8.35%
C) 8.19%
D) 8.32%
303) Find the present value of the deposit. $11,000 at 6% compounded monthly for 10 years. Round to the nearest cent. A) $19,953.36
B) $6,105.96
C) $6,045.96
B) $3,027.34
C) $5,257.16
B) $225
C) $9,059
B) $163,054
C) $50,554
103
305)
D) $2,809
306) Find the present value of the deposit. $9,000 at 8% compounded continuously for 10 years. Round to the nearest dollar. A) $137,853
304)
D) $3,055.34
305) Find the present value of the deposit. $500 at 8% compounded continuously for 10 years. Round to the nearest dollar. A) $7,658
303)
D) $20,013.36
304) Find the present value of the deposit. $4,000 at 4% compounded quarterly for 7 years. Round to the nearest cent. A) $5,285.16
302)
D) $4,044
306)
307) Barbara knows that she will need to buy a new car in 5 years. The car will cost $15,000 by then.
307)
308) Southwest Dry Cleaners believes that it will need new equipment in 10 years. The equipment will
308)
309) An investment of $13,335 earns 8% interest compounded monthly for 4 years. (a) What is the
309)
310) If money can be invested at 7% compounded quarterly, which is larger -- $1000 now or the
310)
311) A certificate of deposit pays 9% interest compounded monthly. What effective interest rate does
311)
How much should she invest now at 5%, compounded quarterly, so that she will have enough to buy a new car? Round to the nearest cent. A) $11,193.23 B) $11,700.13 C) $13,257.81 D) $12,340.54
cost $26,000. What lump sum should be invested today at 6% compounded semiannually, to yield $26,000? Round to the nearest cent. A) $22,224.25 B) $19,427.47 C) $19,282.85 D) $14,395.57
value of the investment after 4 years? (b) If money can be deposited at 4% compounded quarterly, find the present value of the investment. Round to the nearest cent. A) (a) $19,344.51 B) (a) $18,223.02 C) (a) $18,344.51 D) (a) $14,830.78 (b) $17,773.12 (b) $16,773.12 (b) $15,644.59 (b) $15,801.03
present value of $1210 left at 7% interest for 6 years? A) $1000 now B) Present value of $1210 left for 6 years
the CD pay? Round to the nearest tenth when necessary. A) 10.1% B) 181.3% C) 9.4%
D) 8.3%
312) The sales of a new model of notebook computer are approximated by: S(x) = 4,000 - 11,000e-x/7 ,
312)
313) The sales of a mature product (one which has passed its peak) will decline by the function S(t)=
313)
where x represents the number of months the computer has been on the market and S represents sales in thousands of dollars. In how many months will the sales reach $2,000,000? Round to the nearest month. A) 15 months B) 19 months C) 12 months D) 22 months
S0 e-at, where t is time in years. Find the sales after 20 years if a = 0.14 and S0 = 69,900. Round to the nearest sale. A) 4,251 sales
B) 2,126 sales
C) 60,768 sales
D) 3,695 sales
314) The number of books in a small library increases according to the function B = 7,300e0.04t, where t
314)
315) In the formula N = Iekt, N is the number of items in terms of an initial population I at a given time
315)
is measured in years. How many books will the library have after 4 years? Round to the nearest book. A) 10,552 books B) 13,378 books C) 5,810 books D) 8,567 books
t and k is a growth constant equal to the percent of growth per unit time. How long will it take for the population of a certain country to double if its annual growth rate is 7.2%? Round to the nearest year. A) 4 yr B) 1 yr C) 28 yr D) 10 yr
104
316) In the formula N = Iekt, N is the number of items in terms of an initial population I at a given time
316)
317) In the formula N = Iekt, N is the number of items in terms of an initial population I at a given time
317)
318) The number of acres in a landfill decreases according to the function B = 5,000e-0.04t, where t is
318)
t and k is a growth constant equal to the percent of growth per unit time. How long will it take for the population of a certain country to triple if its annual growth rate is 1.4%? Round to the nearest year. A) 214 yr B) 34 yr C) 1 yr D) 78 yr
t and k is a growth constant equal to the percent of growth per unit time. There are currently 77 million cars in a certain country, increasing by 1.6% annually. How many years will it take for this country to have 102 million cars? Round to the nearest year. A) 4 yr B) 16 yr C) 18 yr D) 201 yr
measured in years. How many acres will the landfill have after 3 years? A) 4,435 acres B) 4,604 acres C) 3,793 acres
D) 10,601 acres
319) A bacteria colony doubles in 6 hr. How long does it take the colony to triple? Use N = N0 2t/T,
319)
where N0 is the initial number of bacteria and T is the time in hours it takes the colony to double. (Round to the nearest hundredth, as necessary.) A) 9 hr B) 9.51 hr
C) 2.43 hr
D) 18 hr
320) The population of a small country increases according to the function B = 1,800,000e0.02t, where t
320)
is measured in years. How many people will the country have after 10 years? A) 1,258,146 people B) 2,852,808 people
C) 2,896,988 people
D) 2,198,525 people
321) Use the formula P = Iekt. A bacterial culture has an initial population of 10,000. If its population declines to 4,000 in 2 hours, what will it be at the end of 4 hours? A) 1,600 bacteria B) 800 bacteria C) 3,000 bacteria
321)
D) 6,325 bacteria
322) In the formula A(t) = A0 ekt, A(t) is the amount of radioactive material remaining from an initial
322)
amount A0 at a given time t and k is a negative constant determined by the nature of the material. A certain radioactive isotope has a half-life of approximately 1,950 years. How many years would be required for a given amount of this isotope to decay to 75% of that amount? A) 3,900 yr B) 809 yr C) 734 yr D) 487.5 yr
323) In the formula A(t) = A0 ekt, A(t) is the amount of radioactive material remaining from an initial amount A0 at a given time t and k is a negative constant determined by the nature of the material. An artifact is discovered at a certain site. If it has 47% of the carbon-14 it originally contained, what is the approximate age of the artifact, rounded to the nearest year? (carbon-14 decays at the rate of 0.0125% annually.) A) 4,240 yr B) 6,040 yr C) 3,760 yr D) 2,623 yr
105
323)
324) In the formula A(t) = A0 ekt, A(t) is the amount of radioactive material remaining from an initial
324)
amount A0 at a given time t and k is a negative constant determined by the nature of the material. A certain radioactive isotope decays at a rate of 0.15% annually. Determine the half-life of this isotope, to the nearest year. A) 5 yr B) 333 yr C) 201 yr D) 462 yr
325) The amount of particulate matter left in solution during a filtering process decreases by the
325)
equation P = 700(2)-0.8n , where n is the number of filtering steps. Find the amounts left for n = 0 and n = 5. (Round to the nearest whole number.) A) 1,400, 44 B) 700, 11,200 C) 700, 22 D) 700, 44
326) The decay of 302 mg of an isotope is given by A(t)= 302e-0.03t, where t is time in years. Find the amount left after 40 years. A) 88 mg
B) 46 mg
C) 91 mg
326)
D) 293 mg
327) Newton's law of cooling states that the temperature f(t) of a body at time t is given by:
327)
f(t) = T0 + Ce-kt, where C and k are constants and T0 is the temperature of the environment in which the object rests. If C = -26.8 and k = 0.04 and t is in hours, how long will it take for a frozen roast to thaw to a temperature of 0°C in a refrigerator that is at 5°C? Round your answer to the nearest hour. A) 46 hr B) 40 hr C) 36 hr D) 42 hr
328) Newton's law of cooling states that the temperature f(t) of a body at time t is given by:
328)
f(t) = T0 + Ce-kt, where C and k are constants and T0 is the temperature of the environment in which the object rests. If C = 280 and k = 0.16 and t is in minutes, how long will it take for a glass baking dish containing brownies to cool to a comfortable-to-touch temperature of 93°F in a room that is at 71°F? Round your answer to the nearest minute. A) 11 min B) 13 min C) 16 min D) 20 min
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 329) The graph of y = f(x) has an x-intercept of a and a y-intercept of b. What are the intercepts of the graph of y = f(-x)?
329)
330) A classmate claims that, if a function f(x) has a horizontal asymptote at y = w, then the
330)
331) Suppose the population of deer fluctuates over time. The population increases in the
331)
function can only approach w but cannot actually equal w. Evaluate the classmate's claim.
summer and decreases in the winter. It also varies over many years as well. If you looked at the graph of population versus time, would this relation be a function? Why or why not?
106
332) Consider the linear function f(x) = 5x + 20. What is the domain and range of this
332)
function? Now, suppose the function represents the relationship between studying time and grades on an exam. The variable x represents the number of hours spent studying and f(x) represents the grade on the exam. Does this change the domain and range? If so, what is the new domain and range and why is it different?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2
2
333) True or False. The function y = x - 1 is continuous at x = 1.
333)
x-1
A) False
B) True
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
334) f(x) = ax
334)
The graph of an exponential function with base a is given. Sketch the graph of g(x) = -a x. Give the domain and range of g.
107
335) f(x) = ax
335)
The graph of an exponential function with base a is given. Sketch the graph of h(x) = a-x. Give the domain and range of h.
336) Explain how the graph of y = 2 x + 2 + 1 can be obtained from the graph of y = 2 x.
336)
337) Explain how the graph of y = (1/2)x - 1 can be obtained from the graph of y = 2 x.
337)
108
Answer Key Testname: UNTITLED2
1) A 2) B 3) A 4) B 5) A 6) A 7) B 8) D 9) A 10) D 11) C 12) A 13) B 14) A 15) B 16) C 17) C 18) D 19) B 20) D 21) D 22) B 23) D 24) D 25) B 26) B 27) D 28) C 29) C 30) C 31) B 32) B 33) C 34) B 35) B 36) B 37) A 38) B 39) D 40) C 41) C 42) D 109
Answer Key Testname: UNTITLED2
43) D 44) D 45) B 46) A 47) B 48) D 49) A 50) D 51) A 52) A 53) C 54) B 55) B 56) C 57) A 58) A 59) B 60) B 61) A 62) A 63) A 64) B 65) B 66) A 67) B 68) A 69) A 70) C 71) A 72) B 73) B 74) C 75) D 76) D 77) C 78) C 79) A 80) B 81) B 82) A 83) A 84) A 110
Answer Key Testname: UNTITLED2
85) C 86) A 87) A 88) D 89) A 90) D 91) A 92) C 93) C 94) A 95) B 96) C 97) A 98) D 99) B 100) D 101) D 102) D 103) D 104) D 105) A 106) C 107) C 108) A 109) A 110) C 111) C 112) D 113) C 114) D 115) C 116) D 117) C 118) C 119) C 120) D 121) C 122) A 123) D 124) D 125) A 126) C 111
Answer Key Testname: UNTITLED2
127) C 128) C 129) B 130) C 131) D 132) A 133) D 134) C 135) C 136) D 137) B 138) A 139) D 140) D 141) B 142) C 143) C 144) D 145) C 146) D 147) A 148) A 149) B 150) C 151) D 152) C 153) B 154) B 155) A 156) B 157) A 158) C 159) A 160) D 161) B 162) B 163) C 164) A 165) D 166) D 167) D 168) B 112
Answer Key Testname: UNTITLED2
169) A 170) D 171) C 172) C 173) C 174) D 175) D 176) C 177) A 178) B 179) D 180) C 181) A 182) A 183) A 184) A 185) B 186) B 187) C 188) B 189) A 190) B 191) B 192) D 193) B 194) C 195) C 196) D 197) B 198) D 199) A 200) D 201) B 202) C 203) D 204) A 205) D 206) B 207) D 208) D 209) A 210) A 113
Answer Key Testname: UNTITLED2
211) A 212) C 213) C 214) A 215) C 216) D 217) A 218) D 219) D 220) D 221) D 222) A 223) D 224) A 225) A 226) D 227) A 228) B 229) B 230) B 231) B 232) D 233) B 234) D 235) A 236) A 237) B 238) A 239) A 240) A 241) A 242) C 243) B 244) B 245) B 246) D 247) A 248) B 249) D 250) B 251) A 252) A 114
Answer Key Testname: UNTITLED2
253) C 254) C 255) B 256) D 257) D 258) D 259) D 260) A 261) A 262) D 263) D 264) A 265) C 266) D 267) C 268) B 269) B 270) D 271) D 272) D 273) B 274) D 275) C 276) A 277) A 278) B 279) A 280) A 281) A 282) B 283) A 284) C 285) C 286) C 287) D 288) A 289) C 290) A 291) B 292) C 293) A 294) D 115
Answer Key Testname: UNTITLED2
295) A 296) C 297) A 298) C 299) C 300) B 301) D 302) A 303) C 304) B 305) B 306) D 307) B 308) D 309) C 310) A 311) C 312) C 313) A 314) D 315) D 316) D 317) C 318) A 319) B 320) D 321) A 322) B 323) B 324) D 325) D 326) C 327) D 328) C 329) x-intercept is -a ; y-intercept is b 330) The classmate's claim is wrong. The horizontal asymptote tells us what the behavior of f(x) will be as x approaches the extremes of its domain, but puts no restrictions on the function in between the extremes.
331) This would be a function because at any given time there is only one possible population. Despite the fact that the
population can reach the same level several times this is still a function, but for each point in time, there can be no more than one population. 332) The domain is all real numbers and the range is the set of all real numbers. In the context of exam grades, the domain and range both become the set of nonegative real numbers. In this context, times and grades less than zero do not make sense.
116
Answer Key Testname: UNTITLED2
333) A 334)
domain: (- , ), range: (- , 0)
335)
domain: (- , ), range: (0, )
336) The graph is shifted 2 units to the left and 1 units up. 337) The graph is reflected over the y-axis and then shifted 1 units down.
117
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Decide whether the limit exists. If it exists, find its value. 1) lim f(x) x 1+
A) Does not exist
1)
B) 3 1
C) 3
2
D) 4
2) lim f(x)
2)
x
A) 0
C) Does not exist
B) -2
1
D)
3)
lim f(x) and lim f(x) x (-1)x (-1)+
A) -2, -7
3)
B) -7, -2
C) -7, -5
D) -5, -2
4) lim f(x) and lim f(x) x 0-
A) 3, 1
4)
x 0+
B) 3, -1
C) -1, 3
2
D) -3, -1
5) lim f(x)
5)
x 1
A) 0
B) 2
C) Does not exist
D) 1
6) lim f(x)
6)
x 0
A) Does not exist
C) 0
B) -1
3
D) 1
7) lim f(x)
7)
x 0
A) -1
B) 0
C) -2
D) Does not exist
8) lim f(x)
8)
x 1
A) Does not exist
B) 0
C) -1
4
D) 1
9) lim f(x)
9)
x -1
A) -1 10)
B) 0
C) Does not exist
D) -2
lim f(x) x -1/2
A) -1
10)
B) Does not exist
C) 0
5
D) -2
Complete the table and use the result to find the indicated limit. 11) If f(x) = x2 + 8x - 2, find lim f(x). x 2 x f(x)
1.9
1.99
1.999
2.001
11) 2.01
2.1
A) x 1.9 1.99 1.999 2.001 2.01 2.1 ; limit = 17.70 f(x) 16.692 17.592 17.689 17.710 17.808 18.789
B) x 1.9 1.99 1.999 2.001 2.01 2.1 ; limit = 5.40 f(x) 5.043 5.364 5.396 5.404 5.436 5.763
C) x 1.9 1.99 1.999 2.001 2.01 2.1 ; limit = 18.0 f(x) 16.810 17.880 17.988 18.012 18.120 19.210
D) x 1.9 1.99 1.999 2.001 2.01 2.1 ; limit = f(x) 5.043 5.364 5.396 5.404 5.436 5.763 4
12) If f(x) = x - 1 , find lim f(x). x-1
x f(x)
0.9
x
12)
1
0.99
0.999
1.001
1.01
A) x 0.9 0.99 0.999 1.001 1.01 1.1 ; limit = f(x) 1.032 1.182 1.198 1.201 1.218 1.392
B) x 0.9 0.99 0.999 1.001 1.01 1.1 ; limit = 4.0 f(x) 3.439 3.940 3.994 4.006 4.060 4.641
C) x 0.9 0.99 0.999 1.001 1.01 1.1 ; limit = 1.210 f(x) 1.032 1.182 1.198 1.201 1.218 1.392
D) x 0.9 0.99 0.999 1.001 1.01 1.1 ; limit = 5.10 f(x) 4.595 5.046 5.095 5.105 5.154 5.677
6
1.1
3
13) If f(x) = x - 6x + 8 , find lim f(x). x-2
x f(x)
x
-0.1
13)
0
-0.01
-0.001
0.001
0.01
0.1
A) x -0.1 0.001 0.01 0.1 -0.01 -0.001 ; limit = -2.10 f(x) -2.18529 -2.10895 -2.10090 -2.99910 -2.09096 -2.00574
B) x -0.1 0.001 0.01 0.1 -0.01 -0.001 ; limit = f(x) -1.22843 -1.20298 -1.20030 -1.19970 -1.19699 -1.16858
C) x -0.1 0.001 0.01 0.1 -0.01 -0.001 ; limit = -4.0 f(x) -4.09476 -4.00995 -4.00100 -3.99900 -3.98995 -3.89526
D) x -0.1 0.001 0.01 0.1 -0.01 -0.001 ; limit = -1.20 f(x) -1.22843 -1.20298 -1.20030 -1.19970 -1.19699 -1.16858
14) If f(x) = x - 4 , find lim f(x). x-2
x f(x)
3.9
x
3.99
14)
4
3.999
4.001
4.01
4.1
A) x 3.9 3.99 3.999 f(x) 5.07736 5.09775 5.09978
4.001 4.01 4.1 ; limit = 5.10 5.10022 5.10225 5.12236
x 3.9 3.99 3.999 f(x) 1.19245 1.19925 1.19993
4.001 4.01 4.1 ; limit = 1.20 1.20007 1.20075 1.20745
x 3.9 3.99 3.999 f(x) 3.97484 3.99750 3.99975
4.001 4.01 4.1 ; limit = 4.0 4.00025 4.00250 4.02485
x 3.9 3.99 3.999 f(x) 1.19245 1.19925 1.19993
4.001 4.01 4.1 ; limit = 1.20007 1.20075 1.20745
B) C) D)
7
15) If f(x) = x2 - 5, find lim f(x). x
x f(x)
15)
0
-0.001
0.001
0.01
0.1
-0.1
-0.01
x -0.1 f(x) -4.9900
-0.01 -4.9999
-0.001 -5.0000
0.001 0.01 0.1 ; limit = -5.0 -5.0000 -4.9999 -4.9900
x -0.1 f(x) -1.4970
-0.01 -1.4999
-0.001 -1.5000
0.001 0.01 0.1 ; limit = -15.0 -1.5000 -1.4999 -1.4970
x -0.1 f(x) -1.4970
-0.01 -1.4999
-0.001 -1.5000
0.001 0.01 0.1 ; limit = -1.5000 -1.4999 -1.4970
x -0.1 f(x) -2.9910
-0.01 -2.9999
-0.001 -3.0000
0.001 0.01 0.1 ; limit = -3.0 -3.0000 -2.9999 -2.9910
A) B) C) D)
16) If f(x) = x f(x)
x+1 , find lim f(x). x+1 x 1 0.9
0.99
16) 0.999
1.001
1.01
1.1
A) x 0.9 f(x) 0.21764
0.99 0.999 1.001 1.01 1.1 ; limit = 0.21266 0.21219 0.21208 0.21160 0.20702
x 0.9 f(x) 2.15293
0.99 0.999 1.001 1.01 1.1 ; limit = 2.13640 2.13799 2.13656 2.13624 2.13481 2.12106
x 0.9 f(x) 0.21764
0.99 0.999 1.001 1.01 1.1 ; limit = 0.21213 0.21266 0.21219 0.21208 0.21160 0.20702
x 0.9 f(x) 0.72548
0.99 0.999 1.001 1.01 1.1 ; limit = 0.7071 0.70888 0.70728 0.70693 0.70535 0.69007
B) C) D)
8
17) If f(x) = x - 2, find lim f(x). x
x f(x)
3.9
17)
4
3.99
3.999
4.001
4.01
4.1
A) x 3.9 f(x) 3.9000
3.99 2.9000
3.999 1.9000
4.001 4.01 4.1 ; limit = 2.0000 3.0000 4.0000
B) x 3.9 3.99 3.999 4.001 4.01 4.1 ; limit = 1.50 f(x) 1.47736 1.49775 1.49977 1.50022 1.50225 1.52236
C) x 3.9 f(x) 3.9000
3.99 2.9000
3.999 1.9000
4.001 4.01 4.1 ; limit = 1.95 2.0000 3.0000 4.0000
D) x 3.9 3.99 3.999 4.001 4.01 4.1 ; limit = 0 f(x) -0.02516 -0.00250 -0.00025 0.00025 0.00250 0.02485
Give an appropriate answer. 18) Let lim f(x) = -9 and lim g(x) = -2. Find lim [f(x) - g(x)]. x 2 x 2 x 2
A) -7
B) -9
C) -11
18) D) 2
19) Let lim f(x) = -5 and lim g(x) = 3. Find lim [f(x) · g(x)]. x
x
-10
x
-10
A) -10
19)
-10
B) -2
C) -15
D) 3
20) Let lim f(x) = -8 and lim g(x) = -10. Find lim f(x) . x
3
x
3
A) 4
x
C) 5
B) 3
5
20)
3 g(x)
4
D) 2
21) Let lim f(x) = 4. Find lim log2 f(x). x -1
A) 4
C) 1
B) -1
22) Let lim f(x) = 4. Find lim x
21)
x -1
6
x
6
A) 6
D) 2
f(x).
22)
B) 4
C) 2
D) 1.4142
23) Let lim f(x) = -3 and lim g(x) = -10. Find lim [f(x) + g(x)]2 . x
-1
A) 109
x
-1
x
23)
-1
B) 169
C) 7
9
D) -13
24) Let lim f(x) = 3. Find lim (-2)f(x). x
2
x
A) 4
B) -8
25) Let lim f(x) = 32. Find lim x
24)
2
6
x
5
6
A) 2
C) -2
D) 3
f(x).
25)
B) 6
C) 5
D) 32
26) Let lim f(x) = 4 and lim g(x) = 10. Find lim 10f(x) - 9g(x) . x 5
x 5
A) - 50
B) - 67
7
26)
-3 + g(x)
x 5
C) 130
3
7
D) 5
[f(x)]2 . -1 + g(x) 9
27) Let lim f(x) = -5 and lim g(x) = -8. Find lim x
9
x
A) - 25
9
x
C) 25
B) 9
9
81
27) D) 5 9
Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. 2 28) lim x - 16 x 4 x-4
A) 8
B) 4
C) Does not exist
D) 1
2 29) lim x + 13x + 30
29)
x + 10
x -10
A) 260
C) Does not exist
B) -7
D) 13
2
30) lim x + 4x - 12
30)
x-2
x 2
A) Does not exist
B) 8
C) 4
D) 0
2
31) lim x + 2x - 35 x 5
32) lim
31)
x2 - 25
A) Does not exist
B) - 1
C) 0
5
D) 6 5
x2 - 4
32)
x 2 x2 - 7x + 10
A) 0
28)
B) - 2
C) Does not exist
3
10
D) - 4 3
33) lim
1 1 x+2 2
33)
x
x 0
B) 1
A) 0
C) - 1
4
4
D) Does not exist
2 34) lim x - 64
34)
x -8 x + 8
A) -16
C) 0
B) -8
D) Does not exist
2 35) lim x + 13x + 42
35)
x+6
x -6
A) 13
B) 156
C) 1
D) Does not exist
x-5 x 25 x - 25
36) lim
36)
A) 1
C) 1
B) 5
10 3
37) lim (x + h) - x h 0
D) 0
5
3
37)
h
A) 3x2 + 3xh + h2
B) Does not exist
D) 3x2
C) 0
2
38) lim -3x + 6x - 6
38)
2x2 + 5
x
B) - 3
A)
39) lim x
x -
6
D) 0
5x 3 + 6x -2x4 + 7x3 + 10
39) B) 1
A)
40) lim
C) - 5
2
C) 0
D) - 5 2
x 2x - 10
A) - 1 2
40) B) 1
C) 0
2
11
D)
41) lim 2x + 1
41)
9x - 7
x
B) 2
A)
42) lim x
C) - 1
9
7
D) 0
3x + 1 11x 2 - 7
42)
A) 3
B) - 1
11
D) 0
C)
7
7 43) lim 5x - x + 3
43)
6x3 - x - 7
x
A)
C) 5
B)
6
D) Does not exist
2 5 44) lim 7x + 3x - 5x
44)
5x2 - 2x + 4
x
B) 7
A)
C) Does not exist
5
D)
Use the properties of limits to help decide whether each limit exits. If a limit exists, find its value. 45) Let f(x) = x2 + 1 if x < 0 . Find lim f(x). if x 0 -5 x -4
A) Does not exist
B) 1
C) 17
D) -5
46) Let f(x) = -3x + 6 if x 1 . Find lim f(x). -2x + 5 if x > 1
A) Does not exist 7x - 5
47) Let f(x) = 1
3x - 9
A) -6
46)
x 1
B) 5
C) 6
if x < 1 if x = 1 . Find lim f(x). x 1 if x > 1 B) Does not exist
D) 3
47) C) 0
D) 2
Use a graphing utility to find the limit, if it exists. 4 3 2 48) lim x - 5x + 7x - 6x + 3 x-1 x 1
49) lim (-3 + 2x x
A) 6.5
48)
B) 2
A) -3
45)
C) -2
D) Does not exist.
2/3 + 6x10/3)3
49)
x10
B) Does not exist.
C) 6 12
D) 216
2
50) lim x - 81
50)
x 9 x-9
A) Does not exist
B) 18
C) 1
D) 9
2
51) lim x + 5x - 6
51)
x2 - 1
x 1
A) 7
B) Does not exist
2
C) 0
D) - 5 2
x2 - 1 x 1 x2 - 5x + 4
52) lim
B) - 1
A) 0
53) lim t
52) C) Does not exist
3
D) - 2 3
4t2 - 8 t-2
A) 8
53) B) 4
C) Does not exist
D) 2
4
54) lim x + 4x - 5 x 1
54)
x2 - 1
A) 1 55) lim x
B) 4
x
D)
25x2 + 1 3x
A) 0 56) lim
C) Does not exist
55) C) 1.667
B) -1.667
D) Does not exist
9x2 + 5x + 7 2x
A) Does not exist
56) B) 1.5
C)
D) -1.5
Solve the problem. 57) A company training program determines that, on average, a new employee can do P(s) pieces of 86 + 47s lim work per day after s days of on-the-job training, where P(s) = . Find P(s). s+6 s 1
A) 19
B) Does not exist
C) 133
D) 22
58) The cost of manufacturing a particular videotape is c(x) = 10,000 + 12x, where x is the number of tapes produced. The average cost per tape, denoted by c (x), is found by dividing c(x) by x. Find lim c(x). x 2,000
A) 10
B) Does not exist
C) 17 13
57)
D) 24
58)
59) Given is a graph of a portion of the postage function, which depicts the cost (in cents) of mailing a
59)
letter, p, versus the weight (in ounces) of the letter, x. Find each limit, if it exists: lim p(x), lim p(x), lim p(x) x 3 x 3x 3+
A) 99; 77; does not exist C) 77; 99; does not exist
B) 77; 77; 77 D) 77; 99; 77
60) Suppose that the cost, p, of shipping a 3-pound parcel depends on the distance shipped, x, according to the function p(x) depicted in the graph. Find each limit, if it exists: lim p(x), lim p(x), lim p(x) x 100 x 500 x 1500
A) 5; does not exist; 15 C) 5; 10; 15
B) 5; 5; 15 D) 5; does not exist; does not exist
14
60)
61) Suppose the the cost, C, of producing x units of a product can be illustrated by the given graph.
61)
Find each limit, if it exists:
lim p(x), lim p(x), lim p(x) x 100 x 100x 100+
A) 200; does not exist; does not exist C) 200; 200; 200
B) 200; 300; does not exist D) 200; 300; 200
62) Suppose that the unit price, p, for x units of a product can be illustrated by the given graph. Find
62)
each limit, if it exists:
lim p(x), lim p(x), lim p(x), lim p(x) x 50 x 75 x 50x 50+
A) 8; 8; does not exist; 8 C) 8; 8; 8; 8
B) 10; 8; does not exist; 8 D) 10; 8; 8; 8
63) The blood alcohol level h hours after consumption of 2 ounces of pure ethanol is given by C(h) =
0.55h . Find the blood alcohol level as h approaches infinity. 3 h - h2 + 5
A) .55
B) .11
C) 0
15
D)
63)
64) The current value of an annuity per period is given by P = R i
R
i(1 + i)n
, where n is the number of
64)
periods, i is the interest rate, and R is the amount of the periodic payment. Find the limit of the current value equation as n approaches infinity to derive an expression for the current value for an annuity that makes payments in perpetuity. A) P = 0 B) P = R C) P = D) P = R - R i i
Find all points where the function is discontinuous.
65)
65)
A) x = 4, x = 2
B) x = 4
C) x = 2
D) None
66)
66)
A) x = -2
B) x = -2, x = 1
C) None
D) x = 1
67)
67)
A) x = 0, x = 2 C) x = 2
B) x = -2, x = 0, x = 2 D) x = -2, x = 0
68)
68)
A) x = -2, x = 6
B) None
C) x = -2
16
D) x = 6
69)
69)
A) x = 1, x = 5 C) x = 1, x = 4, x = 5
B) None D) x = 4
70)
70)
A) x = 0
B) x = 1
C) None
D) x = 0, x = 1
71)
71)
A) x = 0, x = 3
B) None
C) x = 3
D) x = 0
72)
72)
A) x = 2
B) x = -2, x = 2
C) x = -2
73)
D) None 73)
A) None C) x = 0
B) x = -2, x = 2 D) x = -2, x = 0, x = 2
17
Find all values x = a where the function is discontinuous. -5x 74) f(x) = (2x - 3)(7 - 2x)
A) a = 3 , - 7 2
74)
B) a = 0, 3 , 7
2
C) a = 3 , 7
2 2
2 2
D) Nowhere
2 75) f(x) = x - 9
75)
x-2
A) a = 3, 2
B) a = -3, 3, 2
C) a = 2
D) Nowhere
2
76) f(x) = x - 49
76)
x+7
A) a = -49 0
B) a = 6
C) a = 7
D) a = -7
if x< 0
77) g(x) = x2 - 5x if 0 x 5 5 A) a = 0
77)
if x > 5
B) a = 0, 5
78) q(x) = x2 + 6x - 13 A) a = 13
C) Nowhere
D) a = 5 78)
B) Nowhere
79) k(x) = e x + 7 A) a > 7
C) a = 6
D) a = 0 79)
B) a > -7
C) Nowhere
D) a < -7
80) f(x) = ln x - 7
80)
x+3
A) a = -3 6
81) f(x) = x + 5 14 A) a = 3
B) a = 7, -3
D) Nowhere
if x < 3 if 3 x 9 if x > 9
9
82) f(x) = x2 - 16 9 A) a = 16
83) f(x) = 6x - 7
C) a = -7, 3
x 2 + 2x - 7 A) a = -7
81) B) a = 9
C) a = -3
D) Nowhere
if x < 5
82)
if 5 x 9 if x > 9
B) Nowhere
C) a = 9
D) a = 5
if x < 0
83)
if x 0
B) a = 2
C) a = 0
18
D) Nowhere
Give an appropriate response. 84) Find the limit of f(x) as x approaches 2 from the right. 1 if x < 2 f(x) = x + 2 if 2 x 4 6 if x > 4 A) 1 B) 4
C) 6
84)
D) The limit does not exist.
85) Find the limit of f(x) as x approaches 3 from the left. -2 if x < 3 f(x) = x + 2 if 3 x 5 7 if x > 5 A) 7
85)
B) 5 D) The limit does not exist.
C) -2
Find the value of the constant k that makes the function continuous. if x 4 86) h(x) = x2 x + k if x > 4 A) k = 20 B) k = -4 C) k = 4
87) g(x) = x2 - 8 2kx
if x < 5 if x 5
x3
4x + k
10
D) k = 5 2
if x < -2
88)
if x -2
5x2 + 16x - 16 x+4
A) k = -8
C) k = 17
B) k = 13
A) k = -8
89) h(x) =
D) k = 12 87)
A) k = 17 2 88) f(x) = x + x + k
86)
B) k = -6
C) k = 2
D) k = -10
if x -4
89)
if x = -4 B) k = 52
C) k = 0
Use a graphing utility to find the discontinuities of the given rational function. 2 90) f(x) = x + 2x + 1 x3 + x2 + 9x - 11
A) 2 B) -1 C) 1 D) The function is continuous for all values of x.
19
D) k = 4
90)
91) f(x) =
x+1 3 x + 2x2 + 6x - 9
91)
A) 1 B) 3 C) -1 D) The function is continuous for all values of x. Solve the problem. 92) The graph below shows the amount of income tax that a single person must pay on his or her income when claiming the standard deduction. Identify the income levels where discontinuities occur and explain the meaning of the discontinuities. Income Tax, 1000's of dollars
Income, 1000's of dollars
A) Discontinuities at x = $44,000 and x = $60,000. Discontinuities represent tax shelters. B) Discontinuities at x = $22,000, x = $44,000, and x = $60,000. Discontinuities represent tax
cheating on the part of high-income earners. C) Discontinuities at x = $44,000 and x = $60,000. Discontinuities represent boundaries between tax brackets. D) Discontinuities at x = $22,000, x = $44,000, and x = $60,000. Discontinuities represent boundaries between tax brackets.
20
92)
93) In order to boost business, a ski resort in Vermont is offering rooms for $125 per night with every fourth night free. Let C(x) represent the total cost of renting a room for x days. Sketch a graph of 1 C(x) on the interval (0, 6] and determine the cost for staying 4 days. 2
A) C 4 1 = $625
B) C 4 1 = $500
C) C 4 1 = $500
D) C 4 1 = $375
2
2
2
2
21
93)
94) Suppose that the cost, p, of shipping a 3-pound parcel depends on the distance shipped, x,
94)
according to the function p(x) depicted in the graph. Is p continuous at x = 50? at x = 500? at x = 1500? at x = 3000?
A) Yes; no; yes; no C) Yes; yes; yes; no
B) No; no; yes; no D) Yes; no; no; no
95) Suppose that the cost, C, of producing x units of a product can be illustrated by the given graph. Is C(x) continuous at x = 50? x = 100? x = 150?
A) No; no; no
B) Yes; yes; yes
C) Yes; no; no
22
D) Yes; no; yes
95)
96) Suppose that the unit price, p, for x units of a product can be illustrated by the given graph. Is
96)
p(x) continuous at x = 50? x = 100? x = 150?
A) No; yes; yes
B) No; no; no
C) No; yes; no
D) Yes; no; yes
97) Consider the learning curve defined in the graph. Depicted is the accuracy, p, expressed as a
97)
percentage, in performing a series of short tasks versus the accumulated amount of time spent practicing the tasks, t. Is p(t) continuous at t = 25? at t = 40? at t = 45?
A) Yes; no; no
B) No; no; no
C) Yes; yes; yes
D) Yes; no; yes
Find the average rate of change for the function over the given interval. 98) y = x2 + 1x between x = 6 and x = 8
A) 15 4
B) 9
99) y = 9x3 + 8x2 + 5 between x = 4 and x = 6 A) 764
C) 15
D) 36
C) 2237 2
D) 2237 6
C) 7
D) 1 3
99)
B) 764 3
100) y = 2x between x = 2 and x = 8 A) - 3 10
98)
100)
B) 2
23
101) y = 3
between x = 4 and x = 7
A) 7
B) - 3
x-2
101) C) 1
10
3
D) 2
102) y = 4x2 between x = 0 to x = 7
102)
4
A) 1
B) - 3
3
C) 7
10
D) 2
103) y = -3x2 - x between x = 5 and x = 6 A) - 1 6
103)
B) 1 2
C) -2
104) y = x3 + x2 - 8x - 7 between x = 0 and x = 2
104)
B) - 1
A) -28
6
C) 1
D) -2
C) -28
D) 1 2
2
105) y = 2x - 1 between x = 1 and x = 5 A) - 1 6
106) y = 3
x+2
D) -34
105)
B) -2
between x = 1 and x = 4
A) - 1 6
106)
B) 1 2
C) -28
D) -2
C) 5
D) - 1 6
107) y = 5x + 7 between x = -1 and x = 0 A) 1 2
107)
B) -28
Suppose the position of an object moving in a straight line is given by the specified function. Find the instantaneous velocity at time t. 108) s(t) = t2 + 5t + 4, t = 3 108)
A) 11 109) s(t) = t2 + 6t + 5, t = 1 A) 8 110) s(t) = 4t2 - 9t - 3, t = 4 A) 23
B) 28
C) 21
D) 10
B) 12
C) 7
D) 13
109)
110) B) 7
C) 25
24
D) 20
111) s(t) = t3 + 6t + 8, t = 4 A) 14
B) 62
C) 54
D) 18
112) s(t) = t3 + 5t + 3, t = 1 A) 8
B) 7
C) 11
D) 9
111)
112)
Find the instantaneous rate of change for the function at the given value. 113) F(x) = x2 + 9x at x = -4
113)
A) -20
B) 5
C) 1
D) -8
114) f(x) = 5x + 9 at x = 2 A) 9
B) 5
C) 10
D) 0
115) s(t) = t2 + 5t at t = 4 A) 13 116) g(t) = 5t2 + t at t = -4 A) -39
114)
115) B) 3
C) 21
116) B) -14
C) -41
117) F(x) = 2x2 + x - 3 at x = 4 A) 17 B) 5
D) 6 117)
C) 15
118) g(x) = x2 + 11x - 15 at x = 1 A) -9 B) 13
D) 19 118)
C) 26
119) s(t) = 3t2 + 5t - 7 at t = -2 A) -1 B) -7 120) g(t) = 3t2 + 6 at t = 4 A) 24
D) 9
D) 11 119)
C) -17
D) 1 120)
C) 8
B) -24
D) 12
Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for the function at the given value. 121) Use a graphing utility to approximate the instantaneous rate of change of f(x) = x1/x at x = 2. 121)
A) 0.2170
B) 0.1085
C) 0.3844
D) 0.3333
122) Use a graphing utility to approximate the instantaneous rate of change of f(x) = x-ln x at x = 6. A) -0.0057 B) -0.0183 C) -0.0241 D) 0.0403
25
122)
Solve the problem. 123) The graph shows the average cost of a barrel of crude oil for the years 1981 to 1990 in constant 1996 dollars. Find the approximate average change in price from 1981 to 1990.
123)
1996 $/Barrel
Year
A) About -$44/year C) About -$1/year
B) About -$4/year D) About -$24/year
124) The graph shows the total sales in thousands of dollars from the distribution of x thousand
124)
catalogs. Find the average rate of change of sales with respect to the number of catalogs distributed from 10 to 50.
Sales (in thousands)
Number (in thousands)
A) 1
B) 3
C) 2
4
D) 1
125) Suppose that the total profit in hundreds of dollars from selling x items is given by P(x) = -x2 + 12x - 32. Find the marginal profit at x = 6.
A) $400 per item C) $0 per item
B) -$1,200 per item D) $200 per item
26
4
125)
126) The total cost to produce x handcrafted wagons is C(x) = 130 + 3x - x2 + 5x3. Find the rate of
126)
change of cost with respect to the number of wagons produced (the marginal cost) when x = 5. A) $498 per wagon B) $615 per wagon
C) $745 per wagon
D) $368 per wagon 2
127) Suppose that the revenue from selling x radios is R(x) = 85x - x dollars. Use the function R (x) to 10
127)
estimate the increase in revenue that will result from increasing production from 115 radios to 116 radios per week. A) $61.80 B) $62.00 C) $108.00 D) $73.50
128) Suppose that the dollar cost of producing x radios is C(x) = 600 + 30x - 0.2x2 . Find the marginal cost when 50 radios are produced. A) -$1,600 B) $10
C) $1,600
D) $50
129) Suppose that the dollar cost of producing x radios is c(x) = 600 + 30x - 0.2x2 . Find the average cost per radio of producing the first 30 radios. A) $1,320.00 B) $720.00
C) $1,270.00
128)
129)
D) $44.00
130) A particular strain of influenza is known to spread according to the function p(t) = 1 (t2 + t), 4
130)
where t is the number of days after the first appearance of the strain and p(t) is the percentage of the population that is infected. Find the instantaneous rate of change of p with respect to t at t = 4. A) 5% per day B) 2% per day C) 9 % per day D) 5 % per day 4 2
131) The graph below shows the number of tuberculosis deaths in the United States from 1989 to 1998.
Deaths
Year Estimate the average rate of change in tuberculosis deaths from 1991 to 1997. A) About -60 deaths per year B) About -460 deaths per year
C) About -1 deaths per year
D) About -120 deaths per year
27
131)
132) The graph shows the population in millions of bacteria t minutes after a bactericide is introduced
132)
into a culture. Find the average rate of change of population with respect to time for the time from 1 to 4 minutes.
Population (in millions)
Time (in minutes)
A) 1 4
B) 4
C) 3
D) 1 3
133) The number of gallons of water in a swimming pool t minutes after the pool has started to drain is Q(t) = 50(20 - x)2 . How fast is the water running out at the end of 10 minutes? A) 1,000 gal/min B) 500 gal/min C) 2,500 gal/min
D) 5,000 gal/min
134) The size of a population of mice after t months is P = 100(1 + 0.2t + 0.02t2 ). Find the growth rate at t = 10 months. A) 60 mice/month
133)
134)
B) 120 mice/month D) 30 mice/month
C) 160 mice/month
135) A ball is thrown vertically upward from the ground at a velocity of 109 feet per second. Its
distance from the ground after t seconds is given by s(t) = -16t2 + 109t. How fast is the ball moving 6 seconds after being thrown? A) -101 ft per sec B) -83 ft per sec C) 13 ft per sec D) 78 ft per sec
28
135)
Estimate the slope of the tangent line to the curve at the given point.
136)
136)
A) 1
B) 1
2
C) 2
D) -1
137)
137)
A) - 1 2
C) - 1
B) -4
4
29
D) 1 4
138)
138)
A) - 1 2
B) - 3
C) 1
2
D) -1
139)
139)
A) 3
B) 1
C) 3
2
2
30
D) 2 3
140)
140)
A) -4
B) 1
C) - 1
4
4
D) - 1 2
141)
141)
A) undefined
B) 3
C) 1
D) 0
C) -27
D) 6
Find f'(x) at the given value of x. 142) f(x) = -3x2 + 12x; Find f (3).
A) -6
142)
B) 18
143) f(x) = -11 ; Find f (-4).
143)
x
A) 11 4
144) f(x) = x ; Find f (81). A) 9
B) 11
C) 16
B) 1 9
C) 1 18
16
11
D) 4
11
144)
31
D) 81
145) f(x) = 2 ; Find f (-1).
145)
x+4
A) 1
B) 2
9
C) - 1
9
D) - 2
9
9
146) f(x) = x + 1; Find f (3).
146) B) 5 3
A) 1 4
C) 5
D)
4
4
3 4
147) f(x) = 32 ; Find f (1).
147)
x
A) 32
B) 64
C) - 32
148) f(x) = x2 - 6x - 5; Find f (-5). A) 50 B) -21 149) f(x) = x3 + 7; Find f (4). A) 55
D) -32 148)
C) -16
D) -10 149)
B) 49
C) -48
D) 48
150) f(x) = -8x2 + 4x + 6; Find f (3). A) -38 B) -36
C) 52
D) -44
150)
151) f(x) = 5 x; Find f (6). A) Does not exist
151) B) 5 6 2
C) 60
D)
5 2 6
Find the equation of the secant line through the points where x has the given values. 152) f(x) = x2 + 4x; x = 5, x = 2
A) y = 10x - 11
B) y = 11x + 10
C) y = 11x - 10
153) f(x) = 3 - x2 ; x = -2, x = -3 A) y = 5x + 9 B) y = 5
152) D) y = 11x 153)
C) y = - 5x - 9
D) y = 5x - 9
154) f(x) = 4 ; x = 4, x = 2
154)
x
A) y = - 4
x2
155) f(x) = 2 x; x = 4, x = 1 A) y = 2 x + 4 3 3
B) y = - 1 x + 3
C) y = - 1 x
2
2
D) y = 1 x - 3 2
155) B) y = - 2 x+ 4 3 3
C) y =
32
2 2 x
D) y = 2 x - 4 3 3
Find the equation of the tangent line to the curve when x has the given value. 156) f(x) = -3 - x2; x = 8
A) y = 8x + 61
B) y = -16x + 61
156)
C) y = 16x - 61
D) y = -2x
157) f(x) = 4 ; x = 4
157)
x+2
A) y = - 1 x + 2 9
9
B) y = - 1 x + 10 9
C) y = 1 x + 2
9
9
D) y = - 2 x + 10
9
9
9
2
158) f(x) = x ; x = -5
158)
4
A) y = -2.5x - 12.5 C) y = -2.5x - 6.25
B) y = -2.5x + 6.25 D) y = -10x - 6.25
3
159) f(x) = x ; x = 5
159)
4
A) y = 25 x - 125 4
2
B) y = 25 x + 125 4
C) y = 125 x + 75
2
2
4
D) y = 75 x - 125 4
2
3
160) f(x) = x ; x = -3
160)
2
A) y = 27x + 27 2
B) y = 27 x + 27
C) y = 27x + 9
2
2
D) y = 9 x + 27 2
161) f(x) = 40 ; x = 4
161)
x
A) y = - 5 x + 20 2
B) y = - 5 x
C) y = - 5x + 30
2
D) y = - 5 x + 10 2
162) f(x) = 18 ; x = 9
162)
x
A) y = - 4 x + 6 9
B) y = - 2 x
C) y = - 2 x + 2
9
9
D) y = - 2 x + 4 9
163) f(x) = x2 - 2 ; x = 2 A) y = 4x - 10
B) y = 4x - 6
C) y = 4x - 12
D) y = 2x - 6
164) f(x) = x2 + 3 ; x = -3 A) y = -6x - 15
B) y = -6x - 6
C) y = -3x - 6
D) y = -6x - 12
165) f(x) = x2 - x ; x = -3 A) y = -7x - 6
B) y = -7x + 9
C) y = -7x + 6
D) y = -7x - 9
163)
164)
165)
33
Use a graphing calculator to find f'(x) when x has the given value. 166) f(x) = -8x2 + 6x; x = 19
A) -310 167) f(x) = 7 x; x = 64 A) - 7 4
B) -298
166)
C) -314
D) -146 167)
B) 7
C) 7
16
64
D) Undefined
168) f(x) = 2 ; x = 3
168)
x
A) -2 9
169) f(x) = x + 1; x = 3 A) 3 4
170) f(x) = 3x2 - 7x; x = 8 A) 55 171) f(x) = 6ex; x = 5 A) 890.479 172) f(x) = 4 ln|x|; x = 9 A) 2.25
B) -4
C) 2
D) -9
B) - 1 4
C) 1 4
D) - 3 4
9
9
169)
170) B) 41
C) 48x - 7
D) 288 171)
B) 81.5485
C) 30
D) 154.4132 172)
B) 1.7778
C) 0.4444
D) 0
173) f(x) = - 4 ; x = 13
173)
x
A) 0.0237
C) 676
B) -0.3077
D) 42.25
Find the x-values where the function does not have a derivative.
174)
174)
A) x = 0
B) x = -1
C) x = 1
34
D) x = 2
175)
175)
A) x = -3, x = 3 C) x = -3, x = 0, x = 3
B) x = -2, x = 0, x = 2 D) x = -2, x = 2
176)
176)
A) x = -2, x = 0, x = 2 C) x = 2
B) x = 0 D) x = -2, x = 2
177)
177)
A) x = 0, x = 1, x = 2 C) x = 2
B) x = 1 D) x = 0
178)
178)
A) x = 1, x = 3 C) x = 1, x = 2, x = 3
B) x = 2 D) Exists at all points
179)
179)
A) x = 5 C) x = 2, x = 5
B) x = 2 D) Exists at all points 35
180)
180)
A) x = 0 C) x = -1, x = 1
B) x = -1, x = 0, x = 1 D) Exists at all points
181)
181)
A) x = 0 C) x = -2, x = 2
B) x = -2, x = 0, x = 2 D) Exists at all points
182)
182)
A) x = 3 C) x = 0, x = 3
B) x = 0 D) Exists at all points
Use a graphing calculator to find f'(x) when x has the given value. 183) f(x) = xx/3; x = 2
A) 1.6675 184) f(x) = x7/x; x = 2 A) -0.8973
183)
B) -0.3555
C) 0.8959
D) 2.0986
B) -2.4868
C) 6.0754
D) 2.5310
184)
Solve the problem.
185) Suppose the demand for a certain item is given by D(p) = -3p2 + 2p + 5, where p represents the price of the item. Find D'(p), the rate of change of demand with respect to price.
A) D'(p) = -3p + 2 C) D'(p) = -3p2 + 2
B) D'(p) = -6p + 2 D) D'(p) = -6p2 + 2
36
185)
186) Suppose the demand for a certain item is given by D(p) = -4p2 + 4p + 8, where p represents the price of the item. Find D'(10). A) -68 B) -76
C) -40
186)
D) -32
187) The profit from the expenditure of x thousand dollars on advertising is given by
187)
P(x) = 930 + 15x - 4x2 . Find the marginal profit when the expenditure is x = 7. A) -41 thousand dollars B) 105 thousand dollars
C) 49 thousand dollars
D) 930 thousand dollars
188) The revenue generated by the sale of x bicycles is given by R(x) = 70.00x - x2 /200. Find the marginal revenue when x = 900 units. A) $12.86 B) $79.00
C) $61.00
188)
D) $70.00
189) The graph shows the amount of potential energy V(x) (in arbitrary energy units) stored in a large
189)
rubber band that is stretched a distance of x inches beyond its relaxed length.
The magnitude of the force required to hold the rubber band at the position x = a is the derivative of the potential energy with respect to x, evaluated at the point x = a. Estimate the force required to hold the band at a stretched position x = 7. (Hint: the force in this problem has units of "energy units per inch".) A) -0.6 energy units per inch B) 2.7 energy units per inch
C) 3.4 energy units per inch
D) 1.6 energy units per inch
190) The force F (in N) exerted by a cam on a lever is given by F = x4 - 10x3 + 45x2 - 62x + 22, where x
(1 x 5) is the distance (in cm) from the center of rotation of the cam to the edge of the cam in contact with the lever. Find the instantaneous rate of change of F with respect to x when x = 2 cm. A) 30 N/cm B) 14 N/cm C) 22 N/cm D) -4 N/cm
37
190)
191) One hundred dollars is deposited in a savings account at 6% interest compounded continuously.
191)
The function defined by f(x) shown in the figure gives the balance in the account after t years. At what rate (in dollars per year) is the balance growing after 15 years?
A) $29/year
B) $7/year
C) $14/year
D) $28/year
192) Refer to the figure, where f(t) is the interest rate (as a percent) on a 6-month certificate of deposit t years after January 1, 1985. The straight lines are tangent to the graph of y = f(t) at t = 2, t = 4, and t = 10. How fast was the interest rate changing on January 1, 1987?
A) 0%/year
B) 2%/year
C) 1%/year
38
D) -2%/year
192)
The graphs of a function f(x) and its derivative f'(x) are shown below. Decide which is the graph of f(x) and which is the graph of f'(x).
193)
193)
A) Neither graph could be the derivative of the other. B) f(x) is the solid line; f'(x) is the dashed line. C) f(x) is the dashed line; f'(x) is the solid line. D) Either graph could be the derivative of the other. 194)
194)
A) f(x) is the solid line; f'(x) is the dashed line. B) f(x) is the dashed line; f'(x) is the solid line. C) Neither graph could be the derivative of the other. D) Either graph could be the derivative of the other.
39
195)
195)
A) Neither graph could be the derivative of the other. B) Either graph could be the derivative of the other. C) f(x) is the dashed line; f'(x) is the solid line. D) f(x) is the solid line; f'(x) is the dashed line. 196)
196)
A) Either graph could be the derivative of the other. B) f(x) is the dashed line; f'(x) is the solid line. C) Neither graph could be the derivative of the other. D) f(x) is the solid line; f'(x) is the dashed line. Sketch the derivative of the graph.
40
197)
197)
A)
B)
C)
D)
198)
198)
41
A)
B)
C)
D)
199)
199)
A)
B)
42
C)
D)
200)
200)
A)
B)
C)
D)
43
201)
201)
A)
B)
C)
D)
44
202)
202)
A)
B)
C)
D)
45
203)
203)
A)
B)
C)
D)
46
204)
204)
A)
B)
C)
D)
47
205)
205)
A)
B)
C)
D)
48
206)
206)
A)
B)
C)
D)
Solve the problem. 207) The graph shows annual sales (in thousands of dollars) of a new video game at a particular store. Sketch a graph of the rate of change of sales as a function of time.
49
207)
A)
B)
C)
D)
208) The graph shows the yearly average interest rates for 30-year mortgages for years since 1998
(Year 0 corresponds to 1998). Sketch a graph of the rate of change of interest rates with respect to time.
50
208)
A)
B)
C)
51
D)
209) The graph shows the amount of potential energy V(x) (in arbitrary energy units) stored in a large rubber band that is stretched a distance of x inches beyond its normal length.
The magnitude of the force required to hold the rubber band at the position x = a is the derivative of the potential energy with respect to x, evaluated at the point x = a. Sketch a graph of the magnitude of the force versus x.
A)
B)
52
209)
C)
D)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response.
210) The graph of the function y = f(x) = x2/3 is shown below. The "V"-shaped graph comes to
210)
211) Is there any difference between the problems "find the derivative of f(x) at x = a" and
211)
212) Explain how the intermediate value theorem can be used to show that there is a zero for
212)
a sharp point at x = 0. Without doing any calculations, decide whether the function has a tangent at x = 0. Give reasons for your answer. [Hint: consider the signs of the tangent lines on either side of x = 0 and what implication this has in terms of the limit definition of the slope of a tangent line.].
"find the slope of the line tangent to f(x) at x = a"? Explain.
x3 - 1.2x2 - 4.2x + 5.7 between 5 and -4.
53
213) A colleague asserts that a calculation of the average rate of change of y with respect to x
213)
214) Does the curve y = x ever have a negative slope? If so, where? Give reasons for your
214)
215) Does the curve y = x3 + 4x - 10 have a tangent whose slope is -2? If so, find an equation
215)
216) Can a tangent line to a graph intersect the graph at more than one point? If not, why not.
216)
217) If functions f(x) and g(x) are continuous for 0 x 4, could f(x) possibly be discontinuous
217)
from x = a to x = b will always be only an approximation to the instantaneous rate of change at x = x. Do you agree with this assertion? Explain your answer using graphs or examples.
answer.
for the line and the point of tangency. If not, why not?
If so, give an example.
g(x)
at a point of [0, 4]? Provide an example.
54
Answer Key Testname: UNTITLED3
1) C 2) B 3) A 4) B 5) C 6) C 7) C 8) D 9) C 10) A 11) C 12) B 13) C 14) C 15) A 16) D 17) D 18) A 19) C 20) A 21) D 22) C 23) B 24) B 25) A 26) A 27) A 28) A 29) B 30) B 31) D 32) D 33) C 34) A 35) C 36) A 37) D 38) B 39) C 40) B 41) B 42) D 55
Answer Key Testname: UNTITLED3
43) B 44) A 45) C 46) D 47) B 48) A 49) D 50) B 51) A 52) D 53) D 54) B 55) B 56) D 57) A 58) C 59) C 60) A 61) B 62) B 63) C 64) B 65) B 66) D 67) B 68) D 69) B 70) C 71) C 72) B 73) C 74) C 75) C 76) D 77) D 78) B 79) D 80) B 81) A 82) C 83) D 84) B 56
Answer Key Testname: UNTITLED3
85) C 86) D 87) C 88) D 89) A 90) C 91) A 92) D 93) B 94) A 95) D 96) C 97) D 98) C 99) A 100) D 101) B 102) C 103) D 104) D 105) D 106) A 107) C 108) A 109) A 110) A 111) C 112) A 113) C 114) B 115) A 116) A 117) A 118) B 119) B 120) A 121) B 122) C 123) B 124) B 125) C 126) D 57
Answer Key Testname: UNTITLED3
127) B 128) B 129) D 130) C 131) D 132) D 133) A 134) A 135) B 136) A 137) C 138) D 139) C 140) C 141) D 142) A 143) B 144) C 145) D 146) A 147) C 148) C 149) D 150) D 151) D 152) C 153) A 154) B 155) A 156) B 157) B 158) C 159) D 160) B 161) A 162) D 163) B 164) B 165) D 166) B 167) B 168) A 58
Answer Key Testname: UNTITLED3
169) C 170) B 171) A 172) C 173) A 174) A 175) D 176) B 177) D 178) D 179) B 180) A 181) B 182) C 183) C 184) C 185) B 186) B 187) A 188) C 189) D 190) A 191) C 192) C 193) B 194) B 195) C 196) B 197) D 198) D 199) A 200) C 201) C 202) C 203) B 204) D 205) A 206) D 207) B 208) B 209) D 59
Answer Key Testname: UNTITLED3
210) The function does not have a tangent at x = 0. The tangents to the left of x = 0 all have negative slopes, whereas those to the right of x = 0 all have positive slopes. Thus, the limit of the slope as x approaches 0 from the left cannot equal the limit of the slope as x approaches 0 from the right and therefore, according to the definition of the slope of the tangent line, no tangent line exists. 211) There is no difference at all. The two quantities are defined exactly the same, namely: f(x) - f(a) slope of tangent at a = lim = f (x). x a x-a
212) Answers will vary. f(5) > 0 and f(-4) < 0. Since the function is a polynomial function it is continuous on the interval between 5 and -4. Thus the function must contain a point between 5 and -4 such that f(x) = 0.
213) No. If y = f(x) is a line, then the average and instantaneous rates of change are identical.
214) The curve y = x never has a negative slope. The derivative of the curve is y' = 1 , which is never negative. A 2 x
curve only has a negative slope where its derivative is negative. Since the derivative of y = x is never negative, the curve never has a negative slope. 215) The curve has no tangent whose slope is -2. The derivative of the curve, y' = 3x2 + 4, is always positive and thus
never equals -2.
216) Yes, a tangent line to a graph can intersect the graph at more than one point. For example, the graph y = x3 - 2x2 has a horizontal tangent at x = 0. It intersects the graph at both (0, 0) and (2, 0). 217) Yes, if f(x) = 1 and g(x) = x - 2, then h(x) = 1 is discontinuous at x = 2. x-2
60
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. 1) f(x) = 2x2 + 4x - 8, find f'(x)
A) 4x2 + 4
1) B) 4x + 4
D) 2x2 + 4
C) 2x + 4
2) f(x) = 3x4 + 3x3 + 9, find f'(x) A) 4x3 + 3x2 B) 12x3 + 9x2
C) 4x3 + 3x2 - 7
3) y = 17x-2 + 11x3 + 5x, find f'(x) A) -34x-1 + 33x2 + 5 C) -34x-3 + 33x2
B) -34x-3 + 33x2 + 5 D) -34x-1 + 33x2
2) D) 12x3 + 9x2 - 7 3)
4) f(x) = 20x1/2 - 1 x20, find f'(x)
4)
2
A) 10x1/2 - 10x10 C) 10x1/2 - 10x19
B) 10x-1/2 - 10x10 D) 10x-1/2 - 10x19
5) f(x) = 9x7/5 - 5x2 + 104 , find f'(x)
5)
A) 63 x2/5 - 10x
B) 63 x6/5 - 10x
C) 63 x6/5 - 10x + 4000
D) 63 x2/5 - 10x + 4000
5
5
5
5
6) g(x) = 8x5 + x4 - 5x2 + 6, find g'(-4) A) 10,280 B) 10,024
6) C) 9,984
D) -216
7) f(x) = 4 - 6 + 9 , find f'(x) x
x
7)
x4
A) 2 - 6 - 36
B) - 2 - 6 - 36
C) - 2 + 6 - 36
D) -2 x + 6 - 36
x1/2
x3/2
x2
x5
x2
x3/2
x5
x2
x2
x3
x3
8) f(x) = x + 9 , find f'(x)
8)
x
A) 1 2 x
9 2 x3/2
C) 1 + 9
B) x3/2 + 9 x
x
1
x3/2
D) 1 - 9 2 x
2x
Find the derivative of the given function. 9) y = (3x2 + 2x)2 A) 36x3 + 36x2 + 8x
9) B) 36x3 + 18x2 + 8x
C) 18x3 + 18x2 + 8x
D) 18x3 + 18x2 + 4x
10) y = (x2 + 3)3 A) 6x5 + 30x3 + 54x C) 6x5 + 36x3 + 54x
10) B) 3x5 + 36x3 + 54x D) 6x5 + 18x3 + 27x
Find the slope of the line tangent to the graph of the function at the given value of x. 11) y = x4 + 4x3 + 2x + 2; x = 3
A) 119
B) 220
12) y = 4x3/2 - 5x1/2; x = 16 A) 197 8
C) 218
A) 3 4
14) y = -x-5 + x-3; x = 1 A) 2 15) y = 9x5/2 - 7x3/2; x = 4 A) 96
D) 121 12)
B) 91
C) 187
D) 101
B) - 3 4
C) - 13 4
D) 13 4
4
13) y = -8x-1 + 5x-2; x = 2
11)
8
4
13)
14) B) -2
D) 8
C) -8
15) B) 6
C) 8
D) 159
16) y = 7 - x; x = 4
16)
x
A) - 11
B) 11
16
C) - 3
16
D) 3
16
16
Find an equation for the line tangent to given curve at the given value of x. 2 17) y = x ; x = -4 2
A) y = -8x - 8
B) y = -4x + 8
17)
C) y = -4x - 16
D) y = -4x - 8
3
18) y = x ; x = 3
18)
4
A) y = 9 x + 27 4
2
B) y = 27 x + 27 2
C) y = 9 x - 27
4
4
2
2
D) y = 27 x - 27 4
2
19) y = x2 - 3; x = -2 A) y = -2x - 7
B) y = -4x - 7
C) y = -4x - 11
D) y = -4x - 14
20) y = x2 + 4; x = -3 A) y = -6x - 5
B) y = -6x - 10
C) y = -3x - 5
D) y = -6x - 14
21) y = x2 - x; x = -4 A) y = -9x - 12
B) y = -9x + 12
C) y = -9x - 16
D) y = -9x + 16
22) y = x - x2; x = -4 A) y = -7x + 16 23) y = x3 - 4x - 3; x = 2 A) y = 5x - 19
19)
20)
21)
22) B) y = 9x + 16
C) y = -9x + 16
D) y = -7x - 16 23)
B) y = 8x - 19
C) y = 8x - 3
D) y = -3
Solve the following.
24) Find all points of the graph of f(x) = 3x2 + 9x whose tangent lines are parallel to the line y - 27x =
24)
0.
A) (3, 54)
B) (4, 84)
C) (6, 162)
D) (5, 120)
25) At what points on the graph of f(x) = 2x3 - 3x2 - 35x is the slope of the tangent line 1? A) (-2, 42), (1, -36) B) (2, -66), (42, 24) C) (-2, 42), (3, -78) D) (0, 0), (3, -78) Find all values of x (if any) where the tangent line to the graph of the function is horizontal. 26) y = x2 + 2x - 3
A) 1 27) y = 2 + 8x - x2 A) 4
C) 1 2
B) 0
27) B) 8
C) -8
D) -4
B) 0, 2
C) -2, 0, 2
D) 0
29) y = x3 - 12x + 2 A) 0
B) 2, -2
C) 0, 2
D) -2, 0, 2
B) - 40 , 40 , 8 3 3
C) 40 , -8 3
A) - 40 , 8 3
26)
D) -1
28) y = x3 - 3x2 + 1 A) 2
30) y = x3 + 8x2 - 320x + 40
25)
28)
29)
30)
3
D) 8
Give an appropriate answer. 31) If g (2) = -1 and h (2) = 2, find f (2) for f(x) = -5g(x) - 3h(x) + 4. A) 3 B) -1 C) 15
32) If g (4) = -4 and h (4) = -6, find f (4) for f(x) = -2g(x) + 2h(x) + 2. A) 22 B) 0 C) -2
31) D) 11 32) D) 20
Use the differentiation feature on a graphing calculator to find the indicated derivative. 33) f(x) = 0.60x3 - 3.09x2 + 4.10x + 0.2; f (3)
A) -35.320
B) 1.960
C) 1.760
33) D) 11.030
Solve the problem.
34) The total cost to produce x handcrafted wagons is C(x) = 130 + 5x - x2 + 4x3. Find the marginal cost when x = 2. A) 38
B) 179
C) 49
D) 168
35) The profit in dollars from the sale of x thousand compact disc players is P(x) = x3 - 4x2 + 10x + 7. Find the marginal profit when the value of x is 6. A) $126 B) $77
C) $133
35)
D) $70
36) If the price of a product is given by P(x) = 1024 + 1,000, where x represents the demand for the x
product, find the rate of change of price when the demand is 32. A) 32 B) 1 C) -1
34)
36)
D) -32
37) For a motorcycle traveling at speed v (in mph) when the brakes are applied, the distance d (in
37)
feet) required to stop the motorcycle may be approximated by the formula d = 0.05 v2 + v. Find the instantaneous rate of change of distance with respect to velocity when the speed is 48 mph. A) 49 mph B) 5.8 mph C) 11.6 mph D) 4.8 mph
38) The power P (in W) generated by a particular windmill is given by P = 0.015 V3 where V is the
38)
39) The energy loss E (in joules/kilogram) due to friction when water flows through a pipe is given by
39)
velocity of the wind (in mph). Find the instantaneous rate of change of power with respect to velocity when the velocity is 5.7 mph. Round your answer to the nearest tenth. A) 5.6 W/mph B) 0.3 W/mph C) 1.5 W/mph D) 3.2 W/mph
E = 0.020(L/D)v2 . In the formula, L is the pipe length (in m), D is the pipe diameter (in m), and v is the water velocity (in m/s). Find a formula for the instantaneous rate of change of energy with respect to velocity. A) dE/dv = 0.04(L/D)v2 B) dE/dv = 0.02(L/D)v
C) dE/dv = (L/D)v
D) dE/dv = 0.04(L/D)v
4
40) The velocity of water in ft/s at the point of discharge is given by v = 12.32 P, where P is the
40)
pressure in lb/in.2 of the water at the point of discharge. Find the rate of change of the velocity with respect to pressure if the pressure is 10.00 lb/in.2 . A) 19.48 ft/s per lb/in.2 B) 1.9480 ft/s per lb/in.2
C) .6,160 ft/s per lb/in.2
D) 3.90 ft/s per lb/in.2
41) A balloon used in surgical procedures is cylindrical in shape. As it expands outward, assume that
41)
the length remains a constant 90.0 mm. Find the rate of change of surface area with respect to
radius when the radius is 0.060 mm. The surface area is given by the formula S(x) = 2 rl + 2 r2 , where l is the length and r is the radius. (Answer can be left in terms of ). A) 180.0 mm2/mm B) 180.24 mm2 /mm
C) 90.24 mm2/mm
D) 90.12 mm2/mm
42) A ball is thrown vertically upward from the ground at a velocity of 143 feet per second. Its
42)
43) Exposure to ionizing radiation is known to increase the incidence of cancer. One thousand
43)
distance from the ground after t seconds is given by s(t) = -16t2 + 143t. How fast is the ball moving 5 seconds after being thrown? A) -17 ft per sec B) -32 ft per sec C) 63 ft per sec D) 315 ft per sec
laboratory rats are exposed to identical doses of ionizing radiation, and the incidence of cancer is recorded during subsequent days. The researchers find that the total number of rats that have developed cancer t months after the initial exposure is modeled by N(t) = 1.19t2.3 for 0 t 10 months. Find the rate of growth of the number of cancer cases at the 7th month. Round your answer to the nearest tenth, if necessary. A) 38.3 cases/month B) 29.3 cases/month
C) 34.3 cases/month
D) 240.4 cases/month
44) The body-mass index (BMI) is calculated using the equation BMI = 703w , where w is in pounds h2
44)
and h is in inches. Find the rate of change of BMI with respect to weight for Sally, who is 59" tall and weighs 120 lbs. If both Sally and her brother Jesse gain the same small amount of weight, who will see the largest increase in BMI? Jesse is 73" tall and weighs 190 lbs. A) 24.234, Jesse B) 0.202, Jesse C) 0.202, Sally D) 24.234, Sally
45) A(x) = -0.015x3 + 1.05x gives the alcohol level in an average person's bloodstream x hours after
drinking 8 oz of 100-proof whiskey. If the level exceeds 1.5 units, a person is legally drunk. Find the rate of change of alcohol level with respect to time. A) dA = -0.045x2 + 1.05x B) dA = -0.045x2 + 1.05 dx dx
C) dA = -0.045x3 + 1.05
D) dA = -0.015x2 + 1.05
dx
dx
5
45)
46) A(x) = -0.015x3 + 1.05x gives the alcohol level in an average person's bloodstream x hours after
46)
47) The median weight, w, of a girl between the ages of 0 and 36 months can be approximated by the
47)
drinking 8 oz of 100-proof whisky. If the level exceeds 1.5 units, a person is legally drunk. Find the rate of change of alcohol level with respect to time when x = 2 hours. A) 1.92 units/hr B) 1.23 units/hr C) 0.87 units/hr D) 0.99 units/hr
function
w(t) = 0.0006t3 - 0.0484t2 + 1.61t + 7.60, where t is measured in months and w is measured in pounds. For a girl of median weight, find the rate of change of weight with respect to time at age 20 months. A) 0.882 lb/mo B) 0.086 lb/mo C) 0.394 lb/mo D) 1.362 lb/mo
48) The polynomial C(x) = -0.006x4 + 0.140x3 - 0.53x2 + 1.79x measures the concentration of a dye in
48)
the bloodstream x seconds after it is injected. Find the rate of change of concentration with respect to time. A) dC = -0.024x3 + 0.420x2 - 1.06x + 1.79 dt
B) dC = -0.018x3 + 0.280x2 - 0.53x + 1.79 dt
C) dC = -0.024x4 + 0.420x3 - 1.06x2 + 1.79x dt
D) dC = -0.006x3 + 0.140x2 - 0.53x + 1.79 dt
Use the product rule to find the derivative. 49) f(x) = (6x - 3)(5x + 1) A) f'(x) = 30x - 9
49) B) f'(x) = 60x - 4.5 D) f'(x) = 60x - 21
C) f'(x) = 60x - 9 50) f(x) = (4x - 3)(4x3 - x2 + 1) A) f'(x) = 48x3 + 48x2 - 16x + 4 C) f'(x) = 16x3 + 16x2 - 48x + 4
50) B) f'(x) = 64x3 - 48x2 + 6x + 4 D) f'(x) = 64x3 - 16x2 + 48x + 4
51) f(x) = (x2 - 4x + 2)(5x3 - x2 + 4) A) f'(x) = 25x4 - 84x3 + 42x2 + 4x - 16 C) f'(x) = 5x4 - 80x3 + 42x2 + 4x - 16 52) f(x) = (6x + 3)2 A) f'(x) = 72x + 36
51) B) f'(x) = 5x4 - 84x3 + 42x2 + 4x - 16 D) f'(x) = 25x4 - 80x3 + 42x2 + 4x - 16 52)
B) f'(x) = 36x + 18
C) f'(x) = 36x + 9
53) f(x) = (5x - 3)( x + 4) A) f'(x) = 7.5x1/2 - 3x-1/2 + 20 C) f'(x) = 3.33x1/2 - 3x-1/2 + 20
D) f'(x) = 12x + 6 53)
B) f'(x) = 7.5x1/2 - 1.5x-1/2 + 20 D) f'(x) = 3.33x1/2 - 1.5x-1/2 + 20
6
54) f(x) = (4x3 + 4)(4x7 - 7) A) f'(x) = 16x9 + 112x6 - 84x2 C) f'(x) = 16x9 + 112x6 - 84x
54) B) f'(x) = 160x9 + 112x6 - 84x D) f'(x) = 160x9 + 112x6 - 84x2
55) f(x) = (6 x - 2)(5 x + 7) A) f'(x) = 30x + 16x1/2 C) f'(x) = 30x + 32x1/2
55) B) f'(x) = 30 + 16x-1/2 D) f'(x) = 30 + 32x-1/2
56) g(x) = (x-5 + 3)(x-3 + 5) A) g'(x) = -8x-9 - 25x-6 - 9x-2 C) g'(x) = -8x-7 - 25x-6 - 9x-4
B) g'(x) = -8x-9 - 25x-4 - 9x-4 D) g'(x) = -8x-9 - 25x-6 - 9x-4
57) f(x) = (3x4 + 8)2 A) f'(x) = 6x4 + 16 C) f'(x) = 72x7 + 192x3
B) f'(x) = 9x16 + 64 D) f'(x) = 144x15 + 96x3
58) (y-2 + y-1)(2y-3 - 3y-4 ) 2 A) 18 - 5y + 8y y7
56)
57)
58) 2 B) 18 + 25y - 8y y7
2 C) 18 + 4y - 2y y7
2 D) 18 + 5y - 8y y7
Use the quotient rule to find the derivative. 59) f(x) = 1 x7 + 2 1 7 (7x + 2)2
A) f'(x) = C) f'(x) =
59) B) f'(x) =
1 7 (7x + 2)2
7x6 (x7 + 2)2
D) f'(x) = -
7x6 7 (x + 2)2
2 60) g(t) = t
60)
t - 11
A) g'(t) =
t2
2
B) g'(t) = t - 22t
(t - 11)2
(t - 11)2
2
C) g'(t) = t + 22t
D) g'(t) =
(t - 11)2
22t
(t - 11)2
2
61) y = x - 3x + 2
61)
x7 - 2
8
7
6
8
A) dy = -5x + 18x - 14x - 4x + 6 dx
(x7 - 2)2
8
7
dx
6
6
(x7 - 2)2
8
C) dy = -5x + 19x - 14x - 4x + 6 dx
7
B) dy = -5x + 18x - 13x - 4x + 6 7
6
D) dy = -5x + 18x - 14x - 3x + 6
(x7 - 2)2
dx
7
(x7 - 2)2
3 62) y = x
62)
x-1 3
A) dy = -2x - 3x dx
2
3
B) dy = -2x + 3x
(x - 1)2 3
C) dy = 2x - 3x dx
dx
2
(x - 1)2 3
D) dy = 2x + 3x
(x - 1)2
dx
2
2
(x - 1)2
2
63) g(x) = x + 5
63)
x2 + 6x 3
2
3
2
A) g'(x) = 2x - 5x - 30x
B) g'(x) = 4x + 18x + 10x + 30
2
D) g'(x) = x + 6x + 5x + 30x
x2 (x + 6)2
x2 (x + 6)2
4
C) g'(x) = 6x - 10x - 30 x2 (x + 6)2
3
2
x2 (x + 6)2
2
64) y = x - 4
64)
x
A) dy = x + 4 dx
x2
B) dy = 1 + 4 dx
C) dy = 1 + 4 dx
x
x2
2
65) y = x + 8x + 3 B) dy = 3x + 8x - 3
C) dy = 2x + 8
D) dy = 3x + 8x - 3
x
dx
x
2
2x3/2
dx
2x3/2
2
66) y = x + 2x - 2
66)
x2 - 2x + 2
A) dy =
-4x2 - 8x (x2 - 2x + 2)2
B) dy =
-4x2 + 8x (x2 - 2x + 2)2
C) dy =
4x2 + 8x 2 (x - 2x + 2)2
D) dy =
4x2 - 8x 2 (x - 2x + 2)2
dx dx
x2
2
A) dy = 2x + 8 dx
dx
65)
x
dx
D) dy = 1 - 4
dx dx
8
67) f(x) = x
1.8 + 6
67)
x3.4 + 1
A) f'(x) = -1.6x
4.2 - 1.6x1.8 - 20.4x2.4 - 20.4
B) f'(x) = -1.6x
4.2 + 1.8x0.8 - 3.4x1.8 + 6x2.4 - 20.4
C) f'(x) = -1.6x
4.2 + 1.8x0.8 - 3.4x1.8 - 20.4x2.4 - 20.4
D) f'(x) = -1.6x
4.2 + 1.8x0.8 - 20.4x2.4
(x3.4 + 1)
2
(x3.4 + 1)
2
x3.4 + 1
(x3.4 + 1)
2
2
68) f(x) = (4x - 1)(4x + 1)
68)
3x + 3
3
2
3
A) f'(x) = 48x + 132x + 24x + 15 (3x + 3)2
3
2
B) f'(x) = 96x + 132x - 24x + 15 (3x + 3)2
3
2
2
D) f'(x) = 96x + 132x - 24x + 15
C) f'(x) = 96x + 144x - 24x + 15
3x + 3
(3x + 3)2
Write an equation of the tangent line to the graph of y = f(x) at the point on the graph where x has the indicated value. 69) f(x) = (-5x2 + 3x + 2)(-2x + 1), x = 0 69)
A) y = -1x - 2
B) y = - 1x + 2
C) y = - 1x - 2
D) y = -1x + 2
2
70) f(x) = -3x - 6 , x = 0
70)
2x + 2
A) y = - 3x + 3
B) y = 3x - 3
C) y = 3x + 3
D) y = - 3x - 3
Using a graphing calculator, find the values of x for which f (x) = 0, to three decimal places. 71) f(x) = 2 - x2 x2 - 2
71)
A) 0, -1.307, 1.307 B) 0, -0.804, 0.804 C) 0 D) There are no real values of x for which f (x) = 0. 2
72) f(x) = x - 4
72)
x4 + 2
A) -2.828, 2.828 B) 0 C) 0, -2.871, 2.871 D) There are no real values of x for which f (x) = 0.
9
Solve the problem. 73) The total cost to produce x units of perfume is C(x) = (2x + 9)(9x + 5). Find the marginal average cost function. A) 36 - 91 B) 18x + 91 + 45 C) 36x + 91 D) 18 - 45 x x x2
74) The total profit from selling x units of cookbooks is P(x) = (5x - 5)(7x - 9). Find the marginal average profit function. A) 35 - 80 x2
B) 35 - 45
C) 35x - 45
x2
74)
D) 35x - 80
75) The demand function for a certain product is given by: D(p) =
73)
75)
7p + 240 . 11p + 19
Find the marginal demand D'(p). A) D'(p) = -2,507 (11p + 19)2
B) D'(p) =
2,507
(11p + 19)2
D) D'(p) = 2,773 + 154p
C) D'(p) = -2,507
(11p + 19)2
11p + 19
76) A rectangular steel plate expands as it is heated. Find the rate of change of area with respect to
76)
temperature T when the width is 1.5 cm and the length is 2.6 cm if dl/dt = 1.3 x 10-5 cm/°C and dw/dt = 8.9 x 10-6 cm/°C. A) 3.9 x 10-5 cm2/°C
B) 4.3 x 10-5 cm2/°C D) 1.2 x 10-5 cm2/°C
C) 2 x 10-5 cm2 /°C 77) The total revenue for the sale of x items is given by: R(x) =
77)
140 x . 9 + x3/2
Find the marginal revenue R'(x). -1/2 + 4x) A) R'(x) = 70(9x (9 + x3/2)2
C) R'(x) = 70(9x
B) R'(x) = 70(9x
-1/2 - 2x)
D) R'(x) = 70(9x
1/2 - 2x)
9 + x3/2
-1/2 - 2x)
(9 + x3/2)2
(9 + x3/2)2
10
78) Murrel's formula for calculating the total amount of rest, in minutes, required after performing a
78)
30(w - 4) particular type of work activity for 30 minutes is given by the formula R(w) = , where w w - 1.5 is the work expended in kilocalories per min. A bicyclist expends 5 kcal/min as she cycles home from work. Find R'(w) for the cyclist; that is, find R'(5). A) 4.9 min2 /kcal B) 8.57 min2 /kcal C) 7.35 min2 /kcal D) 6.12 min2 /kcal
79) Prairie dogs form an important part of the coyote's diet. As coyotes are hunting for prairie dogs,
79)
they must be careful to expend just the right amount of time at each burrow. If a coyote spends too little time at each burrow, it catches very few prairie dogs per kilocalorie of energy expended. Likewise, if the coyote spends too much time digging at a single burrow, it can expend a large amount of energy per prairie dog caught. The relation between energy expended and time spent 1 20 at each burrow is approximated by E = t2 for t > 0.75 minutes, where t is in minutes + t t - 0.75 and E is in kcal expended per prairie dog caught. How much time should a coyote spend at each burrow to minimize the energy expended per prairie dog caught. (Hint: pay close attention to the domain of the above function.) A) 10 minutes B) .8 minutes C) 1.5 minutes D) 2.0 minutes
80) Assume that the temperature of a person during an illness is given by: T(t) =
80)
7t + 98.6, 2 t +1
where T = the temperature, in degrees Fahrenheit, at time t, in hours. Find the rate of change of the temperature with respect to time. 2 2 2 A) dT = 7(1 - t ) B) dT = 7(1 - t ) C) dT = 7 D) dT = 7(t - 1) dt dt dt dt t2 + 1 (t2 + 1)2 (t2 + 1)2 t2 + 1
81) The population P, in thousands, of a small city is given by: P(t) =
81)
800t . 2t2 + 3
where t = the time, in months. Find the growth rate. 2 A) P'(t) = 800(3 + 6t ) (2t2 + 3)2
2
2
B) P'(t) = 800(3 - 2t ) (2t2 + 3)2
2
C) P'(t) = 800(2t - 3)
D) P'(t) = 800(3 - 2t )
(2t2 + 3)2
2t2 + 3
Provide an appropriate response. 82) True or false? The derivative of the quotient of two functions is the quotient of their derivatives. A) True B) False
11
82)
83) True or false? If average product is increasing then the marginal product must be increasing. A) True B) False
83)
84) True or false? If marginal cost is decreasing then the average cost must be decreasing. A) False B) True
84)
85) True or false? If the average cost is decreasing then the marginal cost must be less than the
85)
average cost. A) True
B) False
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
86) Revenues of a company are decreasing. One analyst says it is due to an increase in price.
86)
87) What is true when marginal revenue and marginal cost are equal?
87)
88) What must be true about a demand function so that, at a given price per item, revenue
88)
89) Prove that if average product is increasing, then marginal product is more than average
89)
90) If g(-5) = 2, g (-5) = -2, f(-5) = -3, and f (-5) = -1, what is the value of h (-5) where h(x)
90)
91) Find the error that was committed below when taking the derivative of f(x) = 3x + 10 . Be
91)
Is this necessarily true? Explain.
will decrease if the price per item is increased?
product.
= f(x)g(x)? Show your work.
x2 + 3
specific. Dx
3x + 10 3 x2 + 3 + 3x + 10 (2x) 9x2 + 20x + 9 = = 2 2 x2 + 3 x2 + 3 x2 + 3
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Let f(x) = 8x2 - 5x and g(x) = 7x + 9. Find the composite. 92) f[g(3)] A) 7050
93) g[f(3)] A) 408
92) B) 618
C) 408
D) 1212
B) 7050
C) 618
D) 1212
93)
12
94) f[g(-3)] A) 7050
B) 618
C) 1212
D) 408
95) g[f(-3)] A) 1212
B) 618
C) 7050
D) 408
94)
95)
96) f[g(k)] A) 392k2 - 973k + 603 C) 56k2 - 35k + 9
B) 392k2 + 973k + 603 D) 56k2 + 35k + 9
97) g[f(k)] A) 56k2 - 35k + 9 C) 56k2 + 35k + 9
B) 392k2 - 973k + 603 D) 392k2 + 973k + 603
98) f[g(-k)] A) 56k2 + 35k + 9 C) 56k2 - 35k + 9
B) 392k2 + 973k + 603 D) 392k2 - 973k + 603
99) g[f(-k)] A) 392k2 + 973k + 603 C) 392k2 - 973k + 603
B) 56k2 - 35k + 9 D) 56k2 + 35k + 9
96)
97)
98)
99)
100) f[g(-4)] A) 765
B) 2983
C) 148
D) 37
101) g[f(4)] A) 148
B) 765
C) 2983
D) 37
100)
101)
Find f[g(x)] and g[f(x)]. 102) f(x) = 5x + 9; g(x) = 4x - 7 A) f[g(x)] = 20x - 29 g[f(x)] = 20x + 26 C) f[g(x)] = 20x + 26 g[f(x)] = 20x - 29
102) B) f[g(x)] = 20x - 26 g[f(x)] = 20x + 29
D) f[g(x)] = 20x + 29 g[f(x)] = 20x - 26
103) f(x) = 5x3 + 8; g(x) = 2x A) f[g(x)] = 10x3 + 8
103) B) f[g(x)] = 40x3 + 8
g[f(x)] = 40x3 + 16 C) f[g(x)] = 40x3 + 16
g[f(x)] = 10x3 + 16 D) f[g(x)] = 10x3 + 16
g[f(x)] = 10x3 + 8
g[f(x)] = 40x3 + 8
13
104) f(x) = 2 ; g(x) = 2x3
104)
x
A) f[g(x)] = 4/x3 ; g[f(x)] = 1/x3 C) f[g(x)] = 1/x3 ; g[f(x)] = 16/x3
B) f[g(x)] = 1/x3 ; g[f(x)] = 4/x3 D) f[g(x)] = 16/x3 ; g[f(x)] = 1/x3
105) f(x) = 7/(x)4; g(x) = 2x3 A) f[g(x)] = 7/16x12
105) B) f[g(x)] = 7x12/686
g[f(x)] = 686/(x)12 C) f[g(x)] = 7x12/16
g[f(x)] = x12/16
D) f[g(x)] = 686/7x12
g[f(x)] = x12/686
g[f(x)] = 16/(x)12
106) f(x) = x + 5; g(x) = 4x - 1 A) f[g(x)] = 4x2 + 1 g[f(x)] =
106) B) f[g(x)] = 2 x + 1
4x2 - 5
g[f(x)] = 4 x + 5 - 1
D) f[g(x)] = 4x2 - 5
C) f[g(x)] = 2 x + 5 g[f(x)] = 4 x + 1 - 1
107) f(x) =
g[f(x)] =
4x2 - 5
1 ; g(x) = x + 5 x-5
107)
A) f[g(x)] = x - 5
B) f[g(x)] = (5x - 24)/(x - 5)
g[f(x)] = 1/(x + 5) C) f[g(x)] = 1/(x + 5) g[f(x)] = x - 5
g[f(x)] = 1/x D) f[g(x)] = 1/x g[f(x)] = (5x - 24)/(x - 5)
108) f(x) = 5x2 ; g(x) = x + 3 A) f[g(x)] = 5x2 + 30x + 3
108) B) f[g(x)] = 5x2 + 45
g[f(x)] = 5x2 + 45 C) f[g(x)] = 5x2 + 30x + 45
g[f(x)] = 5x2 + 30x + 3 D) f[g(x)] = 5x2 + 3
g[f(x)] = 5x2 + 3
g[f(x)] = 5x2 + 30x + 49
109) f(x) = x2 + 2x + 3; g(x) = x - 4 A) f[g(x)] = x2 + 2x - 1
109) B) f[g(x)] = x2 + 6x + 11
g[f(x)] = x2 - 6x + 11
g[f(x)] = x2 + 2x - 1 D) f[g(x)] = x2 + 2x - 1
C) f[g(x)] = x2 - 6x + 11 g[f(x)] = x2 + 2x - 1
g[f(x)] = x2 + 6x + 11
110) f(x) = x - 3; g(x) = 5x2 + 5 A) f[g(x)] = 5x2 + 5 - 3
110) B) f[g(x)] = 5x - 10
g[f(x)] = 5x + 2
g[f(x)] =
C) f[g(x)] = 5x2 - 3
D) f[g(x)] =
g[f(x)] = 5(x - 3)2 + 5
5x2 + 2 5x2 + 2
g[f(x)] = 5x - 10
14
111) f(x) = 5 ; g(x) = x - 7
111)
x
A) f[g(x)] =
5 x-7
B) f[g(x)] =
5 x-7
g[f(x)] =
5 - 7x x
g[f(x)] =
5 -7 x
C) f[g(x)] =
5 x-7
D) f[g(x)] =
5 x-7
g[f(x)] =
5 - 7x x
g[f(x)] =
5 -7 x
Write the function as the composition of two functions f and g such that y = f[g(x)]). 112) y = 1 x2 - 4
112)
A) f(x) = 1 , g(x) = x2 - 4
B) f(x) = 1 , g(x) = x - 4
C) f(x) = 1 , g(x) = x2 - 4
D) f(x) = 1 , g(x) = - 1
x2
4
x2
x
4
113) y = 2 + 7
113)
x2
A) f(x) = x, g(x) = 2 + 7
B) f(x) = 2 , g(x) = 7
C) f(x) = x + 7, g(x) = 2
D) f(x) = 1 , g(x) = 2 + 7
x2
x
x2
114) y =
x
x
5 3x + 2
114)
A) f(x) = 5 , g(x) = 3x + 2
B) f(x) = 5 , g(x) = 3x + 2
C) f(x) = 3x + 2, g(x) = 5
D) f(x) = 5, g(x) = 3 + 2
115) y = (7x + 17)3 A) f(x) = 7x + 17, g(x) = x3 C) f(x) = 7x3, g(x) = x + 17
B) f(x) = (7x)3 , g(x) = 17
x
x
115) D) f(x) = x3 , g(x) = 7x + 17
116) y = 6 + 6x2
116) 4
6 + 6x2 , g(x) =
A) f(x) = 6 + 6x2, g(x) = x
B) f(x) =
C) f(x) = x, g(x) = 6 + 6x2
D) f(x) = 6 + 6x, g(x) = x
15
4
6 + 6x2
117) y = (x1/2 + 3)3 + 2(x1/2 + 3) 2 - 1 A) f(x) = x3 + 2x2 - 1, g(x) = x1/2 + 3 B) f(x) = (x + 3)3 + 2(x + 3) 2 - 1, g(x) = x1/2 + 3 C) f(x) = (x + 3)3 + 2x2 - 1, g(x) = x1/2 D) f(x) = x1/2 + 3, g(x) = x3 + 2x2 - 1
117)
Find the derivative.
118) y = (4x + 3)5
118)
A) dy = 5(4x + 3)4 dx
B) dy = 4(4x + 3)4 dx
C) dy = 20(4x + 3)4
D) dy = (4x + 3)4
dx
119) y = 4x + 2
A) dy = dx
dx
119) 2 4x + 2
B) dy = dx
1 4x + 2
C) dy = dx
4 4x + 2
120) f(x) = (x3 - 8)2/3 A) f'(x) =
C) f'(x) =
B) f'(x) =
x3 - 8
x 3
D) f'(x) =
x3 - 8
121) y = (x-2 + x)-3 A) dy = dx
8 4x + 2
B) dy =
(1 + x3 )3 5
x2 3
x3 - 8
2x 3
x3 - 8
121)
3x4 (2 - x3 )
dx
3
3x5 (2 - x3 ) (1 + x3 )3 4
C) dy = 3x (2 - x ) dx
dx
120)
2x2
3
D) dy =
3
D) dy = 3x (2 - x )
(1 + x3 )4
dx
122) y = (x + 1)2(x2 + 1)-3
(1 + x3 )4
122)
A) dy = -2(x + 1)(x2 + 1)-4 (2x2 + 3x - 1) dx
B) dy = 2(x + 1)(x2 + 1)-4(2x2 + 3x - 1) dx
C) dy = 2(x + 1)(x2 + 1)-4(2x2 - 3x - 1)
D) dy = -2(x + 1)(x2 + 1)-4 (2x2 - 3x - 1)
dx
dx
16
123) f(x) =
5
123)
(2x - 3)4
A) f'(x) = C) f'(x) =
-40
B) f'(x) =
(2x - 3)3 5
D) f'(x) =
4(2x - 3)3
124) y = (2x - 1)3 (x + 7)-3
5
8(2x - 3)5 -40
(2x - 3)5
124)
A) dy = 45(2x - 1)3(x + 7)-2 dx
B) dy = 45(2x - 1)2(x + 7)-4 dx
C) dy = 45(2x - 1)2(x + 7)-3
D) dy = 45(2x - 1)3(x + 7)-4
dx
dx
125) y = x x2 + 1 x2 + 1 x2 + 1
A) dy = dx
3
126) y =
125) 2
B) dy = x + 1 dx
C) dy = dx
x2 + 1
x2 + 1 2x2 + 1
2
D) dy = 2x + 1 dx
x2 + 3 x
A) dy =
126) 3
dx
x2 (x2 + 3)
C) dy =
-3
dx
x2 (x2 + 3)
B) dy =
2/3 2/3
x2 + 9
dx
3x2 (x2 + 3)
D) dy =
-x2 - 9
dx
127) y = (3x2 + 5x + 1)3/2
3x2 (x2 + 3)
2/3 2/3
127)
A) dy = (6x + 5)(3x2 + 5x + 1)1/2
B) dy = (3x2 + 5x + 1)1/2
C) dy = 3 (6x + 5)(3x2 + 5x + 1)1/2
D) dy = 3 (3x2 + 5x + 1)1/2
dx dx
x2 + 1
dx
2
dx
2
The table lists the values of the functions f and g and their derivatives at several points. Use the table to find the indicated derivative. x 1 2 3 4 f(x) 1 4 3 2 128) f'(x) -4 5 0 4 128) g(x) 2 4 1 3 g'(x) 3 6 -4 -2 Find Dx(f[g(x)]) at x = 3.
A) -4
C) 0
B) -0
17
D) 16
x 1 2 3 4 f(x) 4 3 1 2 129) f'(x) 4 3 -2 -6 g(x) 3 4 1 2 g'(x) -5 -4 5 3
129)
Find Dx(g[f(x)]) at x = 2.
A) 3
B) 5
C) 15
D) 8
Find the equation of the tangent line to the graph of the given function at the given value of x. 130) f(x) = (x2 + 28)4/5 ; x = 2 A) y = 8 x + 64 B) y = 8 x C) y = 4 x + 64 D) y = 8 x + 96 5 5 5 5 5 5 5
131) f(x) = x3 x3 + 8; x = 1 A) y = 19 x - 13 2 2
131) B) y = 19 x - 11 2 2
C) y = 80 x - 85 9 9
D) y = 80 x + 85 9 9
Find all values of x for the given function where the tangent line is horizontal. 132) f(x) = x2 + 10x + 40
A) -5, 5 133) f(x) =
130)
B) 0, 5
132) D) 0, -5
C) -5
x
133)
3 (x2 + 3)
B) 0, ±
A) 0
15 5
15 5
C) ±
D) ±
3 5
Solve the problem.
134) The total revenue from the sale of x stereos is given by R(x) = 2,000(1 - x )2 . Find the average 600
revenue from the sale of x stereos. A) 2,000x 1 - x 2 600
B) 1,000 1 - x 2 x
C) 1,000x 1 - x 2
600
D) 2,000 1 - x 2
600
x
600
135) The total revenue from the sale of x stereos is given by R(x) = 1,000 1 - x 2 . Find the marginal 600
average revenue. A) 1.67 - 1,000 x2
134)
B) 1.67 - 600
C) 0.003 - 1,000
x2
x2
18
D) 0.003 - 600 x2
135)
136) $2,800 is deposited in an account with an interest rate of r% per year, compounded monthly. At
136)
r 96 the end of 8 years, the balance in the account is given by A = 2,800 1 + . Find the rate of 1200 change of A with respect to r when r = 6. A) dA = 225.12 B) dA = 226.25 dr dr
C) dA = 361.57
D) dA = 359.77
dr
dr
137) The formula E = 1000(100 - T) + 580(100 - T)2 is used to approximate the elevation (in meters)
137)
138) A circular oil slick spreads so that as its radius changes, its area changes. Both the radius r and the
138)
139) When an amount of heat Q (in kcal) is added to a unit mass (in kg) of a substance, the temperature
139)
above sea level at which water boils at a temperature of T (in degrees Celsius). Find the rate of change of E with respect to T for a temperature of 84°C. A) -18,560 m/°C B) -68,280 m/°C C) -19,560 m/°C D) 19,560 m/°C
area A change with respect to time. If dr/dt is found to be 2.6 m/hr, find dA/dt when r = 33.7 m. A) 87.62 m2 /hr B) 175.24 m2/hr C) 43.81 m2 /hr D) 350.48 m2/hr
rises by an amount T (in degrees Celsius). The quantity dQ/dT, called the specific heat, is 0.18 for glass. If dQ/dt = 14.0 kcal/min for a 1 kg sample of glass at 20.0°C, find dT/dt for this same sample. A) 37.9 kcal/min B) 80.8 kcal/min C) 2.52 kcal/min D) 77.8 kcal/min
140) Suppose that a demand function is given by q = D(p) = 23 8 -
p2 , p3 + 1
140)
where q is the demand for a product and p is the price per unit in dollars. Find the rate of change in the demand for the product per unit change in price. 4 4 A) dq = -23p + 92p B) dq = -23p - 92p dp 1/2 dp 1/2 (p3 + 1) 2(p3 + 1) 4
4
C) dq = 23p - 92p dp
2(p3 + 1)
D) dq = -23p - 92p
3/2
dp
2(p3 + 1)
3/2
141) The concentration of a certain drug in the bloodstream t minutes after swallowing a pill
1 containing the drug can be approximated using the equation C(t) = 3t + 1 -1/2, where C(t) is 6 the concentration in arbitrary units and t is in minutes. Find the rate of change of concentration with respect to time at t = 5 minutes. A) - 1 units/min B) - 1 units/min 256 16
C) - 1 units/min
D) - 1 units/min
24
768
19
141)
Provide an appropriate response. 142) What rule is applied first to find the derivative of the function (3x + 2)3 f(x) = (3x2 - 2) ? 3x3 + 2
A) Quotient rule C) Product rule
142)
B) Power rule D) Square root rule
143) What rule is applied first to find the derivative of the function f(x) =
143)
(5x - 1)2 (x3 - 2) ? 5x - 5
A) Product rule C) Square root rule
B) Power rule D) Quotient rule
144) What rule is applied first to find the derivative of the function f(x) =
144)
(4x2 - 4) 5x3 - 6 ? 5x - 5
A) Power rule C) Product rule
B) Quotient rule D) Square root rule
145) What rule is applied first to find the derivative of the function f(x) =
145)
(3x3 - 1) 5x3 - 4 5 2x + 2
A) Quotient rule C) Power rule
B) Square root rule D) Product rule
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
146) How is the graph of y = f(x) = x3 + 4x - 1 related to the graph of
146)
147) How is the graph of y = f(x) = x3 + 3x - 3 related to the graph of
147)
y = g(x) = (x - 4)3 + 4(x - 4) - 1? How is the slope of the graph of g(x) at x = a related to the slope of the graph of f(x) at x = a - 4?
y = g(x) = (2x)3 + 3(2x) - 3? How is the slope of the graph of g(x) at x = a related to the slope of the graph of f(x) at x = a?
148) Find the derivative of f(x) = (4x - 3)3 in two ways. First multiply out and differentiate.
148)
149) Use the chain rule to prove the quotient rule for f(x) = g(x) .
149)
Then use the power rule. Show that the answers are equivalent.
h(x)
20
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. 150) y = 8e12x
150)
A) 8xe96x
B) 8e96x
C) 96xe12x
D) 96e12x
B) 10xe5x2 + 1
C) 10xex2 + 1
D) 10xe2x + 1
B) 8xe
C) 8xe4x2
D) 8xex2
151) y = e5x2 + x
151)
A) 10xe + 1 152) y = 4ex2
152)
A) 8xe2x 153) y = 4ex2
153) B) 8xex2
A) 8xe
C) 8xe2x
D) 8xe4x2
x 154) y = 6e
154)
2ex + 1
A)
ex x (2e + 1)2
B)
155) y = 5x2 e3x A) 10xe3x(2x + 3)
6ex (2ex + 1)
C)
6ex (2ex + 1)3
D)
6ex (2ex + 1)2
155) B) 5xe3x(3x + 2)
C) 5xe3x(2x + 3)
D) 10ex3x(3x + 2)
-x + 1
156) y = e
156)
ex
x
x
A) e - 2 e2x
157) y =
x
B) -e + 2
C) e + 2
e2x
e2x
x
D) -e - 2 e2x
100 2 + 9e.3x
A) 200 + 1170e
157) .3x
(2 + 9e.3x)2
.3x
B) 200 + 630e
C)
(2 + 9e.3x)2
158) y = (x + 9)5e-4x A) -(x + 9)4 (4x + 31) e-5x C) (x + 9)4 (x + 14) e-4x
270e.3x (2 + 9e.3x)2
.3x D) -270e
(2 + 9e.3x)2
158) B) -(x + 9)4 (4x + 31) e-4x D) -20(x + 9)4 e-4x
21
159) y =
ex 3x2 + 9
159)
x-1 (3x2 + 9) - 6x ex
A) e
x-1 (3x2 - 6x + 9)
B) e
2 (3x2 +9) 2
2 (3x2 + 9)
x
C) ex + 3x - 6x + 9
2
D) e (3x - 6x + 9)
2 (3x2 + 9)
2 (3x 2 + 9)
160) y = 103x A) 10 (ln 3) 103x
B) 30 (ln 3) 103x
C) 3 (ln 10) 103x
D) 30 (ln 10) 103x
161) y = 13-x A) -13-x
B) - ln 13 (13-x)
C) 13-x
D) ln 13 (13-x)
160)
161)
162) y = 6(97x - 6) - 2 A) 63 ln 9 (9 7x - 6) C) 42 ln 9 (9 7x - 6)
162) B) 42 ln 54 (97x - 6) D) 63 ln 54 (97x - 6)
163) y = 2(6 x)
163)
A) ln 6 (6
x)
x
B) 2 ln 6 (6 )
x
x
C) 2 ln 6 (6 x) ( x)
D) ln 6 (6 x) ( x)
164) y = 25x - 1 A) 25 ln 25 C) 25x - 1 ln x 165) y = 11x2 A) 2x ln 11
164) B) 25x - 1 ln 25x - 1 D) 25x - 1 ln 25 165) B) 11x2 2x ln x
C) 11x2 2x ln 11
D) 11x2 x ln 11
Solve the problem.
166) The sales in thousands of a new type of product are given by S(t) = 110 - 90e-0.8t, where t
166)
represents time in years. Find the rate of change of sales at the time when t = 6. A) -8,621.5 thousand per year B) -0.6 thousand per year
C) 8,621.5 thousand per year
D) 0.6 thousand per year
167) A company's total cost, in millions of dollars, is given by C(t) = 240 - 70e-t where t = time in years. Find the marginal cost when t = 5. A) 1.62 million dollars per year
B) 0.47 million dollars per year D) 0.87 million dollars per year
C) 2.36 million dollars per year
22
167)
168) The demand function for a certain book is given by the function x = D(p) = 70e-0.002p. Find the marginal demand D'(p). A) D'(p) = -0.14e-0.002p
168)
B) D'(p) = -0.002e-0.002p D) D'(p) = -0.14pe-0.002p-1
C) D'(p) = 0.14e-0.002p
169) Suppose that the amount in grams of a radioactive substance present at time t (in years) is given
169)
by A(t) = 380e-0.39t. Find the rate of change of the quantity present at the time when t = 6. A) 14.3 grams per year B) 10.3 grams per year
C) -10.3 grams per year
D) -14.3 grams per year
170) When a particular circuit containing a resistor, an inductor, and a capacitor in series is connected
170)
171) When a radioactive substance decays, the number N of grams remaining from an initial mass N 0
171)
to a battery, the current i (in amperes) is given by i = 26e-3t(e2.6t - e-2.6t) where t is the time (in seconds). Find the time at which the maximum current occurs. Round to the nearest tenth of a second. A) 1.4 sec B) 0.6 sec C) 0.5 sec D) 1.5 sec
(in grams) is given by N = N 0 (1/2)n , where n is the number of half-lives for which the substance has decayed. Given that the half-life for tritium is 13 years, find the rate in (grams/half-life) at which a 142-gram initial mass of radioactive tritium decays after 41.6 years. A) -3.14 g/half-life B) -10.71 g/half-life
C) -15.45 g/half-life
D) -98.42 g/half-life
172) The nationwide attendance per day for a certain motion picture can be approximated using the
172)
equation A(t) = 13t2 e-t, where A is the attendance per day in thousands of persons and t is the number of months since the release of the film. Find and interpret the rate of change of the daily attendance after 4 months. A) 1.905 thousand persons/day · month; the change in the daily attendance is increasing.
B) 3.81 thousand persons/day · month; the daily attendance is increasing. C) -3.81 thousand persons/day · month; the change in daily attendance is decreasing. D) -1.905 thousand persons/day · month; the daily attendance is decreasing. 173) As a radioactive sample disintegrates, the "parent" atoms are converted into "daughter" atoms.
The number of daughter atoms D(t) that have been formed by a particular time t is given by D(t) = Po(1 - e-kt), where D(t) is the number of daughters, Po is the initial number of parent atoms, k is the decay rate constant in units of s-1 , and t is in seconds. Find an expression for the rate of change of D with respect to time. A) D'(t) = Po + Poe-kt C) D'(t) = kPoe-kt - 1
B) D'(t) = -kPoe-kt D) D'(t) = kPoe-kt
23
173)
174) In one city, 30% of all aluminum cans distributed will be recycled each year. A juice company
174)
distributes 182,000 cans. The number still in use after time t, in years, is given by N(t) = 182,000(0.30)t. Find N'(t).
A) N'(t) = 182,000t(0.30)t-1 C) N'(t) = 182,000(0.30)t
B) N'(t) = 182,000(ln 0.30)(0.30)t D) N'(t) = 182,000(ln t)(0.30)t
175) The pH scale is used by chemists to measure the acidity of a solution. It is a base 10 logarithmic
175)
scale. The pH, P, of a solution and its hydronium ion concentration in moles per liter, H, are related as follows: H = 10-P Find the formula for the rate of change
dH . dP
-P
A) dH = (ln 10 )10-P
B) dH = - 10
C) dH = -(ln 10 )10-P
D) dH = -(ln P)10-P
dP
dP
dP
ln 10
dP
176) Rats are not native to the islands off the western coast of South America. However, rats are often
176)
177) The following formula accurately models the relationship between the size of a certain type of
177)
introduced accidentally to an island by visiting ships. The population of introduced rats follows the logistic function with k = 0.00023 and t in months. Assume that there are 9 rats initially and that the maximum population size is 11,000. Find the rate of growth of the population after 5 months. A) 235 rats/month B) 99 rats/month C) 108 rats/month D) 127 rats/month
tumor and the amount of time that it has been growing: 3 V(t) = 450 1 - e-0.0018t ,
where t is in months and V(t) is measured in cubic centimeters. Calculate the rate of change of tumor volume at 190 months. A) 0.145 cm3 /month B) 0.628 cm3 /month
C) 0.227 cm3 /month
D) 0.106 cm3 /month
178) Researchers have found that the maximum number of successful trials that a laboratory rat can complete in a week is given by P(t) = 50(1 - e-0.2t),
where t is the number of weeks the rat has been trained. What is the maximum number of successful trials that a laboratory rat can complete in a week after being trained for 6 weeks. A) 65 B) 35 C) 9 D) 6
24
178)
179) Researchers have found that the maximum number of successful trials that a laboratory rat can
179)
complete in a week is given by P(t) = 56(1 - e-0.2t),
where t is the number of weeks the rat has been trained. Find the rate of change P'(t). A) P'(t) = 56(1 + 0.2e-0.2t) B) P'(t) = -11.2e-0.2t
C) P'(t) = 11.2e-0.2t
D) P'(t) = 56e-0.2t
180) The natural resources of an island limit the growth of the population to a limiting value of 4,234.
180)
The population of the island is given by the logistic equation 4,234 P(t) = , 1 + 5.03e-0.38t
where t is the number of years after 1980. What is the population of the island in 1,986? A) 2,656 B) 954 C) 2,796 D) 2,516
181) The natural resources of an island limit the growth of the population to a limiting value of 2,936.
181)
The population of the island is given by the logistic equation 2,936 P(t) = , 1 + 4.15e-0.4t
where t is the number of years after 1980. Find the rate of change P'(t). -0.4t -0.4t A) P'(t) = 4,873.8e B) P'(t) = 4,873.8e 2 -0.4t 1 + 4.15e (1 + 4.15e-0.4t)
C) P'(t) =
12,184e-0.4t
-0.4t
D) P'(t) = 2,936 - 4,873.8e
2 (1 + 4.15e-0.4t)
2 (1 + 4.15e-0.4t)
Provide an appropriate response. 182) If Q = 87e-0.9t what happens to Q and to Q' as t increases?
A) Q increases and Q' decreases. C) Q increases and Q' increases.
182)
B) Q decreases and Q' increases. D) Q decreases and Q' decreases.
183) If Q = 146e0.8t what happens to Q and to Q' as t increases? A) Q increases and Q' decreases. B) Q decreases and Q' increases. C) Q decreases and Q' decreases. D) Q increases and Q' increases.
183)
184) If Q = 130 - e-0.6t what happens to Q and to Q' as t increases? A) Q increases and Q' decreases. B) Q decreases and Q' decreases. C) Q increases and Q' increases. D) Q decreases and Q' increases.
184)
185) If Q = 141e0.1t what happens to Q and to Q' as t increases? A) Q increases and Q' increases. B) Q increases and Q' decreases. C) Q decreases and Q' decreases. D) Q decreases and Q' increases.
185)
25
186) Let A(t) represent a quantity which is growing exponentially. The percentage rate of growth A'(t) A(t)
is
186)
.
A) Constant
B) Decreasing
C) Increasing
D) None of these
187) A quantity Q is increasing by 5,000 per year at the present time. This means that
.
187)
(Provide a statement that is always true, involving either Q'(0), Q(0), Q(5,000), or Q'(5,000).) A) Q(0) = 5,000 B) Q'(5,000) = 0 C) Q'(0) = 5,000 D) Q(5,000) = 0
188) A(t) = P(1 + r)nt represents the amount of money in an account paying interest compounded n A'(t) times per year. The percentage rate of growth is A(t)
A) Decreasing
B) Constant
.
C) Increasing
189) The derivative of 10x = A) 10x ln 10
C) 10
x
ln 10
D) x10x - 1
190) B) - 1
C) - 1
x
191) y = ln (x - 3) A)
D) None of these 189)
B) 10x
Find the derivative of the function. 190) y = ln 7x A) 1 7x
188)
7x
D) 1
x
191)
1 x+3
B)
192) y = ln 2x2
1 3-x
C)
1 x-3
D) - 1
x+3
192) B) 2 x
A) 2x x2 + 2
193) y = ln (7 + x2 )
C) 1 2x + 2
193)
A) 14
B)
x
194) y = ln 2x3 - x2
2x 2 x +7
C)
1 2x + 7
D) 2
x
194)
A) 6x - 2
B) 2x - 2
2x2
C) 6x - 2
2x2 - x
195) y = x2 ln x2
D) 4 x
2x3 - x
D) 6x - 2
2x2 - x
195)
2
2
A) 2x (1 + ln x ) x
2
C) 2x + 2
B) 2x(1 + ln x2 )
x
26
D) 2x + ln x2
196) y = ln (x + 4)5
196)
A) 5
B)
x
4 x+4
C)
5 x+5
D)
5 x+4
2 197) y = 5x
197)
ln |3x|
A) ln |3x| - 10x
B) ln |3x| - 5x
C) 5x ln |3x| - 10x
D) 10x ln |3x| - 5x
(ln |3x|)2
(ln |3x|)2
(ln |3x|)2
(ln |3x|)2
198) y = (6x2 + 4) ln(x + 2)
198)
2
2
A) 6x + 4 + 12x ln(x + 2)
B) 6x + 4 + 12x ln(x + 2)
C) 12x ln(x + 2)
D) 12x
ln(x + 2)
x+2
x+2
199) y = 8ln(x + 8)
199)
4 - 2x
A) 32 - 16x + 16ln(x + 8)
B)
C) 32 + 16x + 16(x + 8) ln(x + 8)
D) 32 - 16x + 16(x + 8) ln(x + 8)
(x + 8)(4 - 2x)
(x - 8)(4 - 2x)2
Find the derivative. 200) y = ex ln x, x > 0
x A) e (ln x + x) x
8
(x + 8)(4 - 2x)2 (x + 8)(4 - 2x)2
200) x B) e x
x C) e (x ln x + 1) x
D) ex ln x
x
201) y = e
201)
ln x x
x
x
x
x ln2 x
202) y = ex3 ln x A)
x
C) e + x e ln x
B) e - x e ln x
A) x ex
x
x ln2 x
202)
3x3 ex3 + 1
x3 3 x3 B) e + 3x e ln x x
x3 + 3x2 ex3 ln x
D) e
x
C) e
x
D) x e ln x - e
x3 + 3ex3 ln x
x
27
x
203) y = ln (5x + 7)
203)
e5x + 7
A) 5 - (25x + 35) ln (5x + 7)
B)
(5x + 7) e(5x + 7)
C) 1 - 5 [ln (5x + 7)]
2
(5x + 7) e(5x + 7)
204) B) 1 x
205) y = log (7x - 6) A) 7x - 6
7 ln 10
ln 10
207) y = log |-7x|
1 A) x(ln 10)
208) y = log 3
C) 1 x(ln 4)
1 D) x(ln 10)
205) B)
206) y = log |9 - x| A) - 1
(5x + 7) e(5x + 7)
D) 1 - (5x + 7) ln (5x + 7)
ln [5x + 7] e(5x + 7)
Find the derivative of the function. 204) y = log (4x) A) 1 ln 10
1
7 ln 10
C)
1 ln 10 (7x - 6)
D)
7 ln 10 (7x - 6)
206) B)
1 ln 10 (9 - x)
C) -
1 ln 10 (9 - x)
D) - 9 - x
ln 10
207) B) - ln 10 x
1 C) x (ln 10)
D) x ln 10
5x + 9
A) 5 ln3
5x + 9
208) B)
5 ln 3 (5x + 9)
C) 5
ln 3
D)
5 2(ln 3)(5x + 9)
209) y = log4 |(3x2 - 2x)5/2| A)
5(3x - 1) ln 4(3x2 - 2x)
C)
5 ln 4 (3x2 - 2x)
209) B)
10(3x - 1) ln 4 (3x2 - 2x)
D) ln 4 (3x - 1) (3x2 - 2x)
Solve the problem. 210) Assume the total revenue from the sale of x items is given by R(x) = 32 ln (8x + 1), while the total cost to produce x items is C(x) = x/5. Find the approximate number of items that should be manufactured so that profit, R(x) - C(x), is maximum. A) 211 items B) 261 items C) 51 items D) 160 items
28
210)
211) Suppose that the population of a certain type of insect in a region near the equator is given by
211)
212) Suppose that the demand function for x units of a certain item is p = 90 + 210 ln(x + 5) , where p is
212)
P(t) = 12 ln (t + 12), where t represents the time in days. Find the rate of change of the population when t = 4. A) 0.8 insects B) 1.0 insects C) 1.6 insects D) 3.0 insects
x
the price per unit, in dollars. Find the marginal revenue.
2
A) dR = 90 + 210
B) dR = 210[x - [ln(x + 5) ] ] dx
x2 ln(x + 5)
C) dR = 210 [x - (x + 5) ln(x + 5)]
D) dR = 90 +
210 ln(x + 5)
dx
x+5
dx
dx
x2 (x + 5)
213) The population of coyotes in the northwestern portion of Alabama is given by the formula
213)
p(t) = (t2 + 100) ln(t + 2) , where t represents the time in years since 2000 (the year 2000 corresponds to t = 0). Find the rate of change of the coyote population in 2,002 (t = 2). A) 26 coyotes/year B) 6 coyotes/year
C) 81 coyotes/year
D) 32 coyotes/year
214) Students in a math class took a final exam. They took equivalent forms of the exam in monthly
214)
intervals thereafter. The average score S(t), in percent, after t months was found to be given by S(t) = 77 - 19 ln (t + 1), t 0. Find S'(t). A) S'(t) = - 19 ln 1 B) S'(t) = - 19 t+ 1 t+ 1
C) S'(t) = 19
D) S'(t) = 77 - 19
t+ 1
t+ 1
215) Suppose that the population of a town is given by
215)
P(t) = 8 ln 8t + 5, where t is the time in years after 1980 and P is the population of the town in thousands. Find P'(t). A) P'(t) = 8 B) P'(t) = 4 8t + 5 8t + 5
C) P'(t) = 32 ln 8t + 5
D) P'(t) = 32
8t + 5
8t + 5
29
Answer Key Testname: UNTITLED4
1) B 2) B 3) B 4) D 5) A 6) B 7) C 8) A 9) A 10) C 11) C 12) C 13) A 14) A 15) D 16) A 17) D 18) D 19) B 20) A 21) C 22) B 23) B 24) A 25) C 26) D 27) A 28) B 29) B 30) A 31) B 32) B 33) C 34) C 35) D 36) C 37) B 38) C 39) D 40) B 41) B 42) A 30
Answer Key Testname: UNTITLED4
43) C 44) C 45) B 46) C 47) C 48) A 49) C 50) B 51) A 52) A 53) B 54) D 55) B 56) D 57) C 58) D 59) D 60) B 61) A 62) C 63) C 64) C 65) D 66) B 67) D 68) B 69) D 70) B 71) A 72) C 73) D 74) B 75) A 76) B 77) C 78) D 79) C 80) B 81) B 82) B 83) B 84) A 31
Answer Key Testname: UNTITLED4
85) A 86) Answers will vary. Even if it is true that an increase in price is responsible, an increase in sales may also be partly or wholly responsible. 87) Answers may vary. Maximum profit occurs. 88) Marginal revenue is zero. 89) If f(x) increasing then x f (x) - f(x) > 0 x x2
x f (x) - f(x) > 0 x f (x) > f(x) f(x) f (x) > x
90) Product Rule:
h (x) = f (x)g(x) + f(x)g (x) h (-5) = f (-5)g(-5) + f(-5)g (-5) = (-1)(2) + (-3)(-2) h (-5) = 4 91) When taking the derivative of f(x) = u(x) , the correct formula for the quotient rule is f (x) = v(x)·u (x) - u(x)·v (x) . v(x) [v(x)]2 Notice that the two terms in the numerator are subtracted, not added.
92) A 93) A 94) C 95) B 96) B 97) A 98) D 99) D 100) B 101) B 102) B 103) B 104) C 105) A 106) B 107) D 108) C 109) C 110) D 111) C 112) C 113) C 114) B 115) D 116) C 32
Answer Key Testname: UNTITLED4
117) A 118) C 119) A 120) A 121) C 122) A 123) D 124) B 125) D 126) D 127) C 128) D 129) C 130) A 131) A 132) C 133) C 134) D 135) C 136) D 137) C 138) B 139) D 140) D 141) A 142) C 143) C 144) B 145) C 146) The graph of g is translated 4 units to the right. The slopes are the same. 147) The graph of g is stretched vertically. The slope is 2 times as large. 148) f(x) = (4x - 3)3 = 64x3 - 144x2 + 108x - 27 f'(x) = 192x2 - 288x + 108 149) f(x) = g(x) = g(x) · (h(x))-1 h(x)
f'(x) = g(x)(-1)(h(x))-2 · h'(x) + (h(x))-1 · g'(x) -g(x) h'(x) g'(x) = + h(x) h 2 (x) =
-g(x) h'(x) + h(x) g'(x) h 2 (x)
=
h(x)g'(x) - g(x)h'(x) h 2 (x)
33
Answer Key Testname: UNTITLED4
150) D 151) B 152) D 153) A 154) D 155) B 156) D 157) D 158) B 159) D 160) C 161) B 162) C 163) A 164) D 165) C 166) D 167) B 168) A 169) D 170) C 171) B 172) D 173) D 174) B 175) C 176) C 177) A 178) B 179) C 180) C 181) B 182) C 183) D 184) A 185) A 186) A 187) C 188) B 189) A 190) D 191) C 34
Answer Key Testname: UNTITLED4
192) B 193) B 194) D 195) B 196) D 197) D 198) B 199) D 200) C 201) D 202) B 203) A 204) D 205) D 206) C 207) A 208) D 209) A 210) D 211) A 212) A 213) D 214) B 215) D
35
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the open intervals where the function is changing as requested. 1) Increasing
A) (-2, 2)
B) (-3, )
C) (-2, )
1)
D) (-3, 3)
2) Decreasing
A) (-3, -2)
2)
B) (- , -2)
C) (0, -2)
1
D) (- , -3)
3) Increasing
A) (- , 0)
3)
B) (0, 3)
C) (-1, 0)
D) (- , -1)
4) Decreasing
A) (- , 0)
4)
B) (-1, 0)
C) (0, 3)
D) (3, )
5) Increasing
A) ( ,-1), (2, )
5)
B) (-1, 2)
C) (-1, )
2
D) ( , -1)
6) Decreasing
A) (2, -1)
6)
B) (2, 1)
C) (-1, 2)
D) (1, 2)
7) Increasing
A) (-2, )
7)
B) (3, )
C) (-2, 0)
D) (3, 6)
8) Decreasing
A) (0, -2)
8)
B) (- , -2)
C) (0, 3)
3
D) (- , 3)
9) Increasing
A) (0, 3)
9)
B) (-3, 0), (3, )
C) ( , -3), (0, 3)
D) ( , -3), (3, )
10) Decreasing
A) (-3, 3)
10)
B) (-3, 0), (3, )
C) ( , -3), (0, 3)
4
D) (-3, 0)
Suppose that the function with the given graph is not f(x), but f (x). Find the open intervals where f(x) is increasing or decreasing as indicated. 11) Increasing 11)
A) (0, )
B) (2, )
C) (-2, 2)
D) ( , -2), (2, )
12) Decreasing
A) (0, )
12)
B) ( , -1), (1, )
C) ( , 0)
5
D) (-1, 1)
13) Increasing
A) ( , -2), (2, )
13)
B) (-2, 2)
C) (2, )
D) (0, )
14) Decreasing
A) (-1, 2)
14)
B) ( , -1), (-1, 2)
C) ( , 2)
6
D) ( , 2)
15) Increasing
15)
A) ( , ) C) ( , -3), (-3, 3), (3, )
B) ( , -3), (3, ) D) (-3, 3)
Find all the critical numbers of the function. 16) y = 2.1 - 3.6x + 1.5x2
A) 7 12
16)
B) 1 2
C) - 7 10
D) 6 5
17) f(x) = 2 x3 - 1 x2 - 15x + 2 3
17)
2
A) - 5 , 5 2 2
18) f(x) = 4x3 + 6x2 - 72x + 9 A) 3, -2
B) -3, 5
C) 3
2
D) - 5 , 3 2
18) B) 12
C) -3, 2
D) -2
19) f(x) = 6x
19)
x+1
A) 1
B) -1
C) 6, 0
D) None
C) 10
D) 2 5
20) f(x) = (x + 2)1/5 A) -2
20) B) 2
21) f(x) = xe-2x A) 1 2
21) B) e-2x
C) 0
7
D) -2
22) y = x2/5 - x7/5
22)
A) 0, 2
B) - 2 , 0
7
C) 2
5
D) 2
7
5
Find the open interval(s) where the function is changing as requested. 23) Increasing; y = 7x - 5 A) (- , ) B) (-5, ) C) (-5, 7)
23) D) (- , 7)
24) Increasing; f(x) = 0.25x2 - 0.5x A) (1, ) B) (- , )
C) (- , -1)
D) (-1, 1)
25) Increasing; f(x) = x2 - 2x + 1 A) (0, ) B) (1, )
C) (- , 0)
D) (- , 1)
24)
25)
26) Increasing; y = x4 - 18x2 + 81 A) (- , 0) B) (-3, 3) 27) Increasing; f(x) =
26) C) (-3, 0)
D) (-3, 0), (3, )
1
27)
x2 + 1
A) (- , 0)
B) (0, )
C) (1, )
28) Decreasing; f(x) = - x + 3 A) (-3, ) B) (- , -3)
D) (- , 1) 28)
C) (- , 3)
D) (3, )
29) Decreasing; f(x) = x3 - 4x A) - , - 2 3
29) B) 2 3 ,
3
C) - 2 3 , 2 3
3
3
3
D) (- , )
30) Decreasing; f(x) = x + 1
30)
x-4
A) (- , -4)
B) (- , 4), (4, )
31) Increasing; y = x2 + 10 A) (- , 0) B) (-1, )
A)
D) none
C) (0, )
D) none
31)
32) Decreasing; y = x3/5 + x8/5 3 , - , (0, ) 8
C) (- , 1), (1, )
32)
B) 0, 3 8
C) ( , 0), 3 , 8
8
D)
3 ,8
Solve the problem. 33) Suppose the total cost C(x) to manufacture a quantity x of insecticide (in hundreds of liters) is given by C(x) = x3 - 27x2 + 240x + 750. Where is C(x) decreasing?
A) (0, 750)
B) (10, 750)
C) (8, 10)
33)
D) (8, 750)
34) A manufacturer sells telephones with cost function C(x) = 6.14x - 0.0002x2, 0 x 950 and
34)
35) The cost of a computer system increases with increased processor speeds. The cost C of a system
35)
36) The number of people P(t) (in hundreds) infected t days after an epidemic begins is approximated
36)
revenue function R(x) = 9.2x - 0.002x2 , 0 x 950. Determine the interval(s) on which the profit function is increasing. A) (0, 850) B) (850, 950) C) (50, 800) D) (0, 7,850)
as a function of processor speed is estimated as C(s) = 13s2 - 5s + 1,100, where s is the processor speed in MHz. Determine the intervals where the cost function C(s) is decreasing. A) ( , 0.2) B) Everywhere C) Nowhere D) (0.2, )
by
P(t) =
10 ln(0.79t + 1) . When will the number of people infected start to decline? 0.79t + 1
A) Day 3
B) Day 4
C) Day 2
D) Day 5
37) Suppose a certain drug is administered to a patient, with the percent of concentration in the bloodstream t hr later given by K(t) = drug increasing? A) (1, )
3t . On what time interval is the concentration of the t2 + 1
B) (0, 3)
C) (3, )
D) (0, 1)
38) The percent of concentration of a drug in the bloodstream x hours after the drug is administered is given by K(t) =
t
t2 + 9
37)
38)
. On what time interval is the concentration of the drug increasing?
A) (0, 3)
B) (-3, 3)
C) (1, 3)
D) (0, 4)
39) A probability function is defined by f(x) = 1 e-x2/4 . Give the intervals where the function is 2
increasing and decreasing. A) increasing on ( , )
B) increasing on (0, ); decreasing on ( , 0) D) decreasing on ( , )
C) increasing:on ( , 0); decreasing on (0, )
9
39)
Find the location and value of all relative extrema for the function.
40)
40)
A) Relative minimum of 0 at -2 ; Relative maximum of -1 at 2 ; Relative minimum of 2 at 1. B) Relative minimum of -3 at -1 ; Relative maximum of -1 at 2 ; Relative minimum of 2 at 1. C) Relative minimum of -1 at -3 ; Relative maximum of 2 at -1 ; Relative minimum of 1 at 2. D) Relative minimum of -1 at -3 ; Relative maximum of 2 at -1 ; Relative minimum of 0 at 2. 41)
41)
A) Relative maximum of 0 at 0 ; Relative maximum of 0 at 6. B) Relative maximum of 0 at 0 ; Relative minimum of 3 at -2; Relative minimum of 0 at 6. C) Relative minimum of -2 at 3. D) Relative minimum of -2 at 3 ; Relative maximum of 0 at 6. 42)
42)
A) None B) Relative maximum of 3 at -2 ; Relative minimum of 0 at 2. C) Relative maximum of 3 at -2. D) Relative minimum of 0 at 2.
10
43)
43)
A) Relative minimum of 1 at 0. C) None
B) Relative maximum of 0 at 1. D) Relative minimum of 2 at 1.
44)
44)
A) Relative maximum of 2 at 1. B) Relative minimum of -1 at -2. C) None D) Relative minimum of 1 at 2 ; Relative maximum of -1 at -2. 45)
45)
A) Relative minimum of 3 at 3 ; Relative minimum of 0 at 0 ; Relative maximum of -3 at -3. B) Relative minimum of 3 at 3 ; Relative maximum of -3 at -3. C) Relative minimum of -1 at -3. D) None
11
46)
46)
A) Relative maximum of 5 at -2 ; Relative maximum of 1 at 2. B) None C) Relative maximum of 5 at -2 ; Relative minimum of 0 at 0 ; Relative maximum of 1 at 2. D) Relative minimum of 0 at 0. 47)
47)
A) Relative minimum of 0 at -2 ; Relative maximum of 2 at 0. B) None C) Relative minimum of 0 at -2. D) Relative maximum of 2 at 0. 48)
48)
A) Relative minimum of 1 at 1. B) Relative maximum of -1 at -1 ; Relative minimum of 1 at 1. C) None D) Relative maximum of -1 at -1.
12
49)
49)
A) Relative minimum of -2 at -3 ; Relative maximum of 2 at 3. B) Relative minimum of -2 at -3 ; Relative minimum of 0 at 0 ; Relative maximum of 2 at 3. C) None D) Relative minimum of 0 at 0. Suppose that the function with the given graph is not f(x), but f (x). Find the locations of all extrema, and tell whether each extremum is a relative maximum or minimum.
50)
50)
A) Relative maxima at -2 and 2 B) Relative minimum at -4 C) Relative minimum at -2; relative maximum at 2 D) Relative maximum at -2; relative minimum at 2
13
51)
51)
A) Relative maximum at 1; relative minimum at 3 B) Relative maxima at 1 and 3 C) Relative maximum at 2 D) Relative minimum at 1; relative maximum at 3 52)
52)
A) No relative extrema C) Relative minimum at 0
B) Relative maxima at -3 and 3 D) Relative maximum at 0
14
53)
53)
A) Relative maximum at 1; relative minimum at -1 B) No relative extrema C) Relative maximum at 3; relative minimum at -3 D) Relative minimum at 0 54)
54)
A) Relative maximum at -3; relative minimum at 3 B) Relative maximum at 0 C) Relative minimum at 0 D) No relative extrema
15
55)
55)
A) Relative maximum at 0; relative minimum at -2 B) Relative minima at -3 and 1; relative maximum at -1 C) No relative extrema D) Relative maxima at -3 and 1; relative minimum at -1 56)
56)
A) Relative minimum at 2 C) Relative minima at -1 and 2
B) Relative minimum at 2 D) Relative maximum at 2
16
57)
57)
A) Relative maxima at -3 and 3 C) No relative extrema
B) Relative maxima at 0 D) Relative minima at -3 and 3
Find the x-value of all points where the function has relative extrema. Find the value(s) of any relative extrema. 58) f(x) = x2 + 2x - 3 58)
A) Relative maximum of -4 at -1. C) Relative minimum of -4 at -1.
B) Relative minimum of 0 at -2. D) Relative minimum of -2 at 0.
59) f(x) = 2 + 8x - x2 A) Relative maximum of 18 at 4. C) Relative maximum of 16 at - 4.
59) B) Relative maximum of 50 at 0. D) Relative minimum of 0 at 4.
60) f(x) = x3 - 3x2 + 1 A) No relative extrema. B) Relative maximum of 1 at 0. C) Relative maximum of 1 at 0; Relative minimum of -3 at 2. D) Relative maximum of 0 at 1; Relative minimum of -3 at -2.
60)
61) f(x) = x3 - 12x + 2 A) Relative maximum of 18 at -2; Relative minimum of -14 at 2. B) Relative maximum of 1 at 0; Relative minimum of -3 at 2. C) Relative maximum of 14 at -2; Relative minimum of 0 at 2. D) Relative minimum of -13 at 3.
61)
62) f(x) = 3x4 + 16x3 + 24x2 + 32 A) Relative maximum of 48 at -2; Relative minimum of 32 at 0. B) Relative minimum of 30 at -1. C) No relative extrema. D) Relative minimum of 32 at 0.
62)
17
2
63) f(x) = x + 1
63)
x2
A) Relative maximum of 50 at 0. B) Relative maximum of 50 at 0 ; Relative minimum of 0 at 10. C) Relative minimum of 0 at 10. D) No relative extrema. 64) f(x) =
1
64)
x2 - 1
A) Relative maximum of 0 at 1. C) No relative extrema.
B) Relative minimum of -1 at 0. D) Relative maximum of -1 at 0.
65) f(x) = x2/5 - 1 A) Relative minimum of -2 at 0. C) Relative minimum of -1 at 0.
B) Relative maximum of 2 at 10. D) No relative extrema.
65)
66) f(x) = x4/3 - x2/3
66)
A) Relative maximum of 0 at 0; Relative minimum of - 1 at 2 and - 2 4 4 4 B) Relative maximum of 0 at 0; Relative maximum of - 1 at 4
2 4
C) No relative extrema. D) Relative minimum of of - 1 at 4
67) f(x) =
2 4
1 2 x +1
67)
A) No relative extrema. C) Relative minimum of 0.5 at 0.
B) Relative maximum of 1 at 0. D) Relative maximum of 0 at 1.
68) f(x)= (ln x)2 , x > 0 A) (-1, 0), relative minimum C) (-1, -1) relative maximum
B) (1, -1), relative maximum D) (1, 0), relative minimum
69) f(x) = ln x - x, x > 0 A) (1, 0), relative minimum C) (-1, -1) relative maximum
B) (-1, 0), relative minimum D) (1, -1), relative maximum
68)
69)
70) f(x) = x + ln x A) (1, -1), relative maximum C) (-1, -1) relative maximum
70) B) (-1, 0), relative minimum D) (1, 0), relative minimum
18
71) f(x) = x ln x , x > 0
71)
A) - 1 , 1 , relative minimum e e
B) - 1 , - 1 , relative maximum e e
C) 1 , 1 , relative maximum
D) 1 , - 1 , relative minimum
e e
e
72) f(x) = (ln 3x)2 , x > 0
e
72) 1 , 0 , relative minimum 3
A) (-2, 0), relative minimum
B)
C) (1, 0), relative minimum
D) (3e, 0), relative minimum
73) f(x) = 2xe-x
73)
A) (-1, -2e), relative minimum
B) (-1, -2e), relative maximum
C) 1, 2 , relative maximum
D) 1, 2 , relative minimum
e
e
74) f(x) = xe3x A)
74)
1 1 ,, relative maximum 3 3e
B)
C) - 1 , - 1 , relative minimum 3
1 e , , relative minimum 3 3
D) - 1 , - e , relative maximum
3e
3
3
75) f(x) = x3 ex + 5 A) Relative maximum of 5 at 0; relative minimum of 3.66 at -3 B) Relative minimum of 3.66 at -3 C) No relative extrema D) Relative maximum of 6.34 at -3; relative minimum of 5 at 0 76) f(x) = x
75)
7
76)
6lnx
A) Relative minimum of 7 e at e1/7 6
B) Relative minimum of - 7 e-1 at e-1/7 6
C) Relative maximum of 0 at 0; relative minimum of 7 e at e1/7 6
D) Relative minimum of 0 at 0 Use the derivative to find the vertex of the parabola. 77) y = -2x2 + 8x - 7
A) (2, -1)
77)
B) (-2, -1)
C) (2, 1)
19
D) (-2, 1)
78) y = -3x2 + 12x + 5 A) (-2, -17)
78) B) (-2, 17)
C) (2, 17)
D) (2, -17)
Use a graphing calculator to find the location of all relative extrema (to three decimal places). 79) f(x) = x4 - 3x3 - 21x2 + 74x + 32
79)
A) Relative maximum at x = 1.614; Relative minima at x = -3.07 and x = 3.732 B) Relative maximum at x = 1.604; Relative minima at x = -3.089 and x = 3.735 C) Relative maximum at x = 1.623; Relative minima at x = -3.125 and x = 3.701 D) Relative maximum at x = 1.622; Relative minima at x = -3.008 and x = 3.804 80) f(x) = x4 - 4x3 - 53x2 - 86x - 45 A) Relative maximum at x = 0.997; relative minima at x = -3.251 and x = 7.158 B) Relative maximum at x = 1.043; relative minima at x = -3.278 and x = 7.187 C) Relative maximum at x = 0.876; relative minima at x = -3.24 and x = 7.159 D) Relative maximum at x = -0.944; relative minima at x = -3.192 and x = 7.136
80)
81) f(x) = x5 - 15x4 - 3x3 - 172x2 + 135x - 0.038 A) Relative maximum at x = 0.424; relative minima at x = -0.449and x = -12.469 B) Relative maximum at x= 0.379; relative minimum at x = 12.565 C) Relative maximum at x = 0.379; relative minima at x = -0.472 and x = 12.565 D) Relative maximum at x = 0.331; relative minimum at x = -12.526
81)
Solve the problem. 82) The annual revenue and cost functions for a manufacturer of grandfather clocks are approximately R(x) = 520x - 0.03x2 and C(x) = 160x + 100,000, where x denotes the number of clocks made. What is the maximum annual profit? A) $1,180,000 B) $980,000
C) $1,080,000
82)
D) $1,280,000
83) The annual revenue and cost functions for a manufacturer of precision gauges are approximately
83)
84) Find the number of units, x, that produces the maximum profit P, if C(x) = 75 + 84x and
84)
R(x) = 480x - 0.02x2 and C(x) = 160x + 100,000, where x denotes the number of gauges made. What is the maximum annual profit? A) $1,280,000 B) $1,480,000 C) $1,380,000 D) $1,180,000
p = 88 - 2x. A) 84 units
B) 336 units
C) 1 units
85) Find the maximum profit P if C(x) = 80 + 4x and p = 44 - 2x. A) $200 B) $120 C) $880
D) 4 units 85) D) $800
86) Find the price p per unit that produces the maximum profit P if C(x) = 60 + 60x and p = 96 - 2x. A) $40 B) $74 C) $36 D) $78
20
86)
87) P(x) = -x3 + 15x2 - 48x + 450, x 3 is an approximation to the total profit (in thousands of dollars)
87)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit. A) 10 hundred thousand B) 8 hundred thousand
C) 5 hundred thousand
D) 3 hundred thousand
88) S(x) = -x3 + 6x2 + 288x + 4000, 4 x 20 is an approximation to the number of salmon swimming
88)
upstream to spawn, where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon. A) 8°C B) 4°C C) 20°C D) 12°C
Find f"(x) for the function. 89) f(x) = 4x2 + 2x - 2
A) 4 90) f(x) = 4x4 - 8x2 + 6 A) 16x2 - 16
89) B) 8x + 2
C) 8
D) 0
B) 16x2 - 16x
C) 48x2 - 16x
D) 48x2 - 16
90)
91) f(x) = 2x3/2 - 6x1/2 A) 3x1/2 - 3x-1/2 C) 3x-1/2 + 3x-3/2 92) f(x) =
91) B) 1.5x-1/2 + 1.5x-3/2 D) 1.5x1/2 + 1.5x-1/2
1 2 x -1 2
A) 6x + 2
(x2 - 1)4
92) 2
A)
x3/2
94) f(x) = 6x3 - 8x2 + 6 A) 16x - 36
D) 6x + 2
2x3/2 + 1
8x3/2 + 1
8x3/2 - 1
(x2 - 1)3
9
4(3x - 7)3/2
(x2 - 1)4
(x2 - 1)3
93) B)
C)
x3/2
4x3/2
D)
4x3/2
94) B) 16x - 24
95) f(x) = 3x - 7 A)
2
C) 6x - 2
93) f(x) = x2 + x
2x3/2 - 1
2
B) 6x - 2
C) 36x - 16
D) 24x - 16 95)
B) -
10
C) -
4(3x - 7)3/2
9
4(3x - 7)3/2
D)
10
4(3x - 7)3/2
96) f(x) = x
96)
x+1
A) (x + 1)-2
B) (x + 1)-3
C) -2(x + 1)-2
21
D) -2(x + 1)-3
97) f(x) = 5e-x2
97)
A) 15xe-x2 + 10e-x2 C) 20x2 e-x2 + 5e-x2
B) 10x2 e-x2 D) 20x2 e-x2 - 10e-x2
98) f(x) = ln x
98)
9x
A) -3 - 2 ln x 9x
B) -7 - 2 ln x
C) - ln x
10x3
9x3
D) -3 + 2 ln x 9x3
Find the requested value of the second derivative of the function. 99) f(x) = x4 + 2x3 - 3x + 1; Find f (2).
A) 67
99)
C) 76
B) -71
100) f(x) = 3x4 - 8x2 + 7; Find f (0). A) -16 B) 0
D) 72 100)
C) -6
101) f(x) = 4x2 + 7x - 9; Find f (0). A) -8 B) 0
D) 7 101)
C) 8
D) 4
102) f(x) = x ; Find f (2).
102)
x+1
A) 2 3
B) - 2
C) 1
D) 0
B) 504e-32
C) 512e16
D) 496e-16
27
103) f(x) = 8e-x2 ; Find f (4) . A) 528e-32
9
103)
104) f(x) = ln x ; Find f (1).
104)
3x
A) ln 2
C) - 1
D) 9
C) 18
D) 36
106) f(4)(x) of f(x) = 3x5 - 3x2 - 4x + 1 A) 240x2 + 6 B) 240x + 6
C) 180x
D) 360x
107) f(4)(x) of f(x) = 5x6 - 7x4 + 4x2 A) 1,200x2 - 84x B) 1,200x2 - 84
C) 1,800x2 - 168x
D) 1,800x2 - 168
6
B) - 1
6
Find the indicated derivative of the function. 105) f (x) of f(x) = 6x3 + 4x2 - 5x
A) 36x + 18
105)
B) 18x +36
106)
107)
22
108) f (x) of f(x) = 1
108)
x+1
A) 6(x + 1)-3
B) -6(x + 1)-4
C) 6(x + 1)-4
D) -6(x + 1)-3
109) f (x) of f(x) = x
109)
x+1
A) -6(x + 1)-3
B) 6(x + 1)-4
C) -6(x + 1)-4
D) 6(x + 1)-3
Find the open intervals where the function is concave upward or concave downward. Find any inflection points.
110)
110)
A) Concave upward on (-1, ); concave downward on ( , 2); inflection point at (2, -3) B) Concave upward on (0, ); concave downward on ( , 0); inflection points at (-4, 0), (-1, 0), and
7 ,0 2
C) Concave upward on (0, ); concave downward on ( , 0); inflection point at (0, -1) D) Concave upward on (-1, ); concave downward on ( , 2); inflection points at (-1, 0) and (2, -3)
23
111)
111)
A) Concave upward on ( , -1) and (1, ); concave downward on (-1, 1); inflection points at (-1, -3) and (1, -2)
B) Concave upward on ( , -3) and (2, ); concave downward on (-3, 2); inflection points at
(-1, -3) and (1, -2) C) Concave upward on ( , -1) and (2, ); concave downward on (-1, 2); inflection point at (0, -1) D) Concave upward on ( , -1) and (1, ); concave downward on (-1, 1); inflection points at (-3, -5), (0, -1), and (2, -2)
112)
112)
A) Concave upward on ( , -2); concave downward on (-2, ); inflection point at (-2, 2) B) Concave upward on (-2, ); concave downward on ( , -2); inflection point at (-2, 2) C) Concave upward on ( , -2); concave downward on (-2, ); no inflection points D) Concave upward on (-2, ); concave downward on ( , -2); no inflection points Find the largest open intervals where the function is concave upward. 113) f(x) = x2 + 2x + 1
A) None
B) (-1, )
C) (- , -1)
24
113) D) (- , )
114) f(x) = -3x2 + 18x + 16 A) (3, ) 115) f(x) = 4x3 - 45x2 + 150x A) - 15 , 4
114) B) (- , )
C) (- , 3)
D) None 115)
B) - , - 15
C) - , 15
4
4
D) 15 , 4
116) f(x) = 6
116)
x
A) (- , 0) 117) f(x) = x3 - 3x2 - 4x + 5 A) None
B) (0, ), (- , 0)
C) (0, )
D) (- , ) 117)
B) (- , 1), (1, )
C) (- , 1)
D) (1, )
118) f(x) = 3
118)
x+2
A) (- , ) 119) f(x) =
B) (-2, )
C) None
D) (- , -2)
x 2 x +1
119) B) None D) (- , -1)
A) ( 3, ) C) (- , -1), (-1, ) 120) f(x) = x4 - 8x2 A) (- , -2 3/3) C) None 121) f(x) = 2x - 3e-x A) (- , 0 )
120) B) (- , -2 3/3), (2 3/3, ) D) (2 3/3, ) 121) B) None
C) ( - , )
D) ( 0 , )
Find any inflection points given the equation. 122) f(x) = 5x2 + 30x
122)
A) Inflection point at (-3,-45) C) No inflection points
B) Inflection point at (6,-30) D) Inflection point at (-6,-30)
123) f(x) = 2x3 + 12x2 + 18x A) Inflection point at - 2, - 4 C) Inflection point: (0, 0)
123) B) Inflection points at (-1, -8), (-3, 0) D) No inflection points
25
124) f(x) = 2x
124)
x2 + 1
A) Inflection points at (0, 0), -1 3, - 1 3 , 1 3, 1 3 2
2
B) Inflection points at (-1, -1), (1, 1) C) No inflection points D) Inflection points at (0, 0), (-1, -1), (1, 1) 125) f(x) = ex - 3e-x - 4x
125)
A) Inflection point at (2, -1)
B) Inflection point at (ln 3, 2 - 4 ln 3)
C) Inflection point at 1 ln 3, - 2 ln 3
D) Inflection point at (0, -2)
2
126) f(x) = ln (10 - x2 ) A) Inflection point at (-ln 10, 0) C) Inflection point at (0, -ln 10)
126) B) Inflection point at (0, ln 10) D) No inflection points
Suppose that the function with the given graph is not f(x), but f (x). Find the open intervals where the function is concave upward or concave downward, and find the location of any inflection points.
127)
127)
A) Concave upward on ( , -2) and (2, ); concave downward on (-2, 2); inflection points at -2
and 2 B) Concave upward on ( , -2) and (2, ); concave downward on (-2, 2); inflection points at -120 and 120 C) Concave upward on (-2, 2); concave downward on ( , -2) and (2, ); inflection points at -2 and 2 D) Concave upward on ( , 0); concave downward on (0, ); inflection point at 0
26
128)
128)
A) Concave upward on (-2, 2); concave downward on ( , -2) and (2, ); inflection points at -120 and 120 B) Concave upward on ( , 0); concave downward on (0, ); inflection point at 0
C) Concave upward on ( , -2) and (2, ); concave downward on (-2, 2); inflection points at -2 and 2
D) Concave upward on (-2, 2); concave downward on ( , -2) and (2, ); inflection points at -2 and 2
129)
129)
A) Concave upward on ( , -3) and (3, ); concave downward on (-3, 3); inflection points at -3
and 3 B) Concave upward on ( , -3) and (3, ); concave downward on (-3, 3); inflection points at -20 and 20 C) Concave upward on ( , 0); concave downward on (0, ); inflection point at 0
D) Concave upward on (-3, 3); concave downward on ( , -3) and (3, ); inflection points at -3 and 3
27
130)
130)
A) Concave upward on ( , -1) and (0, 1); concave downward on (-1, 0) and (1, ; ); inflection points at -1, 0, and 1 B) Concave upward on ( , 0); concave downward on (0, ); inflection point at 0
C) Concave upward on (-1, 0) and (1, ); concave downward on ( , -1) and (0, 1); inflection
points at -1, 0, and 1 D) Concave upward on (-1, 0) and (1, ); concave downward on ( , -1) and (0, 1); inflection points at -2, 0, and 2
Decide if the given value of x is a critical number for f, and if so, decide whether the point is a relative minimum, relative maximum, or neither. 131) f(x) = -x2 - 16x - 64; x = 8 131)
A) Critical number, relative maximum at (8, -144) B) Critical number but not an extreme point C) Critical number, relative minimum at (8, -144) D) Not a critical number 132) f(x) = x5 ; x = 0 A) Critical number, relative minimum at (0, 0) B) Not a critical number C) Critical number, relative maximum at (0, 0) D) Critical number but not an extreme point
132)
133) f(x) = (x2 - 6)(2x - 3); x = 1
133)
2
A) Critical number, relative minimum at 1 , 23 2
2
B) Critical number but not an extreme point C) Not a critical number D) Critical number, relative maximum at 1 , 23 2
2
28
134) f(x) = 2x3 - 3x2 - 12x + 18; x = 2 A) Not a critical number B) Critical number, relative minimum at (2, -2) C) Critical number, relative maximum at (2, -2) D) Critical number but not an extreme point
134)
135) f(x) = 3x4 - 4x3 - 12x2 + 24; x = 0 A) Not a critical number B) Critical number, relative minimum at (0, 24) C) Critical number, relative maximum at (0, 24) D) Critical number but not an extreme point
135)
136) f(x) = 4x5 - 5x4; x = 1 A) Critical number, relative maximum at (1, -1) B) Critical number but not an extreme point C) Critical number, relative minimum at (1, -1) D) Not a critical number
136)
137) f(x) = x2 - x - 6; x = 1
137)
2
A) Critical number, relative minimum at 1 ,- 25 2
4
B) Critical number, relative maximum at 1 ,- 25 2
4
C) Critical number but not an extreme point D) Not a critical number 138) f(x) = (x + 4)4 ; x = -4 A) Critical number; relative minimum at (-4, 0) B) Critical number but not an extreme point C) Critical number; relative maximum at (-4, 0) D) Not a critical number 139) f(x) = x19/9 + x10/9 ; x = 10
138)
139)
19
A) Not a critical number B) Critical number, relative maximum at 10 , 0 19
C) Critical number but not an extreme point D) Critical number, relative minimum at 10 , 0 19
29
Solve the problem. 140) The percent of concentration of a certain drug in the bloodstream x hours after the drug is 3x administered is given by K(x) = . At what time is the concentration a maximum? x2 + 64
A) 8 hr
B) 3.2 hr
C) 3 hr
140)
D) 6.4 hr
141) Find the point of diminishing returns (x, y) for the function R(x) = 8,000 - x3 + 30x2 + 700x,
141)
142) The population of a certain species of fish introduced into a lake is described by the logistic
142)
0 x 20, where R(x) represents revenue in thousands of dollars and x represents the amount spent on advertising in tens of thousands of dollars. A) (14 , 20,936) B) (12, 18,992) C) (10 , 17,000) D) (48.26, -746.04)
equation
G(t) =
10,000 , 1 + 19e-1.2t
where G(t) is the population after t years. Find the point at which the growth rate of this population begins to decline. A) (3.53, 5,000) B) (3.37, 7,500) C) (2.45, 5,000) D) (4.85, 7,500)
The function gives the distances (in feet) traveled in time t (in seconds) by a particle. Find the velocity and acceleration at the given time. 143) s = 1 , t = 4 143) t+ 4
A) v = - 2 ft/s, a = 1 ft/s2
B) v = 2 ft/s, a = - 1 ft/s2
C) v = - 1 ft/s, a = 2 ft/s2
D) v = 1 ft/s, a = - 2 ft/s2
512 64
64
512
512
64
64
512
144) s = 9t3 + 7t2 + 2t + 4, t = 3 A) v = 123 ft/s, a = 95 ft/s2 C) v = 287 ft/s, a = 176 ft/s2
B) v = 95 ft/s, a = 123 ft/s2 D) v = 176 ft/s, a = 287 ft/s2
145) s = 9t3 + 6t2 + 6t + 2, t = 1 A) v = 45 ft/s, a = 66 ft/s2 C) v = 15 ft/s, a = 39 ft/s2
B) v = 39 ft/s, a = 15 ft/s2 D) v = 66 ft/s, a = 45 ft/s2
146) s = 2t3 - 3t2 + 8t - 8, t = 3 A) v = 30 ft/s, a = 44 ft/s2 C) v = 0 ft/s, a = 44 ft/s2
B) v = 44 ft/s, a = 30 ft/s2 D) v = 44 ft/s, a = 0 ft/s2
144)
145)
146)
30
147) s = t2 - 5, t = 3
147)
A) v = 3 ft/s, a = - 5 ft/s2 2 8
B) v = 5 ft/s, a = - 3 ft/s2 8 2
C) v = - 3 ft/s, a = 5 ft/s2
D) v = - 5 ft/s, a = 3 ft/s2
2
8
8
2
Provide the proper response. 148) True or false? If the graph of a function f is concave up on its entire domain, then f' is increasing. A) True B) False
149) True or false? If the graph of a function f is concave up on its entire domain, then f' is decreasing. A) True B) False
148)
149)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
150) Explain why every polynomial function of the form
150)
151) Show that a function of the form
151)
152) You are applying the second derivative test and you find that f'(a) = 0 and f''(a) = 0. What
152)
153) Give an example of a function for which the first derivative is always positive and the
153)
154) Give an example of a function for which the first derivative is always negative and the
154)
155) Give an example of a function for which the first derivative and the second derivative are
155)
156) Give an example of a function for which the first derivative and the second derivative are
156)
f(x) = ax3 + bx2 + cx + d, a 0, must have exactly one point of inflection.
f(x) = ax4 + bx3 + cx2 + dx + e has no points of inflection whenever 8ac > 3b2 .
does that tell you? What would be your next step?
second derivative is always negative.
second derivative is always positive.
both always positive.
both always negative.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Sketch the graph and show all extrema, inflection points, and asymptotes where applicable.
31
157) f(x) = 2x3 - 9x2 + 12x
157)
B) Rel max (1, 5), Rel min: (2, 4)
A) No extrema
Inflection point: (0, 0)
Inflection point:
C) Rel min: (1, 10)
3 9 , 2 2
D) Rel max: (0, 0), Rel min: (0, 0)
No inflection points
Inflection point: (0, 0)
32
158) f(x) = 6x2 + 12x
158)
A) Rel min: (-2, -12)
B) Rel min: (-1, -6)
C) Rel min: (2, -12)
D) Rel min: (1, -6)
No inflection points
No inflection points
No inflection points
No inflection points
33
159) f(x) = x1/3(x2 - 63)
159)
A) Rel max: (0,0), Rel min: ± 27, - 27 2
Inflection point: ±3, - 5
B) Rel min: (0, 0)
No inflection points
34
C) No extrema
Inflection point: (0, 0)
D) Rel max: (-3, 54
3
3), Rel min: (3, -54 Inflection point: (0,0)
3
3)
2 160) f(x) = x
160)
x2 + 5
35
B) Rel min: 0, 1
A) Rel min: (0, 0)
5
No inflection points
No inflection points
C) Rel min: 0, - 1
D) Rel min: (0, 0)
5
Inflection points: -
No inflection points
161) f(x) =
1
15 1 , , 3 4
15 1 , 3 4
161)
16 - x2
36
A) Rel max: 0, 1
B) Rel min: 0, 1
4
4
No inflection points
No inflection points
C) Rel min: (0, 1)
D) Rel max: (0, 1)
No inflection points
162) f(x) =
No inflection points
1 2 x - 2x - 15
162)
37
A) Rel min: - 7 , 1
B) Rel min: - 1, 1
5 16
16
No inflection points
No inflection points
C) Rel max: - 1, - 1
D) Rel max: - 7 , - 1
16
5
No inflection points
16
No inflection points
163) f(x) = 5x
163)
x2 - 9
38
B) Rel max: 0, - 1
A) No extrema
9
Inflection point: (0, 0)
No inflection points
C) Rel min: 0, 1
D) No extrema
9
Inflection point: (0, 0)
No inflection points
164) f(x) = ln x - 4
164)
39
A) No extrema
B) No extrema
C) No extrema
D) No extrema
No inflection points
No inflection points
No inflection points
No inflection points
165) f(x) = ln (x2)
165)
40
A) Rel min: (1, 0)
B) No extrema
C) No extrema
D) Rel min: (-3, 0)
Inflection point: (1, 0)
No inflection points
No inflection points
No inflection points
166) f(x) = ex - 5e-x - 6x
166)
41
A) Rel min: 1 ln 5, - 3 ln 5 2
No inflection points
B) Rel min: (2, -4)
No inflection points
C) Rel max: (0, -4), Rel min: (ln 5, 4 - 6 ln 5) Inflection point:
1 ln 5, - 3 ln 5 2
42
D) Rel max: (0, -4), Rel min: (ln 5, 4 - 6 ln 5) No inflection points
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch a graph of a single function that has these properties. 167) a) Continuous and differentiable for all real numbers b) f (x) > 0 on (-3 , -1) and ( 2 , ) c) f (x) < 0 on (- , -3) and ( -1 , 2) d) f (x) > 0 on (- , -2) and ( 1 , ) e) f (x) < 0 on (-2 , 1) f) f (-3) = f (-1) = f (2) = 0 g) f (x) = 0 at (-2 , 0) and (1, 1)
43
167)
168) a) Continuous and differentiable for all real numbers
168)
169) a) Continuous for all real numbers
169)
b) f (x) < 0 on (- , -3 ) and ( 3 , ) c) f (x) > 0 on (-3 , 3) d) f (x) > 0 on (- , 0 ) e) f (x) < 0 on ( 0 , ) f) f (-3) = f (3) = 0 g) An inflection point at (0,0)
b) Differentiable everywhere except x = 0 c) f (x) < 0 on (- , 0) d) f (x) > 0 on ( 0 , ) e) f (x) < 0 on (- , 0) and (0, ) f) f(-2) = f (2) = 5 g) y-intercept and x-intercept at (0,0)
Answer the question. 170) The given graph is that of the derivative of a function f. Using information obtained from the graph of f' and the fact that f(-1) = -1, sketch the graph of f. Explain how you obtained the graph of f.
44
170)
171) The given graph is that of the derivative of a function f. Using information obtained from
171)
172) The given graph is that of the derivative of a function f. Using information obtained from
172)
173) The given graph is that of the derivative of a function f. Using information obtained from
173)
the graph of f' and the fact that f(-2) = 4, sketch the graph of f. Explain how you obtained the graph of f.
the graph of f' and the fact that f(1) = 1, sketch the graph of f. Explain how you obtained the graph of f.
the graph of f' and the fact that f(2) = -4, sketch the graph of f. Explain how you obtained the graph of f.
45
174) The given graph is that of the derivative of a function f. Using information obtained from
174)
4 the graph of f' and the fact that f(-1) = - and f(-3) = 0, sketch the graph of f. Explain how 3 you obtained the graph of f.
175) The given graph is that of the derivative of a function f. Using information obtained from
175)
2 2 the graph of f' and the fact that f(-1) = - and f(1) = , sketch the graph of f. Explain how 3 3 you obtained the graph of f.
176) The graph of f'(x) is a parabola with vertex at (a, 0). The parabola opens up. What does
176)
177) The graph of f'(x) is a parabola with vertex at (a, 0). The parabola opens down. What does
177)
this information tell us about the concavity associated with f(x)? Explain.
this information tell us about the concavity associated with f(x)? Explain.
46
Answer Key Testname: UNTITLED5
1) A 2) A 3) B 4) B 5) A 6) D 7) B 8) D 9) C 10) B 11) D 12) B 13) B 14) B 15) C 16) D 17) D 18) C 19) D 20) A 21) A 22) A 23) A 24) A 25) B 26) D 27) A 28) A 29) C 30) B 31) C 32) D 33) C 34) A 35) A 36) C 37) D 38) A 39) C 40) C 41) C 42) B 47
Answer Key Testname: UNTITLED5
43) C 44) D 45) D 46) D 47) B 48) C 49) A 50) D 51) D 52) A 53) D 54) B 55) D 56) B 57) C 58) C 59) A 60) C 61) A 62) D 63) D 64) D 65) C 66) A 67) B 68) D 69) D 70) C 71) D 72) B 73) C 74) C 75) B 76) A 77) C 78) C 79) B 80) D 81) B 82) B 83) D 84) C 48
Answer Key Testname: UNTITLED5
85) B 86) D 87) B 88) D 89) C 90) D 91) B 92) D 93) D 94) C 95) C 96) D 97) D 98) D 99) D 100) A 101) C 102) B 103) D 104) B 105) D 106) D 107) D 108) B 109) B 110) C 111) A 112) C 113) D 114) D 115) D 116) C 117) D 118) B 119) A 120) B 121) B 122) C 123) A 124) A 125) C 126) D 49
Answer Key Testname: UNTITLED5
127) A 128) D 129) A 130) C 131) D 132) D 133) C 134) B 135) C 136) C 137) A 138) A 139) A 140) A 141) C 142) C 143) C 144) C 145) A 146) B 147) A 148) A 149) B 150) Answers may vary. f''(x) = 6ax + 2b, where a and b are real numbers. Setting f''(x) = 0 and solving for x will give you the x-coordinate of the point of inflection.
151) Answers may vary, but should involve finding f'' and then setting f'' = 0. Attempting to solve the resulting equation by use of the quadratic formula will lead to the desired result.
152) You were applying the second derivative test to determine if you have a local minimum or a local maximum at a. If both f'(a) and f''(a) equal 0, the test is inconclusive. You would then apply the first derivative test. 153) f(x) = x. 154) f(x) = x2 with domain (- , 0).
155) f(x) = x2 with domain (0, ). 156) f(x) = -x2 with domain (0, ). 157) B 158) B 159) D 160) D 161) B 162) D 163) D 164) D 165) C 166) C
50
Answer Key Testname: UNTITLED5
167)
168)
169)
170) Graphs and explanations will vary to some degree. The graph of f should look similar to the following.
51
Answer Key Testname: UNTITLED5
171) Graphs and explanations will vary to some degree. The graph of f should look similar to the following.
172) Graphs and explanations will vary to some degree. The graph of f should look similar to the following.
173) Graphs and explanations will vary to some degree. The graph of f should look similar to the following.
52
Answer Key Testname: UNTITLED5
174) Graphs and explanations will vary to some degree. The graph of f should look similar to the following.
175) Graphs and explanations will vary to some degree. The graph of f should look similar to the following.
176) Concave down on (- , a); concave up on (a, ). 177) Concave up on (- , a); concave down on (a, ).
53
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the location of the indicated absolute extremum for the function. 1) Minimum
A) x = 5
B) x = 3
C) x = -3
1)
D) x = -5
2) Maximum
A) x = -4
2)
B) x = 1
C) No maximum
1
D) x = 4
3) Maximum
A) x = 5
3)
B) x = 3
C) x = 0
D) No maximum
4) Minimum
A) x = 1
4)
B) x = 2
C) x = -1
D) x = -2
5) Minimum
A) x = 2
5)
B) x = 0
C) x = -3
2
D) x = -4
6) Maximum
6)
A) x = 1
B) x = -1
C) No maximum
D) x = 4
7) Minimum
7)
A) x = -1
B) No minimum
C) x = 1
D) x = 2
8) Maximum
8)
A) x = 11 4
B) x = 0
C) x = -4
3
D) No maximum
9) Maximum
A) x = -1
9)
B) x = 2
C) No maximum
D) x = 0
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain. 10) f(x) = x2 - 4; [-1, 2] Maximum A) -3 at x = 1
B) 0 at x = -2
C) 0 at x = 2
D) -3 at x = -1
11) f(x) = x3 - 3x2 ; [0, 4]
11)
Minimum A) No absolute minimum
B) 16 at x = 4 D) 0 at x = 0
C) -4 at x = 2 12) f(x) = 1 x3 - 2x2 + 3x - 4; [-2, 5]
12)
3
Minimum A) - 62 at x = -2 3
10)
B) - 10 at x = 2
C) -4 at x = 0
3
D) - 8 at x = 1
13) f(x) = 1 ; [-4, 1]
3
13)
x+2
Minimum A) 1 at x = 0 2
B) 1 at x = 1 3
C) - 1 at x = -4 2
D) No absolute minimum
14) f(x) = x + 3 ; [-4, 4]
14)
x-3
Maximum
A) No absolute maximum
B) 1 at x = -4
C) 7 at x = 4
D) -1 at x = 0
7
4
15) f(x) = (x2 + 4)2/3; [-2, 2]
15)
Minimum A) No absolute minimum
B) 4 at x = 2 D) 2.5198 at x = 0
C) 2.924 at x = 1 16) f(x) = (x + 1)2 (x - 2); [-2, 1]
16)
Maximum A) -2 at x = 0
B) No absolute maximum D) 0 at x = -1
C) -4 at x = -2 17) f(x) = 3x4 + 16x3 + 24x2 + 32; [-3, 1] Maximum A) 75 at x = 1
17)
B) 59 at x = -3
C) 48 at x = -2
D) 32 at x = 0
18) f(x) = x4/3 - x2/3; [0, 2]
18)
Minimum
A) 0.9324 at x = 2
B) 0 at x = 1
C) - 1 at x =
D) No absolute minimum
4
2 4
19) f(x) = x2 e-0.25x; [3,10]
19)
Maximum A) 8.6615 at x = 8
B) 8.2085 at x = 10 D) 0 at x = 0
C) 4.2513 at x = 3
Graph the function on the indicated domain, and use the capabilities of your calculator to find the location and value of the indicated absolute extremum. 20) f(x) = (x - 3)(x + 3); [0, ) 20) Minimum A) -8.91 at x = 0.3 B) -9 at x = 0 C) -8.75 at x = 0.5 D) -8.51 at x = 0.7
21) f(x) = x(x - 9)2/3; ( , )
21)
Minimum A) 0 at x = 9.0
B) 5.4 at x = 8.5 D) No absolute minimum
C) 4.2 at x = 9.3 3
22) f(x) = x - 4x + 1 ; [-4, 1]
22)
x4 + x 2 + 5
Maximum A) -0.2 at x = 0.6
B) -0.2 at x = -3.9
C) -0.3 at x =0.9
5
D) 0.6 at x = -0.8
Find the absolute extrema if they exist as well as where they occur. 23) f(x) = -3x4 + 20x3 - 36x2 + 8
23)
A) Absolute maximum of -24 at x = 2; no absolute minima B) Absolute maximum of 8 at x = 0; no absolute minima C) No absolute extrema D) Absolute maximum of -11 at x = 1; no absolute minima 24) f(x) =
x-2
24)
x2 + 3x + 6
A) Absolute minimum of - 1 at x = -2; no absolute maxima B) No absolute extrema C) Absolute minimum of - 7 at x = -5; absolute maximum of 1 at x = 6 16
15
D) Absolute minimum of - 1 at x = -2; absolute maximum of 1 at x = 6 15
25) f(x) = 5 - x - 4/x, x > 0 A) Absolute minimum of 1 at x = 2; no absolute maximum B) Absolute maximum of 0 at x = 1; no absolute minimum C) Absolute maximum of 9 at x = -2; absolute minimum of 5 at x = 0 D) Absolute maximum of 1 at x = 2; no absolute minimum
25)
26) f(x) = 4x ln x A) No absolute minimum or maximum B) Absolute minimum of 0.0183 at x = -4; no absolute maximum C) Absolute maximum of 873.5704 at x = e-4; no absolute minimum D) Absolute minimum of 1.4715 at x = e-1 ; no absolute maximum
26)
Solve the problem.
27) P(x) = -x3 + 27 x2 - 60x + 100, x 5 is an approximation to the total profit (in thousands of dollars) 2
27)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit. A) 5 hundred thousand B) 4 hundred thousand
C) 4.5 hundred thousand
D) 5.5 hundred thousand
28) P(x) = -x3 + 12x2 - 36x + 400, x 3 is an approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit. A) 3 hundred thousand B) 6 hundred thousand
C) 2 hundred thousand
D) 7 hundred thousand
6
28)
29) P(x) = -x3 + 24x2 - 144x + 50, x 2 is an approximation to the total profit (in thousands of dollars)
29)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit. A) 10 hundred thousand B) 2 hundred thousand
C) 12 hundred thousand
D) 4 hundred thousand
30) P(x) = -x3 + 12x2 - 21x + 100, x 4 is an approximation to the total profit (in thousands of dollars)
30)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit. A) 10 hundred thousand B) 7 hundred thousand
C) 4 hundred thousand
D) 13 hundred thousand
31) P(x) = -x3 + 15x2 - 48x + 450, x 3 is an approximation to the total profit (in thousands of dollars)
31)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit. A) 8 hundred thousand B) 10 hundred thousand
C) 3 hundred thousand
D) 5 hundred thousand
32) The graph gives the profit P(x) as a function of production level. Use graphical optimization to estimate the production level that gives the maximum profit per item produced.
A) 5 units
B) 3 units
C) 6 units
7
D) 4 units
32)
33) In a certain state, the rate (per 500,000 inhabitants) at which automobiles were stolen each year during the years 1990 - 2000 are given in the figure. Consider the closed interval [1990, 2000].
A (1990, 171) B (1991, 204) C (1992, 255)
D (1993, 281) E (1994, 211) F (1995, 141)
G (1996, 188) H (1997, 255) K (1998, 247)
L (1999, 238) M (2000, 272)
Give all relative maxima and minima on the interval and the years when they occur. A) Relative maxima of 281 in 1993 and 255 in 1997 Relative minima of 171 in 1990, 141 in 1995, 238 in 1999 B) Relative maxima of 281 in 1993, 255 in 1997, 272 in 2000 Relative minima of 171 in 1990, 141 in 1995, 238 in 1999 C) Relative maxima of 281 in 1993, 255 in 1997, 272 in 2000 Relative minima of 141 in 1995 and 238 in 1999 D) Relative maxima of 281 in 1993 and 255 in 1997 Relative minima of 141 in 1995 and 238 in 1999
8
33)
34) In a certain state, the rate (per 500,000 inhabitants) at which automobiles were stolen each year
34)
during the years 1990 - 2000 are given in the figure. Consider the closed interval [1990, 2000].
A (1990, 171) B (1991, 204) C (1992, 255)
D (1993, 281) E (1994, 211) F (1995, 143)
G (1996, 188) H (1997, 258) K (1998, 247)
L (1999, 236) M (2000, 269)
Give the absolute maximum and minimum on the interval and the years when they occur. A) Absolute maximum of 269 in 2000 B) Absolute maximum of 281 in 1993 Absolute minimum of 143 in 1995 Absolute minimum of 171 in 1990 C) Absolute maximum of 258 in 1997 D) Absolute maximum of 281 in 1993 Absolute minimum of 171 in 1990 Absolute minimum of 143 in 1995
35) S(x) = -x3 - 9x2 + 165x + 1300, 5 x 20 is an approximation to the number of salmon swimming
35)
36) S(x) = -x3 - 3x2 + 72x + 900, x 2 is an approximation to the number of salmon swimming
36)
37) Researchers have discovered that by controlling both the temperature and the relative humidity in
37)
upstream to spawn, where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon. A) 20°C B) 19°C C) 6°C D) 5°C
upstream to spawn, where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon. A) 4°C B) 2°C C) 8°C D) 6°C
a building, the growth of a certain fungus can be limited. The relationship between temperature and relative humidity, which limits growth, can be described by R(T) = -0.00006T3 + 0.334T2 - 1.6572T + 97.086,
0 T 46, where R(T) is the relative humidity (in %) and T is the temperature (in °C). Find the temperature at which the relative humidity is minimized. A) 3,708.63°C B) 5.48°C C) -1.52°C D) 2.48°C
9
38) The velocity of a particle (in ft ) is given by v = t2 - 6t + 9, where t is the time (in seconds) for s
which it has traveled. Find the time at which the velocity is at a minimum. A) 6 sec B) 4.5 sec C) 9 sec
D) 3 sec
39) Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of 150 ft. A) 37.5 ft × 150 ft B) 12.5 ft × 37.5 ft
C) 37.5 ft × 37.5 ft
39)
D) 75 ft × 75 ft
40) An architect needs to design a rectangular room with an area of 77 ft2. What dimensions should he use in order to minimize the perimeter? A) 8.77 ft × 8.77 ft
38)
40)
B) 15.4 ft × 77 ft D) 8.77 ft × 19.25 ft
C) 19.25 ft × 19.25 ft
41) A piece of molding 170 cm long is to be cut to form a rectangular picture frame. What dimensions will enclose the largest area? A) 42.5 cm × 42.5 cm
41)
B) 13.04 cm × 42.5 cm D) 34 cm × 34 cm
C) 13.04 cm × 13.04 cm
42) Find two numbers whose sum is 150 and whose product is as large as possible. A) 10 and 140 B) 74 and 76 C) 75 and 75 D) 1 and 149
42)
43) Find two numbers x and y such that their sum is 300 and x2 y is maximized. A) x = 75, y = 225 B) x = 100, y = 200 C) x = 200, y = 100 D) x = 225, y = 75
43)
44) Of all numbers whose difference is 8, find the two that have the minimum product. A) 16 and 8 B) 0 and 8 C) 1 and 9 D) 4 and -4
44)
45) Maximize Q = xy2 , where x and y are positive numbers, such that x + y2 = 7.
45)
A) x =
7 7 ,y= 2 2
B) x = 7 , y = 2
7 2
C) x = 1, y = 6
D) x = 0, y = 7
46) If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a
46)
x certain city, where p(x) = 94 . How many candy bars must be sold to maximize revenue? 36
A) 3,384 thousand candy bars C) 3,384 candy bars
B) 1,692 thousand candy bars D) 1,692 candy bars
47) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $6 per foot for two
opposite sides, and $2 per foot for the other two sides. Find the dimensions of the field of area 670 ft2 that would be the cheapest to enclose.
A) 8.6 ft @ $6 by 77.7 ft @ $2 C) 44.8 ft @ $6 by 14.9 ft @ $2
B) 77.7 ft @ $6 by 8.6 ft @ $2 D) 14.9 ft @ $6 by 44.8 ft @ $2 10
47)
48) If the price charged for a bolt is p cents, then x thousand bolts will be sold in a certain hardware
48)
x store, where p = 46 . How many bolts must be sold to maximize revenue? 32
A) 1,472 thousand bolts C) 736 thousand bolts
B) 736 bolts D) 1,472 bolts
49) A hotel has 250 units. All rooms are occupied when the hotel charges $70 per day for a room. For
49)
50) A baseball team is trying to determine what price to charge for tickets. At a price of $10 per ticket,
50)
51) The stadium vending company finds that sales of hot dogs average 43,000 hot dogs per game
51)
52) Supertankers off-load oil at a docking facility shore point 5 miles offshore. The nearest refinery is
52)
every increase of x dollars in the daily room rate, there are x rooms vacant. Each occupied room costs $24 per day to service and maintain. What should the hotel charge per day in order to maximize daily profit? A) $172 B) $182 C) $160 D) $102
it averages 35,000 people per game. For every increase of $1, it loses 5,000 people. Every person at the game spends an average of $5 on concessions. What price per ticket should be charged in order to maximize revenue? A) $4.00 B) $14.00 C) $6.00 D) $2.00
when the hot dogs sell for $2.50 each. For each 50 cent increase in the price, the sales per game drop by 5000 hot dogs. What price per hot dog should the vending company charge to realize the maximum revenue? A) $3.65 B) $4.30 C) $1.80 D) $3.40
9 miles east of the docking facility. A pipeline must be constructed connecting the docking facility with the refinery. The pipeline costs $300,000 per mile if constructed underwater and $200,000 per mile if over land.
5 mi
9 mi Locate point B to minimize the cost of construction. A) Point B is 4.47 miles from Point A.
C) Point B is 2.50 miles from Point A.
11
B) Point B is 3.51 miles from Point A. D) Point B is 5.66 miles from Point A.
53) Suppose c(x) = x3 - 24x2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost per item of making x items. A) 13 items B) 14 items C) 12 items
53)
D) 11 items
54) Recent research has shown that the population f(S) of cod in the North Sea next year as a function
54)
of this year's population S (measured in thousands of tons) can be described by the Shepherd model, aS f(S) = 1 + (S/b)c
where a, b, and c are constants. The values of a, b, and c are 3.039, 248.71, and 3.23, respectively. Find the approximate value of this year's population that maximizes next year's population using this model. A) 4,000 tons B) 159,000 tons C) 194,000 tons D) 194 tons
55) Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of 143 ft. A) 35.75 ft × 143 ft
C) 35.75 ft × 35.75 ft
B) 11.92 ft × 35.75 ft D) 71.5 ft × 71.5 ft
56) An architect needs to design a rectangular room with an area of 66 ft2. What dimensions should he use in order to minimize the perimeter? A) 16.5 ft × 16.5 ft B) 8.12 ft × 16.5 ft
C) 8.12 ft × 8.12 ft
56)
D) 13.2 ft × 66 ft
57) A piece of molding 160 cm long is to be cut to form a rectangular picture frame. What dimensions will enclose the largest area? A) 12.65 cm × 12.65 cm
55)
57)
B) 12.65 cm × 40 cm D) 32 cm × 32 cm
C) 40 cm × 40 cm
58) A company wishes to manufacture a box with a volume of 32 cubic feet that is open on top and is
58)
59) From a thin piece of cardboard 20 in. by 20 in., square corners are cut out so that the sides can be
59)
twice as long as it is wide. Find the width of the box that can be produced using the minimum amount of material. A) 2.9 ft B) 5.8 ft C) 6.8 ft D) 3.4 ft
folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? Round to the nearest tenth, if necessary. A) 10 in. by 10 in. by 5 in.; 500 in.3 B) 6.7 in. by 6.7 in. by 6.7 in.; 296.3 in.3
C) 13.3 in. by 13.3 in. by 6.7 in.; 1,185.2 in.3
D) 13.3 in. by 13.3 in. by 3.3 in.; 592.6 in.3
12
60) A private shipping company will accept a box for domestic shipment only if the sum of its length
60)
and girth (distance around) does not exceed 102 in. What dimensions will give a box with a square end the largest possible volume?
A) 17 in. × 34 in. × 34 in. C) 34 in. × 34 in. × 34 in.
B) 17 in. × 17 in. × 85 in. D) 17 in. × 17 in. × 34 in.
61) Find the approximate number of batches (to the nearest whole number) of an item that should be
61)
62) A bookstore has an annual demand for 118,000 copies of a best-selling book. It costs $0.80 to store
62)
63) A certain company produces potting soil and sells it in 50 lb bags. Suppose that 200,000 bags are
63)
64) A local office supply store has an annual demand for 50,000 cases of photocopier paper per year.
64)
65) A book publisher wants to know how many times a year a print run should be scheduled.
65)
produced annually if 140,000 units are to be made. It costs $2 to store a unit for one year, and it costs $320 to set up the factory to produce each batch. A) 21 batches B) 23 batches C) 17 batches D) 15 batches
one copy for one year, and it costs $125 to place an order. Find the optimum number of copies per order. A) 6,789 copies B) 5,465 copies C) 6,072 copies D) 6,431 copies
to be produced each year. It costs $4 per year to store a bag of potting soil, and it costs $1000 to set up the facility to produce a batch of bags. Find the number of bags per batch that should be produced. A) 10,000 B) 14,121 C) 9574 D) 100,000
It costs $1 per year to store a case of photocopier paper, and it costs $60 to place an order. Find the optimum number of cases of photocopier paper per order. A) 2,449 B) 1,732 C) 775 D) 6,000,000
Suppose it costs $2,000 to set up the printing process, and the subsequent cost per book is so low it can be ignored. Suppose further that the annual warehouse cost is $3 times the maximum number of books stored. Assuming 6,000 copies of the book are needed per year how many books should be printed in each print run? A) 632 B) 1,000 C) 2,828 D) 2,000
13
66) Find the elasticity of demand E for the demand function q = 1,200 - 17p. 17p A) E = 17p - 1,200
B) E = 17p - 1,200 17p
17p 1,200 - 17p
D) E = 1,200 - 17p
C) E =
66)
17p
67) Find the elasticity of demand E for the demand function q = 40,000 - 10p2 A) E = C) E =
2p2
B) E =
40,000 - p2
-p
D) E =
40,000 - 10p2
4,000 - p2 2p2
4,000 - p2
68) Find the elasticity of demand E for the demand function q = 17 - ln p A) E =
-p 17 - ln p
B) E =
1 17 - ln p
C) E =
67)
-2p2
-p 17p - ln p
68) D) E = 17 - ln p p2
69) Given the demand function q = 310 - 5p, calculate the elasticity of demand when p = 33. A) 0.88 B) 2.86 C) 0.35 D) 1.14
69)
70) Given the demand function q = 169 - 8p, determine the price where demand has unit elasticity. A) p = 5.28 B) p = 26 C) p = 13 D) p = 10.56
70)
71) The demand for ground chuck (hamburger) in a certain region of the United States is given by
71)
q = 1.74p-0.23. Is the demand for ground chuck elastic or inelastic?
A) Elastic C) The demand has unit elasticity.
B) Inelastic D) None of these
72) The demand for boneless chicken breast, in dollars per pound, is given by q = -0.7p + 6, where p
72)
represents the price per pound and q represents the average number of pounds purchased per week per customer. Determine the price at which the demand for boneless chicken breast is unit elastic. A) $8.57 per pound
B) $4.29 per pound C) $5.22 per pound D) The demand is not unit elastic at any price. Find dy/dx by implicit differentiation. 73) x3 + y3 = 5
A)
x2 y2
73) B) -
x2
2 C) - y x2
y2
14
2 D) y x2
74) x4/3 + y4/3 = 1 A) - y
1/3
x
74) B) - x
75) x1/3 - y1/3 = 1 2/3
A) x
D) x
x
2/3
1/3
y
C) - x
x
76) xy2 = 4
1/3
C) y
75) B) y
y
2/3
y
D) - y
2/3
x
76)
A) - y
2x
B) 2x
C) x
D) - 2y
B) x x-y
C) x y-x
D) y y-x
y
77) 2xy - y2 = 1
2y
x
77)
A) y x-y
78) x3 + 3x2y + y3 = 8 A) -
1/3
y
x2 + 2xy x 2 + y2
78) B)
x2 + 2xy
C)
x 2 + y2
x2 + 3xy x 2 + y2
D) -
x2 + 3xy x 2 + y2
79) x + y = x2 + y2
79)
x-y
2
A) x(x - y) - y
x - y(x - y)2
80) y x + 1 = 4 A)
2
2
B) x(x - y) + y
C) x(x - y) + y
x - y(x - y)2
x + y(x - y)2
81) xy + x = 2
B) - 2y
C) -
B) 1 + y x
C) 1 + x y
D) - 1 + x y
2 B) 2xy - y 2x2 y + x
2 C) 2xy + y 2x2 y - x
2 D) 2xy + y + 1 -2x2 y - x - 1
x+1
y 2(x + 1)
D) 2y
x+1
81)
A) - 1 + y x
82) xy + x + y = x2y2
2 A) 2xy - y - 1 -2x2 y + x + 1
82)
83) y3 ex + x = y4x
83)
4 3 x A) dy = y - y e - 1 dx 3y2 ex - 4xy3 dx
x + y(x - y)2
80)
y 2(x + 1)
C) dy =
2
D) x(x - y) - y
y4 - 1 B) dy = dx 3y2 ex - 4xy3 + 1
y4 - 1 3y2 ex - 4xy3
4 3 x D) dy = y - y e - 1 dx
15
3y2 ex - 4xy3 - 1
84) x ln y + y = x6 y6
84)
5 6 - ln y
5 7 - y ln y
A) dy = 6x y dx
B) dy = 6x y
x - 6x6 y6 + y
dx
5 7 - ln y
C) dy = 6x y dx
x - 6x6 y6 + y 5 7 - y ln y - 1
D) dy = 6x y
x - 6x6 y6 + 1
dx
x - 6x6 y6 + y
Find the equation of the tangent line at the given point on the curve. 85) x2 + y2 + 2y = 0; (0, -2)
A) x = 0
B) y = -x
C) y = -2
86) 3x2 + 4xy + y2 + x - 2y = -7; (-1, 3) A) y = 3 B) y = -x - 1 87) x2 + y2 = 25; (-4, 3) A) y = - 4 x + 25 3
3
88) xy2 = 12; (3, -2)
89) 2xy - 2x + y = -14; (2, -2) A) y = - 6 x + 22 5 5
90) x2 + 3y2 = 13; (1, 2) A) y = 1 x + 5 3
86) C) y = -x + 3
D) x = -1 87)
B) y = - 4 x - 25
C) y = 4 x - 25
D) y = 4 x + 25
B) y = 1 x - 3 3
C) y = - 1 x + 3 3
D) y = - 1 x - 3 3
B) y = - 5 x + 22 6 6
C) y = 6 x - 22 5 5
D) y = 5 x - 22 6 6
3
3
3
3
3
3
89)
90) B) y = - 1 x + 7 3
91) xy2 = 4; (4, 1)
3
C) y = 1 x + 11
D) y = - 1 x + 13
C) y = -2x + 9
D) y = - 1 x + 3 2
6
6
6
6
91)
A) y = -8x + 33 92) 2xy - y2 = 1; (1, 1) A) x = 1 93) y x + 1 = 4; (3, 2) A) y = 1 x + 5 4 4
94) xy + x = 2; (1, 1)
B) y = - 1 x + 3 8 2
92) B) y = 1
C) y = x - 1
D) y = -x + 1
B) 2y = - 1 x + 7 2 2
C) y = - 1 x + 11 4 4
D) y = 1 x + 1 2 2
93)
94)
A) y = - 1 x + 3 2
D) y = -x - 2
88)
A) y = 1 x + 3 3
3
85)
2
B) y = 1 x + 1 2
C) y = 2x - 1
2
16
D) y = -2x + 3
Find the equation of the tangent line at the given value of x on the curve. 95) 3x2 + 4xy + y2 + x - 2y = -7, x = -1
A) x = -1
B) y = -x - 1
96) 2xy - 2x + y = -14, x = 2 A) y = 5 x - 22 6
6
C) y = -x + 3
D) y = 3 96)
B) y = - 5 x + 22
C) y = - 6 x + 22
D) y = 6 x - 22
B) x = 1
C) y = 1
D) y = x - 1
B) 2y = 1 x + 1 2 2
C) y = 1 x + 5 4 4
D) y = - 1 x + 11 4 4
C) y = -2x + 3
D) y = - 1 x + 3 2 2
6
97) 2xy - y2 = 1, x = 1 A) y = -x + 1
95)
6
5
5
5
97)
98) y x + 1 = 4, x = 3
98)
A) 2y = - 1 x + 7 2 2
99) xy + x = 2, x = 1
99)
A) y = 1 x + 1 2 2
B) y = 2x - 1
100) y3 + 2xy2 + 3 = 4y2 + x, x = 2 A) y = 1 x - 5 3
5
100)
B) y = 1 x - 7
3
5
C) y = - 1 x + 5
5
3
D) y = - 1 x - 1
3
3
3
101) y (1 - x) + x y + 2x = 5, x = 1
101)
4
A) y = 33 x - 9
B) y = - 11 x + 65
2
6
C) y = - 33 x + 51 2
6
D) y = - 33 x + 105
2
8
8
Solve the problem.
102) The demand equation for a certain product is 3p2 + q2 = 1,700, where p is the price per unit in dollars and q is the number of units demanded. Find dq/dp. A) dq/dp = -3p/q B) dq/dp = -p/3q C) dq/dp = -3q/p
D) dq/dp = -q/3p
103) The demand equation for a certain product is 2p2 + q2 = 1,200, where p is the price per unit in dollars and q is the number of units demanded. Find dp/dq. A) dp/dq = -p/2q B) dp/dq = -2q/p C) dp/dq = -2p/q
17
102)
D) dp/dq = -q/2p
103)
104) The correlation between respiratory rate and body mass in the first three years of life can be
104)
expressed by the function log R(w) = 1.87 - 0.34 log (w), where w is the body weight (in kg) and R(w) is the respiratory rate (in breaths per minute). Find R'(w) using implicit differentiation. A) R'(w) = -25.2w-1.34 B) R'(w) = -25.2w-0.66
C) R'(w) = -25.2w-0.34
D) R'(w) = 74.13w-1.34
105) The position of a particle at time t is given by s, where s3 + 10st + 4t3 - 10t = 0. Find the velocity ds/dt.
A) ds/dt = 10 + 10s - 12t
2
C) ds/dt = 10 - 10s - 12t
2
3s2 - 10t
C) -9
C) - 3
3
4
108) x4/3 + y4/3 = 2; dx/dt = 6, x = 1, y = 1
D) - 4 3
108)
B) 1 6
C) - 1 6
109) xy2 = 4; dx/dt = -5, x = 4, y = 1 8
D) 3 107)
B) 4
A) - 5
2
106)
107) x3 + y3 = 9; dx/dt = -3, x = 1, y = 2
A) 6
D) ds/dt = 10 + 10s - 12t 3s2 + 10t
Assume x and y are functions of t. Evaluate dy/dt. 106) xy + x = 12; dx/dt = -3, x = 2, y = 5 A) 9 B) -3
4
2
3s2 + 10t
3s2 - 10t
A) 3
B) ds/dt = 10 - 10s - 12t
D) -6
109)
B) - 8
C) 8
5
5
D) 5 8
110) x + y = x2 + y2 ; dx/dt = 12, x = 1, y = 0
110)
x-y
A) 1
12
B) - 1
C) -12
12
111) y x + 1 = 12; dx/dt = 8, x = 15, y = 3 A) - 4 3
D) 12
111)
B) 3 4
C) 4
D) - 3
C) 1
D) 5e 4(4 - e)
3
112) x2 ln y = 2 + xey; dx/dt = 5, x = 4, y = 1 A) 0
105)
4
112)
B) 5e 4-e
18
113) x2 ey - y2 ln x = 5; dx/dt = 1, x = 1, y = 2 2
A) 4 + e e2
113) 2
2
C) 4 - 2e
B) 4 - 2e
2
e2
D) 5
Solve the problem. 114) A company knows that unit cost C and unit revenue R from the production and sale of x units are R2 + 5,743. Find the rate of change of revenue per unit when the cost per unit related by C = 270,000 is changing by $14 and the revenue is $4,000. A) $523.40 B) $574.30
C) $280.00
114)
D) $472.50
115) Given the revenue and cost functions R = 36x - 0.9x2 and C = 5x + 10, where x is the daily
115)
production, find the rate of change of profit with respect to time when 15 units are produced and the rate of change of production is 8 units per day. A) $32.00 per day B) $233.60 per day
C) $221.00 per day
D) $72.00 per day
116) A product sells by word of mouth. The company that produces the product has noticed that
116)
revenue from sales is given by R(t) = 4 x, where x is the number of units produced and sold. If the revenue keeps changing at a rate of $900 per month, how fast is the rate of sales changing when 1,100 units have been made and sold? (Round to the nearest dollar per month.) A) $238,797/month B) $14/month
C) $14,925/month
D) $7,462/month
117) The average daily metabolic rate for a hippopotamus living in the wild can be expressed as a
117)
function of weight by m = 132.9w0.75, where w is the weight of the hippopotamus (in kg) and m is the metabolic rate (in kcal/day). Determine dm/dt for a 2,700-kg hippopotamus that is gaining weight at a rate of 13.5 kg/day. A) 9,700 kcal/day2 B) 14 kcal/day2
C) 187 kcal/day2
D) 249 kcal/day2
118) The energy cost of a speed burst as a function of the body weight of a dolphin is given by
118)
E = 43.5w-0.61, where w is the weight of the dolphin (in kg) and E is the energy expenditure (in
kcal/kg/km). Suppose that the weight of a 400-kg dolphin is increasing at a rate of 12 kg/day. Find the rate at which the energy expenditure is changing with respect to time. A) -8.2365 kcal/kg/km/day B) -0.0017 kcal/kg/km/day
C) -30.7749 kcal/kg/km/day
D) -0.0206 kcal/kg/km/day
119) Water is discharged from a pipeline at a velocity v given by v = 1,096p(1/2), where p is the
pressure (in psi). If the water pressure is changing at a rate of 0.251 psi/second, find the acceleration (dv/dt) of the water when p = 32 psi. A) 31 ft/s2 B) 96.87 ft/s2 C) 778.09 ft/s2 D) 24.32 ft/s2
19
119)
120) A zoom lens in a camera makes a rectangular image on the film that is 7 cm × 5 cm. As the lens
120)
121) One airplane is approaching an airport from the north at 163 km/hr. A second airplane
121)
122) A container, in the shape of an inverted right circular cone, has a radius of 3 inches at the top and
122)
123) A man 6 ft tall walks at a rate of 2 ft/s away from a lamppost that is 14 ft high. At what rate is the
123)
zooms in and out, the size of the image changes. Find the rate at which the area of the image begins to change (dA/df) if the length of the frame changes at 0.6cm/s and the width of the frame changes at 0.3 cm/s. A) 1.02 m2 /s B) 5.1 m2 /s C) 7 m2 /2 D) 5.7 m2 /s 5
approaches from the east at 257 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 36 km away from the airport and the westbound plane is 16 km from the airport. A) 1,078 km/hr B) 1,212 km/hr C) 104 km/hr D) 253 km/hr
a height of 6 inches. At the instant when the water in the container is 4 inches deep, the surface level is falling at the rate of -1.8 in./s. Find the rate at which water is being drained. A) -22.62 in.3 /s B) -33.93 in.3 /s C) -45.24 in.3 /s D) -21.6 in.3/s
length of his shadow changing when he is 70 ft away from the lamppost? A) 70 ft/s B) 3 ft/s C) 3 ft/s 3 5 10
D) 3 ft/s 2
124) Boyle's law states that if the temperature of a gas remains constant, then PV = c, where P is the
124)
pressure, V is the volume, and c is a constant. Given a quantity of gas at constant temperature, if V is decreasing at a rate of 14 in.3 /s, at what rate is P increasing when P = 80 lb/in.2 and V = 90 in.3 ?
A) 3600 lb/in.2 -s 7
B) 63 lb/in.2 -s
C) 64 lb/in.2 -s
4
81
D) 112 lb/in.2 -s 9
125) Electrical systems are governed by Ohm's law, which states that V = IR, where V = voltage,
125)
126) The volume of a sphere is increasing at a rate of 2 cm3 /s. Find the rate of change of its surface area
126)
I = current, and R = resistance. If the current in an electrical system is decreasing at a rate of 7 A/s while the voltage remains constant at 16 V, at what rate is the resistance increasing when the current is 28 A? A) 28 ohms/s B) 1 ohms/s C) 4 ohms/s D) 1 ohms/s 7 4
when its volume is
A) 4 cm2 /s
4 cm3 3
B) 4 cm2 /s
C) 8 cm2 /s
3
3
D) 1 cm2 /s 3
Find dy for the given values of x and x. 127) y = x3 + 2x; x = 2, x = 0.01
A) 0.007
127)
B) 0.07
C) 0.14
20
D) 0.014
128) y = 1 ; x = 10, x = -0.003
128)
x
A) 0.03 129) y =
B) 0.00003
C) 0.0003
D) 0.003
x2 ; x = 10, x = 0.1 x2 + 21
A) 146
1331
129)
B) 148
C) 142
1331
1331
D) 144
1331
130) y = x3 - 4x2 + 2x + 1; x = 8, x = -0.3 A) 39 B) -39
C) 37
D) -37
131) y = 2x + 3; x = 18, x = 0.5 A) 0.5 B) 1
C) 5
D) 0.1
130)
131)
132) y = 5x2 - 2x + 3; x = 2, x = - 1
132)
6
A) 3
C) 6
B) -6
D) -3
133) y = 2x5 - 3x2 + x - 1; x = -1, x = 1
133)
3
A) 25 3
B) 22
C) 17
3
3
D) 19 3
134) y = 4 + 3 x; x = 4, x = 0.5
134)
x4
A) 45
128
C) 47
B) .47
128
D) .45
135) y = 1 + 3 x2 + 3; x = 1, x = 0.2
135)
4x3
A) 0.05
B) 0.02
C) 0.15
D) 0.5
136) y = 3x - 7 ; x = 2, x = 0.1
136)
x-1
A) 0.4
B) 6
C) 0.6
Use the differential to approximate the quantity to four decimal places. 137) 101 A) 10.0500 B) 11.0000 C) 9.9500
138) 3.28 A) 1.8200
D) 4
137) D) 10.1000 138)
B) 2.1800
C) 1.6400 21
D) 1.2800
139) e0.49 A) 0.6,126
139) B) 1.6323
140) e0.122 A) 1.1298
C) .5100
D) 1.4900 140)
B) .8,780
141) ln 1.07 A) -0.0726
C) 0.8,851
D) 1.1220 141)
B) 0.0700
C) 0.0677
D) -0.0700
Solve the problem.
142) A company estimates that the revenue (in dollars) from the sale of x units of dog houses is given
142)
by C(x) = 765 +0.03x + 0.0003x2 . Use the differential to approximate the change in revenue from the sale of one more dog house when 300 dog houses are sold. A) $0.21 B) $21.00 C) $33.00
D) $0.33
143) A grocery store estimates that the revenue (in dollars) from the sale of x cases of condensed soup
143)
is given by R(x) = -5,700 + 9.2x - 0.0016x2 . Use the differential to approximate the change in revenue from the sale of one more case of soup when 1,200 cases are sold. A) -$7.28 B) $5.36 C) $3,036.00
D) $7.28
144) The concentration of a certain drug in the bloodstream x hours after being administered is approximately C(x) =
7x
8 + x2
changes from 1 to 1.33. A) 0.46
144)
. Use the differential to approximate the change in concentration as x
B) 0.20
C) 0.49
D) 0.90
145) A tumor is approximately spherical in shape. If the radius of the tumor changes from 7 mm to
145)
146) The weight of a ram can be estimated by the function W(t) = -7.3 + 304.8e-e(-0.00956(t - 131.9)),
146)
147) The edge of a square is measured as 5.42 inches, with a possible error of ±0.04 inch. Estimate the
147)
9 mm, find the approximate change in volume. Round your answer to the nearest hundred. A) 1,400 mm3 B) 2,500 mm3 C) 100 mm3 D) 1,200 mm3
where t is the age of the ram (in days) and W(t) is the weight of the ram (in kg). If a particular ram is 10 days old, use differentials to estimate how much weight it will gain before it is 40 days old. A) 11.3 kg B) 16.4 kg C) 15.1 kg D) 19 kg
maximum error in the area of the square. A) 0.2168 in.2 B) 0.0016 in.2
C) 0.08 in.2
D) 0.4336 in.2
148) A spherical balloon is being inflated. Find the approximate change in volume if the radius increases from 5.9 cm to 6 cm. A) 273.84 cm3 B) 0.236 cm3
C) 139.24 cm3
22
D) 13.924 cm3
148)
Answer Key Testname: UNTITLED6
1) B 2) B 3) C 4) D 5) C 6) B 7) B 8) D 9) B 10) C 11) C 12) A 13) D 14) A 15) D 16) D 17) A 18) C 19) A 20) B 21) D 22) D 23) B 24) D 25) D 26) D 27) A 28) B 29) C 30) B 31) A 32) A 33) B 34) D 35) D 36) A 37) D 38) D 39) C 40) A 41) A 42) C 23
Answer Key Testname: UNTITLED6
43) C 44) D 45) B 46) B 47) D 48) C 49) A 50) C 51) D 52) A 53) C 54) C 55) C 56) C 57) C 58) A 59) D 60) D 61) A 62) C 63) A 64) A 65) C 66) C 67) D 68) B 69) D 70) D 71) B 72) B 73) B 74) B 75) B 76) A 77) D 78) A 79) B 80) C 81) A 82) A 83) A 84) B 24
Answer Key Testname: UNTITLED6
85) C 86) D 87) D 88) B 89) C 90) D 91) B 92) A 93) C 94) D 95) A 96) D 97) B 98) D 99) C 100) D 101) C 102) A 103) D 104) A 105) B 106) A 107) A 108) D 109) D 110) D 111) D 112) D 113) B 114) D 115) A 116) C 117) C 118) D 119) D 120) B 121) D 122) A 123) D 124) D 125) B 126) A 25
Answer Key Testname: UNTITLED6
127) C 128) B 129) C 130) B 131) B 132) D 133) C 134) C 135) C 136) A 137) A 138) A 139) B 140) A 141) B 142) A 143) B 144) B 145) D 146) A 147) D 148) D
26
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the integral.
1)
x8 dx 7
A) x + C 8
2)
3
5
9
3
B) 4 x5/3 + C
C) 12 x5/3 + C
3
5
D) 6x5/3 + C
3) B) 31 x2
C) 31x + C
2
D) 0
4) B) -21x + C
C) x + 7C
D) - 7 + C 2x2
5) B) 2 x9/2 + C
C) 8 x9/2 + C
9
3
D) 24 x9/2 + C 7
6) B) 9 x4/3 + C
C) 27x4/3 + C
4
D) 27 x4/3 4
35 dx x2
A) 35 + C x
8)
2)
x
A) 9x4/3 + C
7)
9
12x3 x dx
A) 11 x9/2 + C
6)
9
D) x + C
7x-3 dx
A) 7x3
5)
C) 9x9 + C
31 dx
A) 31 + C
4)
B) 8x7 + C
4x2/3 dx
A) 8 x5/3 + C
3)
1)
7) B) - 35 + C
C) 35x + C
x
D) -35x + C
13x-7 dx
A) - 91 + C x8
8) B) - 13 + C
C) - 13 + C
6x8
6x6
1
D) - 78 + C x6
9)
(6x2 + 1) dx
9)
A) 2x3 + C 10)
C) 2x3 + x + C
B) 12x + C
(7x2 - 4x) dx
10)
A) 7 x2 + 2x + C
B) 7 x3 + C
C) - 7 x3 - 2x2 + C
D) 7 x3 - 2x2 + C
3
3
3
11)
3
(3x8 - 7x3 + 9) dx
11)
A) 1 x9 - 7 x4 + 9x + C
B) 1 x9 - 7 x4 + 9x + C
C) 9x9 - 7 x4 + 9x + C
D) 9x9 - 7 x4 + 9x + C
3
4
3
3
12)
13)
4
12)
A) 11t3 + 5t2 + 4t + C
B) 11 t3 + 5 t2 + 4t + C
C) 22t + 5 + C
D) 11 t3 + 5t2 + 4t + C
3
2
2
(8x2 + x-4 ) dx 3
3
3
C) - 8x + x 3
-3 3 -3 3
13) 3
+C
B) 8x - x
-3
+C
D) 8x + x
3
-3
3 3
3 3
+C +C
(x4/3 - 3x5/2) dx
14)
A) 3 x7/3 - 4 x7/2 + C
B) 3 x7/3 - 2 x7/2 + C
C) 3 x7/3 - 3 x7/2 + C
D) 3 x7/3 - 6 x7/2 + C
4
7
4
15)
3
(11t2 + 5t + 4) dt
A) - 8x - x
14)
D) x + C
( x+
7
7
3
7
7 7
x) dx
15)
3
3
A) 2 x + 2 x + C
B) 2 x + 3 x + C
C) 1 x3/2 + 2 x4/3 + C
D) 2 x3/2 + 3 x4/3 + C
2
3
3
2
4
16)
5 4 dx 2 x x
16)
A) - 5 - 8 + C
17)
x
x
x+
1 dx x4
B) 2x x - 1 + C
C) 2 x3/2 - 1 + C
D) 2x x - 1 + C
x
3x3
3 x-5 dx x2
18) B) 6 - 5 + C
x
x
C) - 6 - 5 + C
x
x
x
D) - 6 + 5 + C x
x
8e4y dy
19)
A) 1 e4y + C
C) 1 e4y + C
B) 4e4y + C
2
4
D) 2e4y + C
14e0.2x dx
20)
A) 2.8e0.2x + C 0.2x + 1
C) 14e
0.2x + 1
B) 70e0.2x + C D) 14e0.2x + C
+C
(t4 + e2t) dt 5
2t
5
2
A) t + e
22)
D) 5 - 8 x - C
4x4
4x4
x
21)
x
17)
A) 6 + 5 + C
20)
x
3x3
3
19)
C) 5 - 8 + C
x
A) 2 x3/2 - 1 + C 3
18)
B) - 5 - 8 x + C
21) +C
5
3
B) t + e2t + C
C) t + 2e2t + C
5
3
5
3t
5
3
D) t + e
7 3x dx
A) 7
3x 3
+C
22) +C
B) 7
3x
3 ln 7
C) 7
+C
3x
ln 7
3
+C
D) 7
4x 4
+C
23)
(5x-3 - 7x-1 ) dx
23)
A) 5 x-2 + 7 ln x + C
B) - 5 x-2 + 7 ln x + C
C) 5 x-2 - 7 ln x + C
D) - 5 x-2 - 7 ln x + C
3
2
3
24)
1 2 3 dx + + x x2 x3
24)
A) 2 + 6 + 12 + C
B) 2x + 2 ln x2 + 3 ln x3 + C
C) ln x - 2 - 3 + C
D) ln x + 2 ln x2 + 3 ln x3 + C
x2
x3
x
25)
2
x4
2x2
(3x + 5x-1) dx
25)
A) 9 x4 + 25 ln x2 + C
B) 3 x2 + 5 ln x + C
C) 3x3 + 15x - 25 x-1 + C
D) 3x3 + 30x - 25 x-1 + C
4
2
3
26)
3
x2 (3x + x-3 ) dx
26)
A) 3 x4 + ln x + C
B) 3 x4 - ln x + C
C) 3 x4 + ln x + C
D) 3 x4 - ln x + C
4
4
2
27)
28)
2
x 6 dx + 6 x
27)
A) x + C
B) 1 x + C
C) x ln 6 + 6 ln x + C
D) 1 x2 + 6 ln x + C
6
12
4 + 2ex dx x
28)
A) 4 ln x + 2xex-1 + C
B) 8 + 2xex-1 + C
C) 8 + 2ex + C
D) 4 ln x + 2ex + C
x2
x2
4
29)
x5 + 1 dx x
29)
A) 1 x4 + ln x + C
B) 1 x5 + ln x + C
C) 1 x5 - ln x + C
D) 1 x4 - ln x + C
3
5
5
30)
3
x-3 dx 2x x
30)
A) 1 ln x + 3x-1/2 + C
B) 1 x ln x - 3x-1/2 + C
C) 1 ln x - 3x-1/2 + C
D) 1 x ln x + 3x-1/2 + C
2
2
2
31)
2
2 - 2e-0.4x dx x
31)
A) - 2 + 4 e-0.4x + C
B) 4 - 4 e-0.4x + C
C) 2 ln x + 5e-0.4x + C
D) 2 ln x - 5e-0.4x + C
x2
5
x2
5
Solve the problem. 32) The slope of the tangent line of a curve is given by f'(x) = x 2 - 7x + 7.
32)
If the point (0, 10) is on the curve, find an equation of the curve. A) f(x) = 1 x3 - 8x2 + 7x + 1 B) f(x) = 1 x3 - 7 x2 + 7x + 1 3 3 2
C) f(x) = 1 x3 - 8x2 + 7x + 10
D) f(x) = 1 x3 - 7 x2 + 7x + 10
3
3
33) Find C(x) if C'(x) = x and C(9) = 40.
2
33)
A) C(x) = 2x3/2 + 22
B) C(x) = 2 x3/2 + 31 3
C) C(x) = 2x3/2 + 31
D) C(x) = 2 x3/2 + 22 3
34) Find the cost function if the marginal cost function is C'(x) = 12x - 7 and the fixed cost is $11. A) C(x) = 6x2 - 7x + 11 B) C(x) = 12x2 - 7x + 11 C) C(x) = 12x2 - 7x + 10 D) C(x) = 6x2 - 7x + 10
5
34)
35) Find C(x) if C'(x) = 5x2 - 7x + 4 and C(6) = 260.
35)
A) C(x) = 5 x3 - 7 x2 + 4x - 2
B) C(x) = 5 x3 - 7 x2 + 4x + 260
C) C(x) = 5 x3 - 7 x2 + 4x - 260
D) C(x) = 5 x3 - 7 x2 + 4x + 2
3
2
3
3
2
2
3
2
36) The rate at which an assembly line worker's efficiency E (expressed as a percent) changes with
36)
respect to time t is given by E'(t) = 65 - 8t, where t is the number of hours since the worker's shift began. Assuming that E(1) = 96, find E(t). A) E(t) = 65t - 8t2 + 35 B) E(t) = 65t - 4t2 + 157
C) E(t) = 65t - 4t2 + 96
D) E(t) = 65t - 4t2 + 35
37) Suppose that a velocity function is given by v(t) = 2t5 . Find the position function s(t) if s(0) = 5. A) s(t) = 1 t6 + 5 3
C) s(t) = 1 t6 3
B) s(t) = 10t4 + 5
37)
D) s(t) = 2t6 + 5
38) Suppose that an object's acceleration function is given by a(t) = 4t + 7. The object's initial velocity,
38)
v(0), is 2, and the object's initial position, s(0), is 10. Find s(t). A) s(t) = 2 t3 + 7 t2 + 2t B) s(t) = 2t2 + 7t + 2 3 2
C) s(t) = 4 t3 + 7 t2 + 10t + 2 3
D) s(t) = 2 t3 + 7 t2 + 2t + 10
2
3
2
39) Suppose that the acceleration of an object is given by a(t) = 5t2/3 + 2e-t. The object's initial
39)
velocity, v(0), is 4 and the object's initial position, s(0), is -8. Find s(t). 8/3 8/3 A) s(t) = 9t + 2e-t + 4t - 8 B) s(t) = 9t + 2e-t + 6t - 10 8 8 8/3
C) s(t) = 200t 9
D) s(t) = 3t5/3 - 2e-t + 6
+ 2e-t + 2t - 10
40) A company has found that its expenditure rate per day (in hundreds of dollars) on a certain type
40)
41) The approximate rate of change in height of a certain shrub in a nursery is given by f (x) = 10 ,
41)
of job is given by E'(x) = 6x + 5, where x is the number of days since the start of the job. Find the expenditure if the job takes 5 days. A) $3,500 B) $10,000 C) $100 D) $35
x
where the height is measured in inches and x represents the number of years since the shrub was planted. After 2 years, the shrub was 17 inches tall. Find the function that gives the height of the shrub x years after it was planted. Assuming that the shrub will be sold when it is 7 years old, approximate its height when it is sold.
A) f(x) = 10 ln x + 17; 36.5 inches
B) f(x) = 10 ln x + 17; 33.1 inches
C) f(x) = -10 + 39 ; 19.3 inches
D) f(x) = 10 ln x + 10.1; 29.5 inches
x2
2
6
42) The population of a city, in millions, since 1990 has grown at a rate of P (t) = 0.33e0.034t million
42)
people per year, where t is the number of years after 1990. If there were 1.87 million people in 2000, estimate (to two decimal places) the population in 2,009. A) P(19) 18.52 million B) P(19) -11.77 million
C) P(19) 30.28 million
D) P(19) 6.75 million
43) The number of mosquitoes in a lake area after an insecticide spraying decreases at a rate of
43)
M (t) = -7,000e-0.5t mosquitoes per hour. If there were 14,000 mosquitoes initially, how many will there be after 4 hours? A) M(4) 1,895 mosquitoes B) M(4) 14,000 mosquitoes
C) M(4) 764,374 mosquitoes
D) M(4) 103,447 mosquitoes
Find the integral.
44)
4(2x + 5)3 dx
A) 1 (2x + 5)4 + C 4
45)
C) 3 (2x + 5)4 + C
4
8
2
B) 1 (2t + 5)4 + C
C) 3 (2t + 5)4 + C
4
8
D) 3 (2t + 5)4 + C 4
8 dy
46)
(y - 9)3
A)
2
(y - 9)4
+C
B)
-4
(y - 9)2
+C
C)
-2
(y - 9)4
+C
D)
4
(y - 9)2
+C
6x - 7 dx
47)
A) 1 (6x - 7)3/2 + C
B) 1 (6x - 7)3/2 + C
C) 1 (6x - 7)3/2 + C
D) 1 (6x - 7)3/2 + C
3
2
9
48)
D) 1 (2x + 5)4 + C
45)
2
47)
B) 3 (2x + 5)4 + C
3(2t + 5)3 dt
A) 1 (2t + 5)4 + C
46)
44)
6
dr 6r - 7
A) 1 6r - 7 + C 6
48) B) 1 6r - 7 + C
C) 1 6r - 7 + C
3
4
7
D) 1 6r - 7 + C 2
49)
3z2 - 7 dz
9z
49)
A) 1 z(3z2 - 7)3/2 + C
B) z(3z2 - 7)3/2 + C
C) (3z2 - 7)3/2 + C
D) 1 (3z2 - 7)3/2 + C
2
50)
x (7x2 + 3)
A) C)
51)
2
dx
5
-1 56(7x2 + 3)
50) 4
-1
6 14(7x2 + 3)
+C
B)
+C
D)
1
16(y - 1)8
2
9
6
+C
8
D) 1 (y - 1)8 + C
8
8
ex dx ex + e
52) B) x + C
C) e ln(ex + e) + C
e
D) ln(ex + e) + C
6e5x dx
A) 6 e5x + C 5
53) C) 1 e5x + C
B) 6e5x + C
5
5x+1
D) 6e
5x + 1
+C
2 te-7t dt
A) 1 e-7t2 + C 14
55)
3(7x2 + 3)
+C
B) 1 (y - 1)9 + 1 (y - 1)8 + C
+C
A) x + C
54)
-7
4
51)
C) 1 y2 + 1 (y - 1)8 + C
53)
3(7x2 + 3)
(y - 1)7 dy
A)
52)
-7
54) B) 1 e-7t2 + C
C) - 1 e-7t2 + C
7
7
2 (1 - 6x)e3x-9x dx
D) - 1 e-7t2 + C 14
55)
A) 3(1 - 6x)e3x-9x2 + C
B) 3e3x-9x2 + C
C) 1 e3x-9x2 + C
D) 1 (1 - 6x)e3x-9x2 + C
3
3
8
56)
5e1/y dy 3y2
A) - 5e
1/y
3
57)
56) +C
y3
C) 5e
+C
3
B) -11e-4x3 + C
C) 11 e-4x3 + C 12
D) 10ye1/y + C
D) - 11 e-4x3 + C 12
58) B) 3 e z + C
C) -24e z + C
8
D) 3 e z + C 4
x9 dx 10 ex
59) 8
B) 9x + C
10 ex
1
C) -
10 10ex
10 ex
+C
D) -
1 +C 10 10ex -1
4 e1/t dt t5 -1/t4
A) e
61)
+C
57)
A) 1 + C
60)
1/y
3e z dz 8 z
A) -12e z + C
59)
1/y
3 11x2 e-4x dx
A) 12e-4x3 + C
58)
B) 5e
4
60) +C
1/t4
B) - e
4
1/t4
C) - e
+C
4t4
+C
6 (x6 - 2x5 ) (6x5 - 10x4 ) dx
D) - e1/t4 + C
61)
A) (x6 - 2x5 )7 + C
B) 6x5 - 10x4 + C
C) 1 (x6 - 2x5)6 + C
D) 1 (x6 - 2x5)7 + C
6
7
9
62)
7x6 dx 6 (4 + x 7 )
62)
A) 1 (4 + x7 )7 + C
B) -
7
7x6
C) -
63)
5 (4 + x7 )
x3
+C
D) -
+C
B) - 1 (x4 + 2)-1/2 + C
C) 2 (x4 + 2)3/2 + C
D) 8 (x4 + 2)3/2 + C
2
2x2
4
3
5 + 2x3 dx
64)
A) - 4 (5 + 2x3 )-3/4 + C
B) 8 (5 + 2x3 )5/4 + C
C) 4 (5 + 2x3 )5/4 + C
D) 2(5 + 2x3 )5/4 + C
5
15
x2 x3 + 10 dx
65)
A) 2 x3 (x3 + 10)3/2 + C
B)
C) 2 (x3 + 10)3/2 + C
D) 2 (x3 + 10)3/2 + C
9 9
t2 3
1 +C 6 x3 + 10
3
dt
66)
8 + t3
A) 1 t3 (8 + t3 )2/3 + C
B)
C) 1 (8 + t3 )2/3 + C
D) 1 (8 + t3 )2/3 + C
2 4
67)
7 7(4 + x 7 )
A) 1 (x4 + 2)3/2 + C
3
66)
1
+C
63)
3
65)
5 5(4 + x 7 )
x4 + 2 dx 6
64)
1
1
2 6(8 + t3 )
+C
2
x2 + 18x dx (x + 9)2
67)
A) x + 81 + C
B) 81 + C
x+9
C) x +
162
(x + 9)3
x+9
D) x + 9 + C
+C
x+9
10
68)
t2 + 4
t3 + 12t + 9
dt
68)
3
A) ln t + 12t + 9 + C
B) -
C) 3 ln t3 + 12t + 9 + C
D) -
3
69)
6x
(x + 6)4
A) -
3
(x + 6)2
+
12
(x + 6)3
2 3(t3 + 12t + 9)
+C
B) -
4x
(x + 6)3
+C
+C
D) 6 ln x + 6 + C
70)
A) 18 ln 2 + 5y + C
B) 18 ln 2 + 5y + C
C) 19 ln 2 + 5y + C
D) 19 ln 2 + 5y + C
5
(ln x)9 dx x 8
8
1
x(ln x 6)
71) B) (ln x) 10x
10
C) (ln x)
+C
10
10
+C
D) (ln x)10 + C
dx
72)
A) ln x6 + C
B) 1 ln ln x6 + C
C) 1 ln x6 + C
D) ln ln x6 + C
6
6
73)
1
19 dy 2 + 5y
A) (ln x) + C
72)
+C
69)
5
71)
2 (t3 + 12t + 9)
dx
C) 6 ln x + 6 3 - 36 ln x + 6 4 + C
70)
3
ln x4 dx x
A) 1 (ln x4 )2 + C 4
73) B)
1
ln x4
C) 1 (ln x4 )2 + C
+C
2
11
D) 1 (ln x4 )2 + C 8
74)
(ln x)20 dx x
A) (ln x)
21
x
75)
1
x(ln x)18
A)
C) (ln x)
B) 20(ln x)19 + C
+C
21x
x(ln x)19
+C
1
19(ln x)19
B) -
+C
D) -
6
2
1
17x(ln x)17 1
17(ln x)17
+C
+C
2
6
D) (ln t) + C 12
t6 + 4
(9 + ln x)5 dx x
77)
6
A) (9 + ln x) + C
B) 6x2 (9 + ln x)6 + C
6x2
6
6
C) (9 + ln x) + C
D) (9 + ln x) + C
6
6x
1 dx 5x(ln x)
78)
A) ln x + ln ln x + C
B) 5 ln ln x + C
C) ln 5 ln x + C
D) ln ln x + C
5
5
log2 x
79)
x
C)
+C
B) [ln(t6 + 4)]2 + C
C) t ln(t + 4) + C
A)
21
76)
12
79)
21
t5 ln(t6 + 4) dt t6 + 4 6
78)
D) (ln x)
+C
75)
A) [ln(t + 4)] + C
77)
21
dx
1
C) -
76)
74)
(ln 2)(log2 x)2 2 (log2 x)2 2 ln 2
+C
B)
+C
D)
12
(ln x)(log2 x)2 2 (log2 x)2 2
+C
+C
80)
(log9 (6x - 3))3
80)
6x - 3
A) C)
(ln 9)(log9 (6x - 3))4
B)
24 (log9 (6x - 3))4
D)
24
(ln 9)(log9 (6x - 3))4 4 (log9 (6x - 3))4 24 ln 9
Solve the problem.
81) The rate of expenditure for maintenance of a particular machine is given by M'(x) = 12x x2 + 5,
81)
where x is time measured in years. Total maintenance costs through the second year are $111. Find the total maintenance function. A) M(x) = 12(x2 + 5)3/2 + 3 B) M(x) = 4(x2 + 5)3/2 + 3
C) M(x) = 4(x2 + 5)3/2 + 99
D) M(x) = 12(x2 + 5)3/2 + 99
82) The rate of growth of the profit (in millions) from an invention is approximated by P'(x) = xe-x2 ,
82)
83) The work W (in joules) done by a force F (in newtons) moving an object through a distance x (in
83)
where x represents time measured in years. The total profit in year 1 that the invention is in operation is $30,000. Find the total profit function. Round to three decimal places where appropriate. A) P(x) = -0.5e-x2 - 0.214 B) P(x) = -0.5e-x2 + 214,000 C) P(x) = -0.5e-x2 + 0.214 D) P(x) = -0.5e-x2 - 214,000
meters) is given by W =
A) W = kx2 + C
F dx . Find a formula for W, if F = kx and k is a constant. 2
B) W = kx + C
C) W = k + C
2
D) W = kx + C
84) A company has found that the marginal cost of a new production line (in thousands) is C'(x) =
2
9 , where x is the number of years the line is in use. Find the total cost function for the x+e
production line (in thousands). The fixed cost is $20,000. A) C(x) = ln(x + e) + 20 B) C(x) = ln(x + e) + 11 9 9
C) C(x) = 9 ln(x + e) + 11
D) C(x) = 9 ln(x + e) + 20
13
84)
85) The current (in amperes) in an inductor of constant inductance L (in henries) is given by 1 i= L
85)
V dt , where V is the voltage (in volts) and t is the time (in seconds). Find a formula for i, if
V = 9t(t2 - 6). A) i = 1 9 t4 - 27t2 + C L 4
B) i = 1 9 t4 - 54t + C
C) i = 1 9 t4 - 54t2 + C
D) i = L 9 t4 - 27t2 + C
L 4
L 4
4
Answer the question, concerning the use of substitution in integration. 86) If we decide to use u = x + e, then which of the following are correct? i) du = dx ii) x = u - e iii) du = e dx iv) dx = e du A) Both i and iv
B) Both ii and iii
C) Both i and ii
86)
D) Only i
87) If we decide to use u = x2 + e, then which of the following are correct?
87)
i)
du = 2x dx ii) x2 = u - e iii) du = 2ex dx 2x iv) du = dx 3
A) Both ii and iii
B) Only i
88) If we use u = x2 as a substitution to find
C) Both i and ii
D) Both i and iv
2 xex dx, then which of the following would be a
88)
correct result?
2eu du
A)
du
B)
C)
eu du 2
D) None of the above
89) If we decide to use u = ex2 , then which of the following are correct? i)
du = 2e dx u ii) x2 = e iii) du = 2ex dx 2x iv) du = dx e
A) Both i and iii C) Only iii
B) Both ii and iii D) None of the above
14
89)
90) If we use u = x2 + 2 as a substitution to find
(x2 + 2) dx, then which of the following would be a
correct result? A) u du 2
B)
u du
C)
90)
(u - 2) du
D) None of the above
91) If we use u = x + e as a substitution to find
(x + e)n dx, then which of the following would be a
91)
correct result?
A)
un+1 du
B)
un dx
C)
un du
D) None of the above
92) Suppose that we are using substitution to find an antiderivative. If, after making the substitution,
92)
we find that there is still an x-term left in the integrand, what should we do? A) This would never happen if we made the correct substitution.
B) Use an alternative method, because substitution will never give the antiderivative. C) Go back to the equation relating x and u, solve for x, and substitute in the integrand. D) None of the above Approximate the area under the graph of f(x) and above the x-axis using n rectangles. 93) f(x) = 2x + 3 from x = 0 to x = 2; n = 4; use right endpoints A) 11 B) 17 C) 15
94) f(x) = 3x2 - 2 from x = 1 to x = 5; n = 4; use right endpoints A) 144 B) 140 C) 154
93) D) 13 94) D) 150
95) f(x) = 3 from x = 2 to x = 10; n = 4; use right endpoints
95)
x
A) 4.72
B) 6.25
C) 7.20
D) 3.85
96) f(x) = 1 ; interval [1, 5]; n = 4; use left endpoints
96)
x2
A) 2.0833
B) 1.4236
C) 1.4636
D) 0.4636
97) f(x) = x2 + 2; interval [0, 5]; n = 5; use left endpoints A) 66 B) 32 C) 65
D) 40
98) f(x) = x2 from x = 0 to x = 4 n = 2; use midpoints A) 20 B) 40
D) 8
97)
98) C) 38.75
15
99) f(x) = e-x + 5 from x = -2 to x = 2; n = 4; use right endpoints A) 20.54 B) 26.96 C) 24.22
99) D) 31.48
100) f(x) = 1 from x = 3 to x = 8; n = 2; use midpoints
100)
x
A) 40720
632043
B) 352
C) 20360
459
210681
101) f(x) = 25 - x2 from x = -5 to x = 5; n = 2; use midpoints A) 187.5 B) 62.5 C) 10 102) f(x) = 2x3 - 1 from x = 1 to x = 6; n = 5; use right endpoints A) 850 B) 800 C) 825 Find the exact value of the integral using a formula from geometry. 9 103) 81 - x2 dx -9 A) 162 B) 81 C) 81 2 7
104)
D) 176 459
101) D) 93.75 102) D) 875
103) D) 81
4
(2 + 3x) dx
104)
3
A) 34 3
105)
B) 68
C) 56
D) 28
(3 - x) dx
105)
0
A) 9 9
106) 0
B) 2.25
C) 4.5
D) 18
81 - x2 dx
A) 81
4
106) C) 81
B) 162
2
16
D) 81
3
107)
f(x)dx for the indicated region.
107)
1
A) 5 9
108)
B) 10
C) 12.5
D) 7.5
f(x)dx for the indicated region.
108)
2
A) 14
B) 49
C) 42
17
D) 7
Solve the problem. 109) The table below shows the velocity of a remote-controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with left endpoints. Time Velocity (sec) (in./sec) 0 0 1 10 2 14 3 10 4 20 5 23 6 25 7 12 8 5 A) 228 in.
B) 114 in.
C) 119 in.
D) 104 in.
110) The table below shows the velocity of a remote-controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with right endpoints. Time Velocity (sec) (in./sec) 0 0 1 8 2 20 3 27 4 26 5 24 6 27 7 22 8 4 A) 158 in.
B) 148 in.
C) 154 in.
18
109)
D) 164 in.
110)
111) Joe wants to find out how far it is across the lake. His boat has a speedometer but no odometer.
111)
The table below shows the boat's velocity at 10-second intervals. Estimate the distance across the lake using right endpoints. Time Velocity (sec) (ft/sec) 0 0 10 12 20 30 30 51 40 48 50 53 60 50 70 53 80 43 90 15 100 0 A) 5,300 ft
B) 355 ft
C) 3,550 ft
D) 3,650 ft
112) A piece of tissue paper is picked up in gusty wind. The table below shows the velocity of the
112)
paper at 2-second intervals. Estimate the distance the paper traveled using right endpoints. Time Velocity (sec) (ft/sec) 0 0 2 8 4 12 6 6 8 29 10 34 12 24 14 10 16 2 A) 250 ft
B) 227 ft
C) 125 ft
D) 230 ft
113) In the table below, the velocity of a projectile fired straight into the air is given every half second. Use right endpoints to estimate the distance the projectile traveled in four seconds. Time Velocity (sec) (m/sec) 0 132 0.5 127.1 1.0 122.2 1.5 117.3 2.0 112.4 2.5 107.5 3.0 102.6 3.5 97.7 4.0 92.8 A) 459.4 m
B) 918.8 m
C) 439.8 m
19
D) 879.6 m
113)
114) A swimming pool has a leak, and the leak is getting worse. The table below gives the leakage rate
114)
every 6 hours. Use right endpoints to estimate the number of gallons lost in 48 hours. Time Leakage (hr) (gal/hr) 0 0 6 0.6 12 1.3 18 1.9 24 3.0 30 4.5 36 5.9 42 7.0 48 8.1 A) 253.8 gal
B) 32.3 gal
C) 145.2 gal
D) 193.8 gal
115) The graph below shows the rate of natural gas usage (in therms per day) in one household for a 30-day period. Estimate the total number of therms used during this period. Use left endpoints and rectangles with widths of 3 days.
A) 165 therms
B) 136 therms
C) 159 therms
20
D) 147 therms
115)
116) The graph below shows the rate of change of the price of a stock (in dollars per share per week)
116)
over a period of 6 weeks. Estimate the total change in dollars per share of the stock during this period. Use rectangles with widths of 1 week, and let the function value at the midpoint of the rectangle give the height of the rectangle.
A) $1.60/share
B) $0.90/share
C) $2.20/share
Provide the proper response. 117) The definite integral represents area only if the function involved is the interval [a, b]. A) negative
B) positive
D) $2.00/share
?
at every x-value in
C) nonnegative
117)
D) rational
118) If f(a) = 3 and f(x) is increasing on the interval [a, b], then which method of estimating the area
118)
under the graph of f(x) and above the x-axis will yield the highest value? Assume that n = 10 in all cases. A) Using midpoints B) Using left endpoints
C) Using right endpoints
D) Cannot be determined
119) If f(b) = 4 and f(x) is decreasing on the interval [a, b], then which method of estimating the area
119)
under the graph of f(x) and above the x-axis will yield the lowest value? Assume that n = 10 in all cases. A) Using left endpoints B) Using right endpoints
C) Using midpoints
D) Cannot be determined
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
120) What is the difference between the indefinite integral and the definite integral?
21
120)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
121) In estimating the definite integral for f(x) on the interval [5, 10], we compute x = b - a . If n = 5, n
then what is the value of x? A) 2.5
121)
B) 2 D) None of the above
C) 5
b
122) If f(x) gives the rate of change of F(x) for x in [a, b], then
f(x) dx represents
122)
a
i) the total change in F(x) as y goes from a to b. ii) the total change in F(x) as x goes from a to b. iii) the total change in f(x) as x goes from a to b. A) Only i is correct. B) Only ii is correct.
C) Either ii or iii could be correct. Evaluate the definite integral. 5 123) 4 dx -2 A) 28 15
124) 2
(z-
A) - 15 15 5
123) B) 12
C) 14
D) 7
15) dz
2
125)
D) Only iii is correct.
124) C) - 19 15
B) - 15
2
D) - 19 + 2 15 2
(x + 4) dx
125)
-1
A) 25
126)
1
C) 36
B) -36
(x2 + 1) dx
126)
-1
A) 8 3
127)
0
D) 29
C) - 2
B) 0
3
D) - 8 3
(4 + x2 ) dx
127)
-1
A) 0
C) 4
B) -2
22
D) 13 3
9
128) 0
(2x2 + x + 8) dx
A) 1197 2
16
129)
128) C) 537
B) 98
2
D) 1173
3 x dx
129)
0
A) 288 4
130)
B) 192
C) 24
D) 128
x-1/2 dx
130)
1
A) 1 4
131) 1
C) 0
A) 224 -1
131) B) 44
C) 46
3
A) 1 6
4 1
A) 14 2
1
132)
1
C) 7
B) 28
6
D) 7
24
133) B) 17
C) 17
3
5
D) 14 5
x3 - x-1 dx x2
A) 9 4
135)
15
x1/2 dx
3
134)
D) 226
4x-4 dx
-2
133)
D) 3
(x3/2 + x1/2 - x-1/2) dx
15
132)
B) 2
134) B) 9
C) 15
8
16
D) 17 8
2 8 4 t -t dt t6
A) 9 2
135) B) 11
C) 5
6
6
23
D) 19 6
136) 1
4 2 t +1 dt t
136)
A) 77
B) 72
5
6
137)
C) 92
5
D) 32
5
6x dx
137)
0
A) 6 e
138) 1
B) 24
e 1
138)
0
B) 16
12x -
A) 31 6
2
6
139) B) 6e2 - 13
C) 6e2 - 6
D) 6e2 - 19
140) B) 3
D) 31
C) 31
2
10
dt 1+t
141) B) 4
A) ln 3
142)
D) -16e2
4 x(x2 + 1) dx
5
141)
C) -16
13 dx x
A) 12e2 - 13 1
140)
D) 54
16 dx x
A) 0
139)
C) 36
C) ln 7
5
3
D) - 24 25
(x - 4)3 dx
142)
0
B) 18
A) -240 1
143) 0
4x3
5 (1 + x4 )
A) 1 4
C) - 60
D) 4
dx
143) B) 31
C) 15
128
64
24
D) 15 16
1
144) 0
x + 16 dx
A) 34 17 - 128 3
1
145)
144)
5x
3
4
C) 34 17
B) 17 17 - 64
D) 51 17 - 51
3
2
2
1 + x2 dx
145)
0
A) 5 (2 5/4 - 1) 2
1
146)
6r 16 + 3r2
0
4
147) 2
A) 1 (e6 - e12) 2
1
1 0
B) 2 19 - 8
C) - 2 19 + 8
D) 19 - 4
147) B) 1 (e12 - e6 )
C) e12 - e6
3
D) 3(e12 - e6 )
2 dx x(6 + ln x)
A) -0.090
149)
146)
e3x dx
3
148)
D) 4(2 5/4 - 1)
C) 4 2
dr
19 -2 2
A)
4
B) 2(2 5/4 - 1)
148) B) 9.803
C) 0.219
D) 7.386
et
149)
2 (4 + et)
A) 1 - 1 4
4+e
B) 1 - 1 5
C) 1 - 1
4+e
4+e
5
D) 1 - 1 4+e
4
Use the definite integral to find the area between the x-axis and f(x) over the indicated interval. 150) f(x) = x-1/2; [1, 4]
A) 2 151) f(x) = ex - 1; [-2, 3] A) -e3 - e-2 - 5 152) f(x) = 2x + 7; [1, 5] A) 18
B) 1 2
C) 1 4
D) 4
B) e-2 - e3 - 5
C) e3 + e-2 - 5
D) e3 - e-2 - 5
150)
151)
152) B) 9
C) 52
25
D) 26
153) f(x) = x2 - 6x + 9; [2, 4] A) 2 3
154) f(x) = 2x - x2; [0, 2] A) 2 3
153) B) 1
C) 7
D) 4
B) 4 3
C) 5 3
D) 7 3
3
3
3
154)
155) f(x) = 3 ; [1, 3]
155)
x3
A) 4 3
156) f(x) = -x2 + 9; [0, 5] A) 10 9
157) f(x) = x2 + 1; [0, 1] A) 1 3
158) f(x) = x4 - 4x3 + 4x2 ; [0, 2] A) 15 16
B) 1
C) 3
3
D) 1 2
156) B) 98
C) 10
D) 5
B) 5 3
C) 4 3
D) 2 3
B) 15 17
C) 17 15
D) 16 15
3
3
9
157)
158)
Find the area of the shaded region. y = 2x + 1
159)
A) 7.5
159)
B) 5
C) 10
26
D) 12.5
y = x2 + 3
160)
A) 23 3
160)
B) 22
C) 25
3
3
D) 26 3
161)
161)
y = 4 - x2
A) 5
C) 23
B) 3
3
27
D) 5 3
162)
162) y=
A) 16 3
x-3
B) 22
C) 38
3
3
D) 29 3
y = x3 - 4x
163)
A) 17 4
163)
B) 41
C) 9
4
4
28
D) 33 4
164)
y=
1 x
164)
A) ln 4.5
B) ln 4
C) ln 5.5
D) ln 5
y = (x - 3)2
165)
A) 4
165)
B) 2
3
C) 1
3
3
D) 5 3
y = ex
166)
166)
(2, e2 )
(1, e)
A) e2 - e
B) e2 + e
C) e2 + e - 1 29
D) e2 - e + 1
167)
167)
y=7
A) 7
B) 42
C) 49
D) 14
Solve the problem. 168) A company has found that its rate of expenditure (in hundreds of dollars) on a certain type of job is given by E'(x) = 10x + 11, where x is the number of days since the start of the job. Find the total expenditure if the job takes 5 days. A) $18,000 B) $61 C) $6,100 D) $180
169) After a new firm starts in business, it finds that its rate of profit (in hundreds of dollars) after t
168)
169)
years of operation is given by P'(t) = 3t2 + 4t + 9.
Find the profit in year 4 of the operation. A) $7,200 B) $10,200
C) $6,000
D) $11,600
170) A certain object moves in such a way that its velocity (in m/s) after time t (in s) is given by v = t2 + 5t + 10.
4
Find the distance traveled during the first four seconds by evaluating
170)
(t2 + 5t + 10) dt. Round
0
your answer to the nearest tenth of a meter. A) 80.0 m B) 46.0 m
C) 101.3 m
D) 61.3 m
171) For a particular circuit, the current (in amperes) after time t (in seconds) at a certain point P is given by
i = 0.005t0.2. Find the charge (in coulombs) that passes point P during the first second by evaluating 1 0.005t0.2 dt. 0 A) 0.005 coulombs B) 238 coulombs
C) 1.2 coulombs
D) 0.0042 coulombs
30
171)
172) A force acts on a certain object in such a way that when the object has moved a distance of r (in m), the force f (in newtons) is given by f = 4r2 + 5r.
4
Find the work (in joules) done through the first four meters by evaluating
172)
(4r2 + 5r) dr.
0
A) 125.3 joules
B) 40 joules
C) 64 joules
D) 90.3 joules
173) An object is traveling with a velocity (in feet per second) given by
173)
174) A population of bacteria grows at a rate of P'(t) = 18 et where t is time in hours. Determine how
174)
175) The rate of change in a person's body temperature, with respect to the dosage of x milligrams of a
175)
v(t) = 7t3 - 5t2 + 2t, where t is time in seconds. Find the object's average velocity from t = 0 to t = 5 seconds. A) 910.4 ft/sec B) 239.2 ft/sec C) 152.0 ft/sec D) 182.1 ft/sec
much the population increases from t = 0 to t = 3. Round your answer to two decimal places. A) 361.54 B) 343.54 C) 705.08 D) 352.54
2 drug, is given by D'(x) = . One milligram raises the temperature 2.8°C. Find the function x+7 giving the total temperature change.
B) D(x) = ln
A) D(x) = 2 ln x + 7 - 1.4 C) D(x) = ln
2 - 2.8 x+7
2 - 1.4 x+7
D) D(x) = 2 ln x + 7 + 2.8
176) The number of books in a small library increases at a rate according to the function
176)
177) For a certain drug, the rate of reaction in appropriate units is given by R'(t) = 9 + 2 , where t is
177)
B (t) = 160e0.05t, where t is measured in years after the library opens. How many books will the library have 4 year(s) after opening? A) 708 B) 195 C) 35 D) 3,200
t
t2
measured in hours after the drug is administered. Find the total reaction to the drug from t = 4 to t = 8. Round to two decimal places, if necessary. A) 27.7 B) 30.44 C) 6.49 D) 15.22
31
Provide the proper response.
b
178) If f(x) 0 on the interval [a, b], then
f(x) dx represents
178)
a i) the area to the right of the y-axis between y = a and y = b. ii) the area above the x-axis between x = a and x = b. iii) the area below the x-axis between x = a and x = b. A) Only ii is correct. B) Only i is correct.
C) Either ii or iii could be correct.
D) None of the above is correct. b
179) If f(x) 0 on the interval [a, b], then
f(x) dx represents
179)
a i) the area to the right of the y-axis between y = a and y = b. ii) the area above the x-axis between x = a and x = b. iii) the area below the x-axis between x = a and x = b. A) Only i is correct. B) Only ii is correct.
C) Only iii is correct. b
180) If F'(x) = f(x), then
D) Either ii or iii could be correct.
f(x) dx =
180)
a i) F(a) - F(b). ii) F(b) - F(a). iii) F(b) + F(a). A) Only iii is correct.
B) Either i or ii could be correct. D) Only i is correct.
C) Only ii is correct. b
181) If r(t) is the rate of change of revenue, then
r(t) dt is
a i) the total revenue up to time b. ii) the total revenue from time a to time b. iii) the change in revenue at any time. A) Only i is correct.
B) Only iii is correct. D) None of the above is correct.
C) Only ii is correct.
32
181)
182) Which integral or integrals have a value of zero? b
i)
r(t) dt
ii)
b
t3 dt
-b b
iii)
b
182)
t3 dt , where b > 0
10
iv)
0
t3 dt , where b < 0
b
A) Both i and iv Find the area between the curves. 183) y = 2x - x2 , y = 2x - 4
A) 31 3
184) y = x2 - 5x + 4, y = -(x - 1)2 A) 9 8
B) Only i
C) Both i and ii
D) All of these
B) 37 3
C) 32 3
D) 34 3
183)
184) B) 8 7
187) y = x, y = x2 A) 1
D) 8
C) 1 - e-2
D) e-2 - 1
8
185) x = 0, x = -2, y = ex, y = 0 A) e-2 B) 1 + e-2 186) y = x3, y = 4x A) 4
C) 7
9
185)
186) B) 16
C) 2
D) 8 187)
B) 1
C) 1
D) 1
188) x = 0, x = 1, y = x2 + 6, y = x2 + 2 A) 12 B) 16
C) 8
D) 4
C) 37 3
D) 32 3
3
189) y = x2, y = 4 A) 34 3
2
6
12
188)
189) B) 31 3
190) y = 1 x2 , y = -x2 + 6
190)
2
A) 8
B) 32
C) 4
33
D) 16
191) y = x3, y = x2
191)
A) 1
B) 5
6
C) 5
12
D) 1
6
12
192) x = 2, x = 5, y = 1 , y = 1 x
A) ln 5 + 3
192)
x2
C) ln 5 - 1
D) ln 5 + 1
193) x = 1, x = 4, y = ln x , y = ln 2x A) ln 2 B) ln 16
C) ln 8
D) ln 8 - 6
194) x = -3, x = 3, y = 10x/(1 + x2 ), y = 0 A) ln 10 B) 10 ln 10
C) 10 e10
2
10
B) ln 5 - 3 2
10
2
10
2
10
193)
194) D) 0
Solve the problem.
195) Find the producers' surplus if the supply function is given by S(q) = q2 + 4q + 20. Assume supply and demand are in equilibrium at q = 24. A) 10,386 B) 10,638
C) 10,836
D) 10,368
196) Find the producers' surplus if the supply function of some item is given by S(q) = q2 + 2q + 8. Assume supply and demand are in equilibrium at q = 30. A) 19,800 B) 12,700 C) 17,200
C) 432
3
A) 16
B) 8
C) 16 3
3
given by D(q) = 27 - q2/3. Find the producers' surplus. (Hint: The equilibrium quantity q0 is a
B) 64
C) 32
34
199)
D) 8
200) Suppose the supply function of a certain item is given by S(q) = 2q + 7, and the demand function is perfect cube.) A) 64 5
198)
D) 216
199) Suppose the supply function of a certain item is given by S(q) = 4q + 2, and the demand function is given by D(q) = 14 - q2 . Find the producers' surplus.
197)
D) 128
198) Find the consumers' surplus if the demand for an item is given by D(q) = 72 - q2 , assuming supply and demand are in equilibrium at q = 6. A) 144 B) 72
196)
D) 18,900
197) Find the consumers' surplus if the demand function for an item is given by D(q) = 30 - q2 , assuming supply and demand are in equilibrium at q = 4. A) 64 B) 128 C) 64 3
195)
D) 32 5
200)
201) Suppose the supply function of a certain item is given by S(q) = 2q + 7, and the demand function is
201)
given by D(q) = 27 - q2/3. Find the consumers' surplus. (Hint: The equilibrium quantity q0 is a perfect cube.) A) 32 5
C) 64
B) 32
5
D) 64
202) Suppose the supply function of a certain item is given by S(q) = 2eq, and the demand function is
202)
given by D(q) = 8e-q. Find the producers' surplus. Round your answer to three decimal places. A) 0.664 B) 1.228 C) 1.337 D) 0.773
203) Suppose the supply function of a certain item is given by S(q) = 2eq, and the demand function is
203)
given by D(q) = 8e-q. Find the consumers' surplus. Round your answer to three decimal places. A) 0.773 B) 0.664 C) 1.337 D) 1.227
204) A company determines that its marginal revenue per day is given by R'(t) = 60et, and that R(0) = 0,
204)
where R(t) = revenue, in dollars, on the tth day. The company's marginal cost per day is given by C'(t) = 140 - 0.3t, and that C(0) = 0, where C(t) = cost, in dollars, on the tth day. Find the total profit from t = 0 to t = 8 (the first 8 days). Round your answer to the nearest dollar. T Note: P(T) = R(T) - C(T) = [R'(t) - C'(t)] dt. 0 A) $177,697 B) $177,747 C) $177,687 D) $177,668
205) A company determines that its marginal revenue (in dollars per day) is given by MR(t) = 80et. The
205)
206) The flow of blood in a blood vessel is faster toward the center of the vessel and slower toward the
206)
company's marginal cost (in dollars per day) is given by MC(t) = 90 - 0.1t. Find the total profit from t = 0 to t = 9 (the first 9 days). A) $647,365 B) $647,361 C) $647,353 D) $647,441
p outside. The speed of the blood is given by V = (R2 - r2 ), where R is the radius of the blood 4Lv
vessel, r is the distance of the flowing blood from the center of the blood vessel, and p, v, and L are physical constants related to the pressure and viscosity of the blood and the length of the blood p vessel. If R is constant, we can think of V as a function of r: V(r) = (R2 - r2 ). The total blood 4Lv R
flow Q is given by Q(R) =
A) 2048
p 1875 Lv
0
2 rV(r) dr . Find Q for a blood vessel of radius R = 1.6 mm.
B) 512
p 625 Lv
C) 1024
p 625 Lv
35
D) 512
p 375 Lv
207) In town A, the birth rate is given by b'(t) = 58e0.20t (births per year), where t is the number of
207)
208) The velocity of particle A, t seconds after its release is given by
208)
years since 1990. In town B, the birth rate is given by B'(t) = 86e0.39t (births per year), where t is the number of years since 1990. How many more births are there in town B than in town A during the 1990s (from t = 0 to t = 10)? A) 10,673 births B) 3,792 births C) 8,751 births D) 8,821 births
va (t) = 9.0t - 0.6t2 meters per second. The velocity of particle B, t seconds after its release is given by vb(t) = 11.2t - 0.3t2 meters per second. If velocity is measured in meters per second, how much farther does particle B travel than particle A during the first ten seconds (from t = 0 to t = 10)? Round to the nearest meter. A) 6 m B) 520 m C) 210 m D) 410 m
209) The velocity of particle A, t seconds after its release is given by va (t) = 2.6e0.6t meters per second.
209)
The velocity of particle B, t seconds after its release is given by vb(t) = 11.4t - 0.3t2 meters per second. If velocity is measured in meters per second, how much farther does particle A travel than particle B during the first ten seconds (from t = 0 to t = 10)? Round to the nearest meter. A) 1,278 m B) 1,174 m C) 1,474 m D) 1,274 m
Use n = 4 to approximate the value of the integral by the trapezoidal rule. 6 210) x dx 0
A) 9 3
211)
B) 18
C) 36
A) 28 2 0
A) 15 3
1
2
211) B) 14
C) 7
D) 35 4
2x2 dx
2
213)
D) 45
(2x + 3) dx
1
212)
210)
212) B) 16
C) 11
3
D) 11 2
3 dx x2
A) 423 400
213) B) 423
C) 423
200
100
36
D) 213 100
1
214)
(x2 + 5) dx
-1
A) 43
B) 43
2
2
215) 0
0
A) 743
9
1 + x2
D) 241 8
6x
216) C) 1171
280
140
D) 1747 840
dx
A) 84627 5
16
B) 1171
6800
218)
C) 241
12
6 dx 1+x
1 0
8
215) B) 173
140
217)
3
D) 55
(x4 + 4) dx
A) 337 1
C) 32
4
16
216)
214)
217) B) 29691
C) 47907
3400
6800
D) 47907 3400
2x - 1 dx
218)
1
A) 162.1 2
219)
B) 28.8
C) 172.5
D) 220.5
4 - x2 dx
219)
0
A) 3.0
B) 12.0
C) 6.0
Use n = 4 to approximate the value of the integral by Simpson's rule. 7 220) x dx 0 A) 49 B) 245 C) 49 2 12 3
221) 1
D) 1.5
220) D) 49 4
(10x + 2) dx
A) 110 3
221) B) 88
C) 44
37
D) 22
2
222) 0
4x2 dx
222)
A) 32 3
3
223) 1
B) 83
0
0
9
1 + x2
C) 177 16
D) 67 8
14x
226) B) 1171
C) 1171
420
280
D) 1747 420
dx
A) 47907 5
4
6 dx 1+x
6800
228)
D) 19
225) 24
A) 1747 1
6
B) 125
840
227)
900
(x4 + 2) dx
A) 125 1
C) 23
3
12
226)
D) 1813
224) B) 14
3
0
200
(x2 + 2) dx
A) 7 2
C) 423
50
-1
225)
3
223)
A) 1813 1
3
D) 28
3 dx x2
1800
224)
C) 16
B) 11
227) B) 24033
C) 24033
1700
3400
D) 47907 3400
2x - 1 dx
228)
1
A) 399.5
B) 393.9
C) 28.5
D) 368.2
Use your calculator to approximate the integral using the method indicated, with n = 100. Round your answer to four decimal places. 3 229) 229) ex dx (trapezoidal rule) 0 A) 19.0855 B) 20.0855 C) 19.0462 D) 19.0870
38
4
230)
x + 4 dx (trapezoidal rule)
230)
0
A) 9.7732 1
231)
1 x+3
dx (trapezoidal rule)
A) 4.3473
B) 4.3325
4+
0
3
232)
B) 9.6544
C) 9.8032
D) 9.7516
231) C) 4.2877
D) 4.1985
x ln x dx (trapezoidal rule)
232)
1
A) 2.8682 1
233) 0
5+
1 x+3
B) 2.9436
4
D) 2.9202
dx (Simpson's rule)
A) 5.2323
234)
C) 2.8518
233)
B) 5.3305
C) 5.2877
D) 5.3695
x ln x dx (Simpson's rule)
234)
1
A) 7.3404 3
235)
B) 7.2500
C) 7.2488
D) 7.2516
ex dx (Simpson's rule)
235)
0
A) 19.0855 1
236)
B) 20.0855
C) 19.0870
D) 19.1506
x + 4 dx (Simpson's rule)
236)
0
A) 2.1691
B) 2.1202
C) 2.0923
39
D) 2.0325
Solve the problem. Round your answer, if appropriate. 237) Suppose that the accompanying table shows the velocity of a car every second for 8 seconds. Use the trapezoidal rule to approximate the distance traveled by the car in the 8 seconds. Time (sec) Velocity (ft/sec) 0 15 1 16 2 17 3 19 4 18 5 20 6 17 7 15 8 16 A) 137.5 feet B) 209.5 feet
C) 275 feet
237)
D) 153 feet
238) Suppose that the accompanying table shows the velocity of a car every second for 8 seconds. Use
238)
Simpson's rule to approximate the distance traveled by the car in the 8 seconds. Time (sec) Velocity (ft/sec) 0 20 1 21 2 22 3 24 4 23 5 25 6 22 7 20 8 21 A) 176.33 feet B) 178.33 feet
C) 177.50 feet
D) 132.00 feet
239) The following table shows the rate of water flow (in gal/min) from a stream into a pond during a 30-minute period after a thunderstorm. Use the trapezoidal rule to estimate the total amount of water flowing into the pond during this period. Time (min) Rate (gal/min) 0 225 5 275 10 325 15 275 20 245 25 225 30 175 A) 8,725 gallons
B) 7,725 gallons
C) 7,733.3 gallons
40
D) 7,050.0 gallons
239)
240) The following table shows the rate of water flow (in gal/min) from a stream into a pond during a
240)
30-minute period after a thunderstorm. Use Simpson's rule to estimate the total amount of water flowing into the pond during this period. Time (min) Rate (gal/min) 0 275 5 325 10 375 15 325 20 295 25 275 30 225 A) 10,600 gallons
B) 9,233.3 gallons
C) 9,225 gallons
D) 8,383.3 gallons
241) A surveyor measured the length of a piece of land at 100-ft intervals (x), as shown in the table.
241)
Use the trapezoidal rule to estimate the area of the piece of land in square feet. x Length (ft) 0 60 100 70 200 90 300 65 400 60 A) 28,000 ft2
B) 29,000 ft2
C) 34,500 ft2
D) 28,500 ft2
242) A surveyor measured the length of a piece of land at 100-ft intervals (x), as shown in the table.
242)
Use Simpson's rule to estimate the area of the piece of land in square feet. x Length (ft) 0 60 100 70 200 90 300 65 400 60 A) 34,500 ft2
B) 29,000 ft2
C) 28,000 ft2
D) 28,500 ft2
243) The growth rate of a certain tree (in feet) is given by 2 2 y= + e-t /2 , t+ 1
where t is time in years. Estimate the total growth of the tree through the end of the second year by using the trapezoidal rule with n = 2. A) 5.41 feet B) 3.51 feet C) 1.75 feet D) 2.70 feet
41
243)
244) The growth rate of a certain tree (in feet) is given by
244)
2 2 y= + e-t /2 , t+ 1
where t is time in years. Estimate the total growth of the tree through the end of the second year by using Simpson's rule with n = 2. A) 5.11 feet B) 3.68 feet C) 2.34 feet D) 3.41 feet
Provide the proper response. 245) Why might we want to use numerical integration?
245)
i) The integral cannot be evaluated by any technique. ii) The antidifferentiation is complicated. iii) f(x) is not known. A) Both i and iii B) Both ii and iii
C) Both i and ii
D) All of the above
246) To apply the trapezoidal rule, how should the sum of rectangles found by using left endpoints
246)
and the sum of rectangles found by using right endpoints be combined? A) left endpoint sum - right endpoint sum 2
B) left endpoint sum + right endpoint sum 2
C) left endpoint sum + right endpoint sum D) left endpoint sum - right endpoint sum 247) When we use Simpson's rule, it is necessary that the number of subintervals be ? . A) a multiple of 4 B) even C) odd D) a multiple of 3
247)
248) Simpson's rule approximates consecutive portions of the curve with portions of ? . A) parabolas B) rectangles C) triangles D) trapezoids
248)
249) Which of the following expressions is associated with Simpson's rule?
249)
A) a - b 3n
B) b - a 2n
C) b - a 3n
D) b - a n
250) Given that n is the number of subdivisions, the difference between the true value of an integral
and the value given by the trapezoidal rule or Simpson's rule depends on which of the following? i) n
ii) n 2 iii) n 3
iv) n 4 A) Either ii or iv
B) Only ii
C) Either i or ii
42
D) Either iii or iv
250)
251) Which of the following numerical integration methods generally gives the best approximation for the same number of subintervals? A) Simpson's rule
B) summation of areas of rectangles D) No method is generally best.
C) the trapezoidal rule
43
251)
Answer Key Testname: UNTITLED7
1) D 2) C 3) C 4) D 5) C 6) D 7) B 8) C 9) C 10) D 11) A 12) B 13) B 14) D 15) D 16) B 17) A 18) D 19) D 20) B 21) A 22) B 23) D 24) C 25) B 26) A 27) D 28) D 29) B 30) A 31) C 32) D 33) D 34) A 35) D 36) D 37) A 38) D 39) A 40) B 41) D 42) D 44
Answer Key Testname: UNTITLED7
43) A 44) D 45) C 46) B 47) C 48) B 49) C 50) A 51) D 52) D 53) A 54) D 55) C 56) A 57) D 58) D 59) C 60) B 61) D 62) B 63) A 64) C 65) C 66) D 67) A 68) A 69) A 70) C 71) C 72) B 73) D 74) D 75) D 76) A 77) C 78) D 79) A 80) A 81) B 82) C 83) B 84) C 45
Answer Key Testname: UNTITLED7
85) A 86) C 87) C 88) C 89) B 90) D 91) C 92) C 93) A 94) C 95) D 96) B 97) D 98) A 99) C 100) B 101) A 102) D 103) B 104) B 105) C 106) A 107) B 108) B 109) B 110) A 111) C 112) A 113) C 114) D 115) C 116) C 117) C 118) C 119) B 120) Answers may vary. One possible answer: The indefinite integral is a set of functions. The definite integral represents a number.
121) D 122) B 123) A 124) D 125) C 46
Answer Key Testname: UNTITLED7
126) A 127) D 128) A 129) D 130) B 131) D 132) C 133) A 134) B 135) B 136) B 137) B 138) B 139) D 140) D 141) C 142) C 143) C 144) A 145) B 146) B 147) B 148) C 149) B 150) A 151) D 152) C 153) A 154) B 155) A 156) B 157) C 158) D 159) C 160) D 161) C 162) C 163) B 164) B 165) B 166) A 167) C 47
Answer Key Testname: UNTITLED7
168) A 169) C 170) C 171) D 172) A 173) D 174) B 175) A 176) A 177) C 178) A 179) C 180) C 181) C 182) C 183) C 184) A 185) C 186) D 187) C 188) D 189) D 190) D 191) D 192) B 193) C 194) B 195) D 196) D 197) D 198) A 199) B 200) B 201) C 202) D 203) D 204) C 205) B 206) B 207) D 208) C 209) D 48
Answer Key Testname: UNTITLED7
210) B 211) B 212) D 213) B 214) B 215) C 216) B 217) C 218) C 219) A 220) A 221) C 222) A 223) D 224) B 225) A 226) D 227) C 228) A 229) D 230) D 231) C 232) B 233) C 234) A 235) C 236) B 237) A 238) B 239) B 240) B 241) D 242) C 243) B 244) D 245) D 246) B 247) B 248) A 249) C 250) A 251) A 49
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use integration by parts to find the integral.
1)
3xex dx
1)
A) 3ex - 3xex + C 2)
C) 3ex - ex + C
2)
A) 1 x2 e2x - 1 xe2x + 1 e2x + C
B) 1 x2 e2x - xe2x + 1 e2x + C
C) 1 x2 e2x - 1 xe2x + C
D) 1 x2 e2x - 1 xe2x + 1 e2x + C
4
2
4
2
2
4
3) 2
2
2
4
B) x ln x - x + C
2
C) 3x2 ln x - x + C
D) 3x ln x - 3 x + C 2
4
ln 3x dx
4)
A) 3x ln x - x + C C) x ln 3x - 3x + C
B) x ln 3x + x + C D) x ln 3x - x + C
x - 3 ln x dx
5)
A) 1 x2 ln x - 1 x2 - 3x + C
B) 1 x2 ln x - 3xlnx - 1 x2 + 3x + C
C) ln x - 1 x2 + C
D) 1 x2 ln x - 1 x2 + C
2
4
2
4
6)
2
6x ln x dx 2
5)
4
2
A) 3x2 ln x - 3 x2 + C
4)
D) 3xex - 3ex + C
e2x x2 dx 2
3)
B) xex - 3ex + C
2
4
4
x4 ln 4x dx
6)
A) 1 x5 ln 4x - 1 x5 + C
B) 1 x5 ln 4x - 1 x6 + C
C) ln 4x - 1 x5 + C
D) 1 x5 ln 4x + 1 x5 + C
5
25
5
5
5
1
30 25
ln 4x dx x6
7)
7)
A) ln 4x + 1 x-5 + C
B) - 1 x-5 ln 4x - 1 x-4 + C
C) - 1 x-5 ln 4x - 1 x-5 + C
D) - 1 x-5 ln 4x + 1 x-5 + C
5
5
5
25
20
5
25
x 2 - x dx
8)
8)
A) - 2 x(2 - x)3/2 - 4 (2 - x)5/2 + C
B) - 2 x(2 - x)3/2 - 2 (2 - x)5/2 + C
C) - 2 x(2 - x)3/2 + 4 (2 - x)5/2 + C
D) 2 x(2 - x)3/2 + 4 (2 - x)5/2 + C
3
15
3
3
15
5
3
15
x - 6 e3xdx
9)
9)
A) 1 (x - 6)e3x + 1 e3x + C
B) 3(x - 6)e3x - 9 e3x + C
C) 1 (x - 6)e3x - 1 e3x + C
D) (x - 6)e3x - e3x + C
3 3
9 9
(6x + 4) e-2x dx
10)
10)
A) -12x e-2x - 32 e-2x + C
B) - 3x e-2x - 7 e-2x + C
C) - 3 x e-2x - e-2x + C
D) 3x e-2x + 7 e-2x + C
2
2
Use integration by parts to find the integral. Round the answer to two decimal places if necessary. 4 11) 7x ln x dx 2 A) 46.93 B) 11.06 C) 64.9 D) 6.70 3
12)
ln 3x dx
11)
12)
1
A) 3.49 1
13) 0
C) 7.8
B) -0.51
D) 7.49
x dx x+1
A) -1.33
13) B) 0.39
C) -0.94
2
D) -2.27
2
14)
x4 ln 6x dx
14)
1
A) 16.87 2
15)
B) 14.52
C) -4.23
D) 14.31
(x - 7)e3xdx
15)
1
A) -759.61 4
16)
B) -9,139.98
C) -2,279.97
D) -674.80
(x - 4) ln x dx
16)
0
A) -3.61 5
17)
C) 0.91
B) -8.91
D) 7.09
x 5 - x dx
17)
1
A) 13.87 3
18)
C) 9.60
B) -13.87
xex dx
18)
0 Give your answer in exact form. A) 4e3 + 1 B) 2e3 - 1 7
19)
C) 2e3 + 1
1 0
D) 2e3
xe- x dx
19)
0 Give your answer in exact form. A) -8e-7 - 1 B) -8e-7
20)
D) -3.20
C) -6e-7 + 1
D) -8e-7 + 1
7x + 3 dx ex
A) -6.25
20) B) 12.15
C) 3.75
D) 6.32
Find the integral by using integration by parts or other techniques. Round the answer to four decimal places if necessary.
21)
20x2 e2x dx
21)
A) 5e2x(x2 - x + 1) + C C) 10e2x(2x2 - 2x + 1) + C
B) 5e2x(2x2 - 2x + 1) + C D) 20e2x(2x2 - 2x + 1) + C
3
(2x - 1) ln(15x) dx
22)
22) 2
A) (x2 - x) ln 15x - x + x + C
B) (x2 - x) ln 15x - x2 + x + C
2
2
2
2
4
2
C) ( x - x) ln 15x - x + x + C
x2
23)
D) (x2 - x) ln 15x - x + 2x + C 2
x + 21 dx
23)
A) (30x - 504x + 7,056) (x + 21) + C
B)
(30x2 - 504x + 336) (x + 21)3 +C 105
(15x2 - 252x + 3,528) (x + 21)3 +C 105
D)
(30x2 - 504x + 7,056) (x + 21)3 +C 105
2
105
C)
x2 dx x2 + 21
24)
24)
A) x x2 + 21 - ln (x + x2 + 21) + C
B) 3x x2 + 21 - 21 ln (x + x2 + 21) + C
C) x x2 + 21 - 21 ln (x + x2 + 21) + C
D) x x2 + 21 + 21 ln (x + x2 + 21) + C
2
2
2
2
2
2
18xex2 dx
25)
A) 9 ex2 + C 2
2
26)
25) B) 9x2 ex2 + C
C) 18ex2 + C
D) 9ex2 + C
(1 - x2 ) e3x dx
26)
1
A) -256.9864 5
27)
x2
3
B) -254.0107
C) -403.4288
D) -200.1948
x3 + 6 dx
27)
0
A) 117.7945 28)
B) 163.6035
C) 166.3292
x dx (7x2 + 3)5
D) 215.9566 28)
A) - 7 (7x2 + 3)-6 + C
B) - 7 (7x2 + 3)-4 + C
C) - 1 (7x2 + 3)-6 + C
D) - 1 (7x2 + 3)-4 + C
3
3
14
56
4
20s3 ds 1 - s4
29)
29)
A) -10 1 - s4 + C
B) -10s3 1 - s4 + C
10s4
C)
D)
1 - s4
-5 2 1 - s4
+C
Use the table of integrals or a computer or calculator with symbolic integration capabilities to find the integral. 3x dx 30) 30) x2 (1 + 3x2 ) x2
A) ln
1 + 3x2
C) ln 1 + 3x
2
x2
31)
B) - 3 ln 1 + 3x
+C
D) - 3 ln 1 + 3x
x2
2
x2
+C
31)
5 + 4x + 5 + 4x -
C) 1 ln
5 +C 5
5 + 4x 5 + 4x +
5
5 + 4x 5 + 4x +
B) ln
5 +C 5
D) 1 ln 5
5 +C 5
5 + 4x 5 + 4x +
5 +C 5
x dx 1+x
32)
A) 2(x - 2) 1 + x + C
B) (x - 2) 1 + x + C
C) 4(x - 2) 1 + x + C
D) (x - 2) 1 + x + C
3
3
3
33)
2
+C
dx x 5 + 4x
A) ln
32)
2
+C
5 dx x2 + 9
33)
A) 5 ln x + x2 + 9 + C
2 B) - 5 ln 3 + x + 9 + C
C) 5 tan-1 x + C
D) 5 ln x + x2 + 3 + C
3
3
3
5
x
34)
2
81 - x2
dx
34)
A) 1 ln x - 9 + C
B) 2 tan-1 x + C
C) 2 sin-1 x + C
D) 1 ln x + 9 + C
9
x+9
9
9
35)
9
8
35)
1 +C 8x + 7
B) 1 tan-1 x + C 8
C) 1 ln 8x2 + 7x
7
x +C 8x + 7
1 dx x2 - 25
36)
A) 1 ln x - 5 + C
B) ln x + x2 - 25 + C
C) 1 ln 5 + x + C
D) ln x + x2 + 25 + C
10
x+5
10
5-x
1 x 9 + x2
dx
37)
2 A) - 1 ln 3 + 9 + x + C
2 B) 1 ln 3 + 9 + x + C
C) ln x + x2 + 9 + C
2 D) - 1 ln 3 + 9 - x + C
3
38)
8
D) 1 ln
8
37)
x-9
dx 8x2 + 7x
A) 1 ln
36)
9
x
3
x
3
x
2 dx 3x 4x + 9
A) 2 ln
38)
x +C 4x + 9
B) 2 ln
x +C 4x + 9
C) 9 + x - 9 ln(4x + 9) + C
D) 1 ln
x +C 4x + 9
9
16
4
27
16
9
6
39)
9x2 + 18 dx
39)
A) 3 x x2 + 18 + 18 ln x + x2 + 18 + C 2
B) 1 x x2 + 2 + 2 ln x + x2 + 2 + C 2
C) 3 x 9x2 + 18 + 9ln(x + 9x2 + 18) + C 2
D) 1 x 9x2 + 18 + 18 ln x + 9x2 + 18 + C 2
Solve the problem.
40) A particle moves so that its velocity (in m/s) is given by v = 2te-t, where t is the time (in seconds). Find the distance traveled between t = 0 and t = 4. A) 0.65 B) 2.11
C) 5.86
D) 1.82
41) The rate of growth of a microbe population is given by m'(x) = 30xe2x, where x is time in days. What is the growth after 1 day? A) 62.52 B) 62.92
C) 55.42
C) 15,062
C) 222,613,533
A) e7 - e3
B) 3e7 + e3
C) e7 + e3
A) 3 ln 3 - 2
C) 3 ln 3 - 3
45)
D) ln 3
46) The rate of water usage for a business, in gallons per day, is given by W(t) = 689te-t, where t
represents the number of hours since midnight. Approximately how many gallons of water does the business use in the first 4 hours of the day? A) 626 gallons B) 651 gallons C) 63 gallons D) 752 gallons
7
44)
D) 3e7
45) Find the area between y = ln x and the x-axis from x = 1 to x = 3. Give your answer in exact form. B) 2 3
43)
D) 222,613,544
44) Find the area between y = (x - 3)ex and the y-axis from x = 3 to x = 7. Give your answer in exact form.
42)
D) 30,161
43) The rate of growth of a microbe population is given by m'(x) = 30xe2x, where x is time in days. What is the net growth between day 3 and day 7? A) 117,238,789 B) 111,306,789
41)
D) 110.84
42) The rate of growth of a microbe population is given by m'(x) = 30xe2x, where x is time in days. What is the net growth between day 1 and day 3? A) 15,073 B) 30,175
40)
46)
47) A person's metabolic rate tends to go up after eating a meal and then, after some time has passed,
47)
it returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food, and the effect (in kJ/hr) for one individual is F(t) = -10.28 + 175.9te-t/1.3 where t is the number of hours that have elapsed since eating a meal. Find the total thermic energy of a meal for the next four hours after a meal by integrating the thermic effect function between t = 0 and t = 4. A) 128.4 kJ B) 200.3 kJ C) 186.5 kJ D) 150.1 kJ
Find the volume of the solid of revolution formed by rotating about the x-axis the region bounded by the curves. 48) y = x + 1, y = 0, x = -1, x = 7 48) 63 512 A) 4 B) 64 C) D) 2 3
49) f(x) = x, y = 0, x = 1, x = 23 A) 11 B) 528
49) C) 264.5
50) f(x) = x2 , y = 0, x = 1, x = 8 A) 6,553.6 B) 32,767
50) C) 6,553.4
51) f(x) = 2x + 5, y = 0, x = 0, x = 3 A) 24 B) 221.83
D) 170.33 51)
C) 402.00
52) f(x) = 9x + 3 , y = 0, x = 1, x = 5 A) 228 B) 120
D) 201.00 52)
C) 228
D) 120
1 , y = 0, x = -3, x = 7 Give your answer in exact form. x+4
53) f(x) =
B) 2 ( 11 - 1)
(ln 11 - 1)
A)
54) y = 1 , y = 0, x = 1, x
55) y = 1 ,
y = 0, x = 1,
x
A) 1
B) 1
56) y = ex, y = 0, x = -1, 2
(e3 - e-1 )
C) 1 ln 7
7
ln 11
D)
2
54) ln 7
D)
x=5
55) C) 2
ln 5
B)
5
C) ln 11
53)
x = 7 Give your answer in exact form.
A) 7
A)
D) 264
5
D) 4 5
x = 3 Give your answer in exact form.
B)
2
(e6 - e-2 )
C) (e6 - e-2)
8
56) D) 2 (e3 - e-1 )
57) y = 81 - x2 , y = 0, x = 0, x = 9 A) 324 B) 972
57) C) 486
D) 18
Find the average value of the function on the given interval. 58) f(x) = 3x2 - 4; [0, 6]
A) 32 3
59) f(x) = ex/2; [0, 17] A) 288.79 60) f(x) = x + 2; [1, 13] A) 3.070 61) f(x) = x + 4; [1, 11] A) 10 62) f(x) = 4 - x2 ; [1, 17] A) -303
58)
B) 33
C) 32
59) B) 577.70
C) 578.09
4
64) f(x) = (8x + 1)1/2 ; [0, 3] A) 31 9
B) 2.939
C) 2.713
20
B) 16.050
C) 10.450
D) 9.091 62)
B) -98.104
C) -92.549
D) -98.333 63)
B) 0
C) 81
B) 31 3
C) 31 3
D) 81
2
64)
B) 41 e10 - 1 4
4
C) 41 e10 - 1 20
4 B) 2 (4e + 1) (e2 - 1)
D) 13
65)
66) f(x) = 8x ln x ; [1, e2 ] Give your answer in exact form. 2 A) 2 (3e + 1) (e2 - 1)
D) 4.408 61)
65) f(x) = x2 e2x; [0, 5] Give your answer in exact form. A) 41 e10
D) 144.40 60)
63) f(x) = x3 - 9x2 + 27x - 27; [0, 6] A) 81
D) 36
20
4 C) 2 (3e - 1) (e2 + 1)
D) 17 e10 - 2 5
5
66) 4 D) 2 (3e + 1) (e2 - 1)
Solve the problem. 67) The amplitude of an alternating voltage V = V(t) is sometimes indicated by giving the rms (root mean square) voltage, which is the square root of the average value of V2 . Find the rms voltage if V(t) = 6,600t over the period t = 0 to t = 1/60 s. A) 63.5 V B) 36.7 V
C) 0.1 V
9
D) 1,346.9 V
67)
68) Suppose the number of items a new worker on an assembly line produces daily after t days on the
68)
69) The voltage v (in volts) induced in a tape head is given by v = t2 e3t, where t is the time (in
69)
70) Suppose the number of items a new worker on an assembly line produces daily after t days on the
70)
71) The design of an electric power generating station depends on both the peak and the average
71)
72) The price per share of a stock can be approximated by the function S(t) = t(25 - 3t) + 22, where t is
72)
73) The intensity of the reaction to a certain drug, in appropriate units, is given by R(t) = te-0.04t,
73)
job is given by 25 + 2t. Find the average number of items produced daily in the first 10 days. A) 40 B) 35 C) 38 D) 350
seconds). Find the average value of v over the interval from t = 0 to t = 2. Round to the nearest volt. A) 194 volts B) 1,564 volts C) 40 volts D) 6 volts
job is given by 25 + 2t. Find the average number of items produced daily in the first 20 days. A) 45 B) 40 C) 48 D) 900
power that it must produce. If a community uses 112 + 48t - 2t2 megawatts at time t (in hours) during the period t = 0 to t = 24, find the average level of power consumption for that day. A) 688 MW B) 304 MW C) 7,296 MW D) 16,512 MW
time (in years) since the stock was purchased. Find the average price of the stock over the first 7 years. A) $423.50 B) $33.00 C) $60.50 D) $78.00
where t is time (in hours) after the drug is administered. Find the average intensity during the 8th hour. A) 825e-0.32 - 850e-0.36 B) 800e-0.28 + 625e-0.32
C) -0.72e-0.28 - 825e-0.32
D) 800e-0.28 - 825e-0.32
74) Find the volume of a right circular cone with a height of 10 meters and a base radius of 5 meters. A) 250 3
cubic m
B) 250 cubic m 3
C) 61 cubic m
D) 50 3
cubic m
75) An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve y=1-
x2 25
, -5 x 5, about the x-axis (dimensions are in feet). How many cubic feet of fuel will
the tank hold to the nearest cubic foot? A) 5 cubic ft B) 21 cubic ft
C) 17 cubic ft
10
74)
D) 8 cubic ft
75)
76) An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve y=1-
x2 16
76)
, -4 x 4, about the x-axis (dimensions are in feet). If a cubic foot holds 7.481 gallons
and the helicopter gets 3 miles to the gallon, how many additional miles will the helicopter be able to fly once the tank is installed (to the nearest mile)? A) 100 mi B) 150 mi C) 75 mi D) 301 mi
The function represents the rate of flow of money in dollars per year. Assume a 10-year period and find the present value. 77) f(x) = 500 at 2% compounded continuously 77) A) $5,535.07 B) $45,468.27 C) $20,468.27 D) $4,531.73
78) f(x) = 2000 at 7% compounded continuously A) $14,188.15 B) $42,759.58
C) $14,383.28
D) $28,964.36
79) f(x) = 500e0.04x at 5% compounded continuously A) $4,758.13 B) $95,241.87
C) $5,258.55
D) $55,258.55
80) f(x) = 1000e-0.04x at 5% compounded continuously A) $27,328.92 B) $16,217.81 C) $15,628.55
D) $6,593.67
81) f(x) = 0.5x at 4% compounded continuously A) $605.77 B) $186.81
C) $38.46
D) $19.23
82) f(x) = 0.01x + 400 at 8% compounded continuously A) $7,249.47 B) $2,118.20
C) $2,294.71
D) $2,753.65
78)
79)
80)
81)
82)
83) f(x) = 1,800x - 120x2 at 6% compounded continuously A) $35,267.04 B) $1,257,489.26 C) $86,634.34
83) D) $645,057.74
The function represents the rate of flow of money in dollars per year. Assume a 10-year period and find the accumulated amount of money flow at t = 10. 84) f(x) = 500 at 6% compounded continuously 84) A) $8,333.33 B) $12,483.32 C) $23,517.66 D) $6,850.99
85) f(x) = 2000 at 7% compounded continuously A) $86,107.22 B) $28,964.36
85) C) $28,571.43
86) f(x) = 500e0.04x at 7% compounded continuously A) $8,698.80 B) $11,742.15
D) $58,327.06 86)
C) $45,304.70
87) f(x) = 1000e-0.04x at 4% compounded continuously A) $10,268.81 B) $22,853.65 C) $41,501.46 11
D) $58,426.29 87) D) $27,026.81
88) f(x) = 0.5x at 5% compounded continuously A) $29.74 B) $629.74
88) C) $229.74
D) $59.48
89) f(x) = 0.08x + 900 at 6% compounded continuously A) $10,280.60
B) $42,407.83
89) C) $12,336.72
D) $11,215.20
Solve the problem. 90) An investment is expected to produce a uniform continuous rate of money flow of $500 per year for 10 years. Find the present value at 3% compounded continuously. A) $12,346.97 B) $29,013.64 C) $4,319.70 D) $5,830.98
90)
91) The rate of a continuous money flow starts at $500 and increases exponentially at 4% per year for
91)
92) The rate of a continuous money flow starts at $1000 and decreases exponentially at 4% per year for
92)
93) A money market fund has a continuous flow of money at a rate of f(x) = 0.01x + 300 for 10 years.
93)
94) A money market fund has a continuous flow of money at a rate of f(x) = 2,700x - 160x2 for 10
94)
95) A real estate investment is expected to produce a uniform continuous rate of money flow of $2000
95)
96) The rate of a continuous money flow starts at $500 and increases exponentially at 4% per year for
96)
97) The rate of a continuous money flow starts at $1000 and decreases exponentially at 4% per year for
97)
98) A money market fund has a continuous flow of money at a rate of f(x) = 0.5x for 10 years. Find the
98)
10 years. Find the present value if interest is earned at 7% compounded continuously. A) $5,830.98 B) $22,497.65 C) $29,013.64 D) $4,319.70
10 years. Find the present value if interest is earned at 3% compounded continuously. A) $21,379.79 B) $28,767.90 C) $14,482.18 D) $7,191.64
Find the present value of this flow if interest is earned at 5% compounded continuously. A) $1,816.29 B) $2,361.18 C) $9,646.82 D) $1,967.65
years. Find the present value of this flow if interest is earned at 7% compounded continuously. A) $117,703.93 B) $53,999.52 C) $817,847.92 D) $517,286.10
per year for 10 years. Find the final amount at an interest rate of 8% compounded continuously. A) $68,187.29 B) $25,000.00 C) $30,638.52 D) $80,638.52
10 years. Find the final amount if interest is earned at 5% compounded continuously. A) $7,844.83 B) $91,105.94 C) $157,027.30 D) $8,669.88
10 years. Find the final amount if interest is earned at 6% compounded continuously. A) $11,517.99 B) $31,309.14 C) $49,530.32 D) $24,924.39
final amount if interest is earned at 6% compounded continuously. A) $61.70 B) $197.52 C) $30.85
12
D) $475.29
99) A money market fund has a continuous flow of money at a rate of f(x) = 1,900x - 140x2 for 10
99)
years. Find the final amount if interest is earned at 8% compounded continuously. A) $48,333.33 B) $68,614.77 C) $30,830.60 D) $13,853.08
Determine whether the improper integral is convergent or divergent.
100) 2
21 dx x2
100)
A) Convergent 11
101) 2
(x + 1)2
B) Divergent
dx
101)
A) Convergent
102) 1
B) Divergent
11 dx x
102)
A) Divergent
103) -
B) Convergent
35x4 + 2 dx 7x5 + 2x + 7
103)
A) Convergent
104) 0
B) Divergent
dx x1/9
104)
A) Divergent
105) 6
B) Convergent
dx x7/4
105)
A) Divergent
106) 7
B) Convergent
dx 7 x
106)
A) Divergent
B) Convergent
dx
107) 4
107)
x + 6 9/8
A) Divergent
B) Convergent
13
3x + 6 3 9x + 2x2 + 1
108) 1
108)
A) Divergent
B) Convergent
2x + 3 2 x + 3x + 1
109) 1
109)
A) Divergent
B) Convergent
2ex dx
110)
110)
0
A) Convergent 0
111)
B) Divergent
7e5x dx
111)
-
A) Divergent
B) Convergent
ln x dx
112)
112)
1
A) Divergent
113) 1
B) Convergent
ln x dx x
113)
A) Convergent
114) 1
B) Divergent
x dx ex
114)
A) Convergent
B) Divergent
4 x3 e-x dx
115)
115)
-
A) Divergent
B) Convergent
e-7x dx
116)
116)
2
A) Divergent
B) Convergent
14
0
117)
e9x dx
117)
-
A) Divergent 0
118)
B) Convergent
2 4xe-x dx
118)
-
A) Divergent
B) Convergent
2 2xe-x dx
119)
119)
-
A) Convergent
B) Divergent
Evaluate the improper integral. If the integral does not converge, state that the integral is divergent. -3 3 120) dx 5 x A) - 1 B) 0 C) 1 D) - 1 108 27 972
25
121)
(x + 1)2
0
dx
A) 0 0
122)
120)
121) C) 25
B) -25
D) Divergent
25xe3x dx
122)
-
A) 0
C) 3.7778
B) -2.7778
D) Divergent
17xe2x dx
123)
123)
0
A) -4.25
B) 4.25
C) 0
D) Divergent
6 x5 e-x dx
124) -
A) - 1 3
124) B) 1
C) 0
6
15
D) Divergent
0
125)
4 e3x dx
125)
-
B) 4
A) 0
x dx ex
126) 1
D) Divergent
Give your answer in exact form.
A) 2
126) C) - 1
B) 0
e
e-4x dx
127)
C) -4
3
e
D) Divergent
Give your answer in exact form.
127)
2 -8
A) e
B) -e-8
4
0
128)
C) 0
D) Divergent
2 8xe-x dx
128)
-
A) -4
C) 0
B) -8
D) Divergent
Find the area between the graph of the function and the x-axis over the given interval, if possible. 129) f(x) = 5 for (- , 0] (x - 1)2
A) 1 130) f(x) =
C) 5
B) -5 24
(x - 1)3
D) Divergent
for (- , 0]
A) -0.5 131) f(x) = 21e-x for (- , e] A) -1.386
130) B) 12
C) -12
D) Divergent
B) -318.234
C) 1.386
D) Divergent
131)
132) f(x) = 15 for (- , 0]
132)
x-1
A) 0 133) f(x) =
C) 15
B) -15 x
5 (1 + x2 )
A) - 1 8
129)
D) Divergent
for (- , )
133) C) 1
B) 0
8
16
D) Divergent
134) f(x) = x2 e-x3 for (- , ) A) 1 3
134) C) - 1 3
B) 0
D) Divergent
135) f(x) = 1 for (-1, )
135)
x+2
A) 1
B) ln 2
4
136) f(x) =
1
(x + 5)3
C) 1
D) Divergent
for (-5, )
A) 3 5
136) B) 5
C) 1
9
10
D) Divergent
137) f(x) = 1 for (1, )
137)
x1.9
A) 10 29
B) 19
C) 10
29
9
D) Divergent
Solve the problem. Round your answer to the nearest whole number.
138) The capital value of an asset is defined as
R(t)e-rt dt , where k is the annual rate of interest
138)
0 compounded continuously and R(t) gives the annual rate at which earnings are produced by the asset at time t. Find the capital value of an asset that produces $5000 yearly income at 4% compounded continuously. A) $100,000 B) $150,000 C) $125,000 D) $130,000
139) The capital value of an asset is defined as
R(t)e-rt dt , where k is the annual rate of interest
139)
0 compounded continuously and R(t) gives the annual rate at which earnings are produced by the asset at time t. Find the capital value of an asset that produces $5000 yearly income at 5% compounded continuously. A) $100,000 B) $95,000 C) $83,333 D) $125,000
140) The capital value of an asset is defined as
R(t)e-rt dt , where k is the annual rate of interest
0 compounded continuously and R(t) gives the annual rate at which earnings are produced by the asset at time t. Find the capital value of an asset that produces $5000 yearly income at 6% compounded continuously. A) $100,000 B) $83,333 C) $80,000 D) $85,000
17
140)
141) The capital value of an asset is defined as
R(t)e-rt dt , where k is the annual rate of interest
141)
0 compounded continuously and R(t) gives the annual rate at which earnings are produced by the asset at time t. Find the capital value of an asset that produces $5000 yearly income at 7% compounded continuously. A) $75,000 B) $71,429 C) $83,333 D) $70,000
142) The capital value of an asset is defined as
R(t)e-rt dt , where k is the annual rate of interest
142)
0 compounded continuously and R(t) gives the annual rate at which earnings are produced by the asset at time t. Find the capital value of an asset that produces $5000 yearly income at 8% compounded continuously. A) $60,500 B) $62,500 C) $71,429 D) $65,000
143) The capital value of an asset is defined as
R(t)e-rt dt , where k is the annual rate of interest
143)
0 compounded continuously and R(t) gives the annual rate at which earnings are produced by the asset at time t. Find the capital value of an asset that produces $5000 yearly income at 9% compounded continuously. A) $55,000 B) $50,000 C) $62,500 D) $55,556
144) The capital value of an asset is defined as
R(t)e-rt dt , where k is the annual rate of interest
144)
0 compounded continuously and R(t) gives the annual rate at which earnings are produced by the asset at time t. Suppose an asset produces a perpetual stream of income with a flow rate of R(t) = 1200e0.03t . Find the capital value at an interest rate of 7% compounded continuously.
A) $17,142
B) $30,000
145) The capital value of an asset is defined as
C) $12,000
D) $40,000
R(t)e-rt dt , where k is the annual rate of interest
145)
0 compounded continuously and R(t) gives the annual rate at which earnings are produced by the asset at time t. Suppose income from an investment starts (at time 0) at $8000 a year and increases linearly and continuously at a rate of $300 per year. Find the capital value at an interest rate of 6% compounded continuously. A) $216,667 B) $2,222,222 C) $133,333 D) $138,333
146) The rate of a reaction to a drug is given by r'(t) = 3t2 e-t, where t is the number of hours since the drug was administered. Find the total reaction to the drug over all the time since it was administered, assuming this is an infinite time interval. (Hint: lim tke-t = 0 for all real numbers t k.)
A) 0
B) 6
C) 9
18
D)
146)
147) In an epidemiological model used to study the spread of drug use, a single drug user is
147)
introduced into a population of N non-users. Under certain assumptions, the number of people expected to use drugs as a result of direct influence from each drug user is given by 2(1 - e-kt) -bt e dt , k
S=N
0 where b and k are constants. Find the value of S. A) 2Ne-bkt
B) 2N/[b(b + k)] D) N(1 - e-kt)e-bt
C) 2N/(b + k)
148) Radioactive waste is entering the atmosphere over an area at a decreasing rate. Use the improper integral
148)
Pe-kt dt with P = 15 to find the total amount of waste that will enter the atmosphere
0 for k = 0.02. A) 75
B) 7,500
C) 30
D) 750
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide the proper response.
149) A student wishes to find the integral
f(x) dx of a function that has the property limit
149)
0 lim f(x) = 1. Why can this not be done? x
150) A student wishes to take the integral over all real numbers of f(x) = claims this is zero because - +
151) A student claims that
b
x2 if x < 0 1 , if x > 0 , and x
150)
equals zero. What is wrong with this thinking?
f(x) dx always exists, as long as a and b are both positive.
151)
a Refute this by giving an example of a function for which this is not true.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
152) A student knows that
-1
f(x) dx = 56. Can a
f(x) dx be found, and if so, what is it?
-
A) Yes, -56
B) No
19
152)
153) A student knows that
f(x) dx diverges, but needs to investigate
a f(x) g(x) = . Does this integral necessarily also diverge? 7
A) Yes
g(x) dx , where
153)
a
B) No
154) A student knows that
a
f(x) dx converges. Does a
f(x) dx also necessarily converge?
154)
-
A) Yes
B) No
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
155) A student needs -
1 dx . Is this integral the same as 2 ex
20
0
1 dx , and if so, why? ex
155)
Answer Key Testname: UNTITLED8
1) D 2) D 3) A 4) D 5) B 6) A 7) C 8) A 9) C 10) B 11) A 12) A 13) B 14) D 15) D 16) C 17) A 18) C 19) D 20) C 21) B 22) A 23) D 24) C 25) D 26) A 27) B 28) D 29) A 30) D 31) C 32) A 33) A 34) D 35) D 36) B 37) A 38) B 39) C 40) D 41) B 42) A 21
Answer Key Testname: UNTITLED8
43) A 44) B 45) A 46) A 47) B 48) D 49) D 50) C 51) D 52) D 53) D 54) D 55) D 56) B 57) C 58) C 59) C 60) B 61) A 62) D 63) B 64) A 65) C 66) D 67) A 68) B 69) A 70) A 71) B 72) C 73) D 74) A 75) C 76) D 77) D 78) C 79) A 80) D 81) D 82) D 83) A 84) D 22
Answer Key Testname: UNTITLED8
85) B 86) A 87) A 88) A 89) C 90) C 91) D 92) D 93) B 94) B 95) C 96) A 97) A 98) C 99) B 100) A 101) A 102) A 103) B 104) A 105) B 106) A 107) B 108) B 109) A 110) B 111) B 112) A 113) B 114) A 115) B 116) B 117) B 118) B 119) A 120) A 121) C 122) B 123) D 124) C 125) B 126) A 23
Answer Key Testname: UNTITLED8
127) A 128) A 129) C 130) C 131) D 132) D 133) B 134) D 135) D 136) D 137) C 138) C 139) A 140) B 141) B 142) B 143) D 144) B 145) A 146) B 147) B 148) D 149) The only way the limit of the integral can exist is if the limit of the function is zero. 150) Infinity cannot be added like this. 151) Answers will vary, but f(x) =
1
(x - c)d
where a < c < b and d a positive integer is a family of examples.
152) B 153) A 154) B 155) Yes, the function is symmetric about the y-axis.
24
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the function. 1) Find f(3, 4) when f(x, y) = 3x + 2y - 7. A) 7 B) 8
1) C) 17
D) 10
2) Find g(2, 1) when g(x, y) = 4y2 - 8xy. A) -14 B) 1
C) -12
D) 0
3) Find h(3, 6) when h(x, y) = 3x + y2 . A) 3 5 B) 5 3
C) 9
D) 10
4) Find f(100, 3) when f(x, y) = y log x. A) 60 B) 3
C) 6
D) 30
2)
3)
4)
5) Find g(3, 4) when g(x, y) = x - 6y .
5)
x2 + y2
A) - 25 21
B) - 5
C) - 21
21
25
6) Find h(9, 1) when h(x, y) = (x + y)3 . A) 1,000 B) 30
D) - 21 5
6) C) 100
7) Find f(5, 0, 9) when f(x, y, z) = 5x2 + 5y2 - z 2 . A) 44 B) -56
D) 730 7)
C) 53
8) Find f(0, 1, -1) when f(x, y, z) = 2 x - 3yz + 8x. A) -2 B) -3
D) 206 8)
C) 4
1
D) 3
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Graph the first-octant portion of the plane. 9) 3x + 3y + 2z = 6
9)
10) x + z = 3
10)
2
11) 9x + 12y + 18z = 36
11)
12) 6y + 3z = 12
12)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the level curves in the first quadrant of the xy-plane for the given function at heights of z = 0, z = 2, and z = 4. 13) 4x + 2y - 4z = 4 13)
3
A) z=0
z=4
z=2
B) z=4
z=2
z=0
C) z=4
z=2
z=0
D) z=4
z=2
z=0
4
2
14) - x + y = 2z
14)
5
A)
z=4 z=2 z=0
B)
z=4 z=2 z=0
5
C)
z=4 z=2 z=0
D)
z=4
z=2 z=0
2
15) - y + x = 2z
15)
5
6
A)
z=4 z=2 z=0
B)
z=4
z=2 z=0
C)
z=4
z=2
z=0
7
D)
z=4
z=2 z=0
Choose the graph that matches the equation. 16) z = -x2 - y2
16)
A)
B)
C)
D)
8
17) z = 3 - x2 A)
B)
C)
D)
17)
9
18) z = 4 - x2 - y2 A)
B)
C)
D)
18)
10
19) z = 4x2 + 4y2 + 2 A)
19) B)
C)
D)
11
20) z = 1 - x - 2y A)
B)
C)
D)
20)
21) x2 + y2 + (z - 1)2 = 1 A)
21) B)
C)
D)
12
Solve the problem.
22) Let f(x, y) = 9x - 2y2 - 4 Find
22)
lim f(x + h, y) - f(x, y) h 0 h
A) 9 - 2y2 - 4
B) 9 - 4y
C) 9
D) 9x
23) Let f(x, y) = 8x - 3y2 - 5 Find
23)
lim f(x, y + h) - f(x, y) h 0 h
A) -6y
B) 8x - 6y2
C) 8 - 6y2
D) -6y - 5
24) Let f(x, y) = 5 - 4x2 - 2y3 Find
24)
f(x + h, y) - f(x, y) h
A) -8x - 4h 2
B) -8x2 - 4h
C) -8x - 4h
D) -8x - 4h - 2y3
25) Let f(x, y) = 5 + 10x2 + 9y3 Find
25)
f(x, y + h) - f(x, y) h
A) 9y2 h + 27yh + 9h2 C) 27y2 + 27yh + 9h 2
B) 10x2 + 27y2 + 27yh + 9h 2 D) 27y2h + 27yh + 27h2
26) Production of television sets is given by P(x,y) = 100 2 x-2/3 + 2 y-1/3 -3, where x is work hours 3
5
26)
and y is the amount of capital. If 125 work hours and 64 units of capital are used, what is the production output? A) 2,536 B) 492 C) 25 D) 49,205
27) The number of cows that can graze on a ranch is approximated by C(x,y) = 9x + 5y - 6, where x is
27)
28) The surface area of a human body (in square meters) is approximated by
28)
the number of acres of grass and y the number of acres of alfalfa. If the ranch has 80 acres of alfalfa and 70 acres of grass, how many cows may graze? A) 1,064 cows B) 1,024 cows C) 1,030 cows D) 1,070 cows
A = 0.202W(0.425) H(0.725), where W is the weight of the person in kilograms and H is the height in meters. Find A if W = 71 and H = 1.3. Round your answer to two decimal places. A) 1.61 m2 B) 1.50 m2 C) 1.66 m2 D) 1.60 m2
13
n
29) The multiplier function M = (1 + i) (1 - t) + t compares the growth of an Individual Retirement 1 + (1 - t)i n
29)
Account (IRA) with the growth of the same deposit in a regular savings account. The function M depends on the three variables n, i, and t, where n represents the number of years an amount is left at interest, i represents the interest rate in both types of accounts, and t represents the income tax rate. Values of M > 1 indicate that the IRA grows faster than the savings account. Find the multiplier when funds are left for 19 years at 5% interest and the income tax rate is 29%. A) 61.962 B) 1.074 C) 1.263 D) 1.426
30) The volume of a flower pot is given by V = 1 h r1 2 + r2 2 + r1r2 where r1 is the major radius and 3
30)
r2 is the minor radius and h is the height of the pot (see figure below).
If the dimensions of the pot are r1 = 7 inches, r2 = 4 inches and h = 8 inches, find the volume of potting soil required to fill the pot to the top. Round to the nearest cubic inch. A) 790 in.3 B) 779 in.3 C) 327 in.3
D) 1,523 in.3
31) The price-earnings ratio of a stock is given by R(P, E) = P , where P is the price per share of a E
31)
stock, and E is the earnings per share of the same stock. If the price per share of a given stock is $120, and the earnings per share of the same stock are $37.65, what is the price-earnings ratio of the stock? Give decimal notation to the nearest tenth. A) 157.7 B) 0.3 C) 3.2 D) 4,518.0
32) The yield of a stock is given by Y(D, P) = D , where D is the dividends per share of a stock, and P P
32)
is the price per share of the stock. If the price per share of a stock is $145, and the dividends per share of the same stock are $4.65, then what is the yield of the stock to the nearest tenth of a percent? A) 3.2% B) 31.2% C) 1.6% D) 674.3%
33) The sum of all the forces acting on an object can be described mathematically by F(m, a) = ma,
where m is the mass of the object (in kg) and a is the acceleration felt by the object (in m/s2 ) as a result of the forces (in newtons, N) acting on it. What is the force required to accelerate a 3 kg object at 11 m/s2 ?
A) 3 N
B) 33 N
C) 121 N
14
D) 14 N
33)
34) Newton's Universal Law of Gravitation states that the attractive force exerted on one spherically
34)
symmetric object of mass M by a second spherically symmetric object of mass m is given by mM F=G , where r is the distance between the centers of the two objects and G is a constant equal r2 to 6.67 × 10-11 Nm 2 /kg2 when F is in newtons (N), M and m are in kilograms (kg), and r is in meters (m). What is the magnitude of the gravitational force exerted on an object of mass 60 kg by an object of mass 1 × 1012 kg if the objects are 18 m apart?
A) 123.5 N
B) 222.3 N
C) 12.4 N
D) 36,018.0 N
35) The intelligence quotient in psychology is given by
35)
m Q(m, c) = 100 · , c
where m is a person's mental age, and c is his or her chronological, or actual, age. Find Q(21, 19) and round your answer to the nearest whole number. A) 100 B) 111 C) 90 D) 11
Find the partial derivative.
36) Find fx(7, -9) when f(x,y) = 7x2 - 9xy. A) 17 B) 179
36) C) -224
37) Find fy(1, 5) when f(x, y) = 7xy - 6y. A) 35 B) 0
D) -17 37)
C) 1
38) Find fx(3, -2) when f(x, y) = (6x + 4y)2 . A) 60 B) 20
D) 7 38)
C) 120
39) Find fx(3, -2) when f(x, y) = x3 + 2x2 y. A) 15 B) 3
D) 216 39)
C) 21
40) Find fx(-5, -1) when f(x, y) = 6x3 - 3xy - y. A) 453 B) 3
D) 33 40)
C) 2
D) 452
41) Find fx(-4, -2) when f(x, y) = e5x + 3y. Leave your answer in terms of e. A) 5e-14
B) e-26
C) 5e-26
41) D) 3e-26
42) Find fx(4, 2) when f(x, y) = ln xy + x + y + 1 . A) 1 5
B)
42)
5 ln 15
C) 1
15
D) ln 15
43) Find fy(5, 4) when f(x, y) = yexy. Leave your answer in terms of e. A) e20 + 20 e20
B) 4e20 + 20 e20
C) 20 e20
15
43) D) e20 + 4 e20
3 44) Find fy(-1, 2) when f(x, y) = 2xy .
44)
x2 + y2
A) 48 25
B) 184
C) - 56
25
25
45) Find gx(2, 1) when g(x, y) = ln 1 + 5x2 y3 . A) 21 11
D) - 56 5
45)
B) 11
C) 20
21
21
46) f(x, y) = 8x - 8y2 - 2. Find fx(x, y). A) 8x B) 8
D) 21 20
46) C) 6
D) -16y
47) Let z = f(x,y) = 5x2 - 18xy + 4y3 . Find z .
47)
x
A) 10x2 - 18x C) -18x + 12y2
B) 10x - 18y + 12y2 D) 10x - 18y
48) f(x,y) = 2x + 4x2 y2 - 8y2 . Find fx(x, y). A) 8xy2 - 16y
48)
B) 2 + 8x2 y
C) 2 + 8xy2
D) 8x2 y - 16y
49) Let z = g(x,y) = 6x + 8x2 y2 - 3y2 . Find z .
49)
y
A) 16yx - 16y
C) 6 + 16yx2
B) 16yx - 6y
D) 16yx2 - 6y
50) Let z = f(x, y) = x3 - 9x2 y - 6xy3 . Find z .
50)
x
A) -9x2 - 6xy2 C) 3x2 - 18x - 6
B) 3x2 - 18xy - 6y3 D) 3x2 - 18xy - 18xy2
51) f(x, y) = x3 + 2x2y - 9xy3. Find fy(x, y).
51)
A) x3 + 4xy - 27xy2 C) 2x2 - 9xy2
B) 2x2 y - 27y2 D) 2x2 - 27xy2
52) Let z = f(x, y) = (x + y)6 . Find z .
52)
x
A) 6(x + y)5
B) 6y(x + y)5
C) 6(x + y)
53) Let z = f(x, y) = 5(x + 5y - 2)2 . Find z .
D) 6x(x + y)5 53)
y
A) 25x + 125y C) 50x + 250y + 100
B) 25x + 125y - 50 D) 50x + 250y - 100
16
54) f(x, y) = 6(x + 2y - 8)2. Find fx(x, y). A) 12x + 24y + 96 B) 12x + 24y
54) C) 12x + 24y - 96
D) 6x + 12y - 48
Find fx (x, y).
55) f(x, y) = ex-y A) fx(x, y) = xex-y
B) fx(x, y) = -ex-y
C) fx(x, y) = -yex-y
D) fx(x, y) = ex-y
55)
56) f(x, y) = e2x - 4y A) fx(x, y) = 2e2x - 4y
56) B) fx(x, y) = e2x - 4
C) fx(x, y) = -4e2x - 4y
D) fx(x, y) = 2e2x
57) f(x, y) = e3xy A) fx(x, y) = 3ye3x
57) B) fx(x, y) = 3(x + y)e3xy
C) fx(x, y) = 3ye3xy
D) fx(x, y) = 3e3xy
58) f(x, y) = x ln (9x + 5y)
58)
A) fx(x, y) = 9x + ln (9x + 5y) 9x + 5y
B) fx(x, y) = ln (9x + 5y)
9 9x + 5y
C) fx(x, y) = ln (9x + 5y) +
D) fx(x, y) =
59) f(x, y) = y ln (6x + 6y)
59)
A) fx(x, y) = 6xy 6x + 6y C) fx(x, y) =
9x 9x + 5y
B) fx(x, y) = ln (6x + 6y) + 6 6x + 6y
6y 6x + 6y
D) fx(x, y) = y ln (6x + 6y)
60) f(x, y) = (x + y) ln (xy)
60)
A) fx(x, y) = ln (xy) + (x + y)
B) fx(x, y) = y ln (xy) + (x + y)
C) fx(x, y) = x ln (xy) + (x + y)
D) fx(x, y) = ln (xy) + (x + y)
xy
x
x
x
61) f(x, y) = 5x - y y
61)
5x
A) fx(x, y) = - 5 - 5
B) fx(x, y) = 5 + y
C) fx(x, y) = 5 - y
2 D) fx(x, y) = 5x + y
y2
y
y
y
5x 2
y
17
5x 2 5x2
62) f(x, y) = y ln 1
62)
x
A) fx(x, y) = y 2
C) fx(x, y) = - y
B) fx(x, y) = - y
x2
x2
D) fx(x, y) = - 1
x
x
2
63) f(x, y) = x + y
63)
xy
2
2
3
2 2
xy
x 2 y2
2
2
B) fx(x, y) = 2x + 2y - x y
A) fx(x, y) = yx - y
2
2
2
D) fx(x, y) = x - y
C) fx(x, y) = x - y
xy
x2
64) f(x, y) = x
64)
x+y
A)
y
B) -
(x + y)2
65) f(x, y) = A) C) -
x
C) 2x + y
(x + y)2
D) -
(x + y)2
y
(x + y)2
1 2 x + y2
65)
x
B) -
3/2 (x2 + y2 ) x
D)
3/2 2(x2 + y2 )
1
3/2 2(x2 + y2 ) y
3/2 (x2 + y2 )
9
66) f(x, y) = ln y
66)
x2 9
A) -ln 2
x
C) - 2
D) - 2y
B) y xy
C) y x
D) 1 xy
x
x3
67) f(x, y) = ln xy A) y ln x
9
B) -ln 2y
x
67)
-x 68) f(x, y) = e
68)
x2 + y2
2 -x 2 A) - e (x + y + 2x)
B) -
2 -x 2 C) e (x + y + 2x)
2 -x 2 D) - e (x + y + x)
2 (x2 + y2 )
2 (x2 + y2 )
2xe-x
2 (x2 + y2 ) 2 (x2 + y2 )
18
69) f(x, y) = xye-x A) y(e-x - x) 2
69) B) e-x(1 - x)
C) -ye-x
D) ye-x(1 - x)
2
70) f(x, y) = x - y
70)
x2 + y2
A)
4xy2
B)
2 (x 2 + y2 )
71) f(x, y) =
6
C)
4 (x 2 + y2 )
2 (x 2 + y2 )
D)
2y2
2 (x 2 + y2 )
71) 6
7 4 C) 7 x y
5
6 4
B) 7x6 x7 y4
6x
x7 y4 x-1 + 1+x
72) f(x, y) =
6xy4
72) B)
(1 + x)2
1+x 2 x-1
6
7 4 D) 7 x y
2y
A) x - 1 + 2y C)
2x2 y2
x7 y4
A) 7x y 6
4xy2
2y (1 + x)-2
x-1 -
D)
73) f(x, y) = x14 - 5x6 y23 + 4y-7 A) 14x13 - 30x6 y23 C) 14x13 - 5x6y23 + 4y-7
1
x + 1 (1 + x)2 2y 1+x
73) B) 14x13 - 30x6 y22 - 28y-8 D) 14x13 - 30x5 y23
4 3
3
74) f(x, y) = x - y
74)
x2 + y2
4 2 x4 + 3x2 y2 + 2xy4 2 x2 + y2 (x2 + y2)
4 2 x4 + 3x2 y2 + 2xy4 4 x2 + y2 (x2 + y2 )
3
3
A) 3 x - y 3
4 3
C) x - y
x 2 + y2
B) 3 x - y 4
4
D) x - y
(x4 + 3x2 y2 + 2y4 )(x2 + y2 )
x4 + y4
Find the second-order partial derivative. 75) Find fxy when f(x,y) = 8x3y - 7y2 + 2x.
A) 24x2
75) C) 48xy
B) -14
76) Find fyx when f(x,y) = 8x3y - 7y2 + 2x. A) 24x2 B) -14
D) -28 76)
C) -28
19
D) 48xy
77) Find fxx when f(x,y) = 8x3 y - 7y2 + 2x. A) 48xy B) -28
77) C) 24x2
78) Find fyy when f(x,y) = 8x3y - 7y2 + 2x. A) 48xy B) -14
78) C) -28
79) Find fxy when f(x,y) = 10x2 y4 - 7x3 y5. A) 80xy3 - 21x2 y4 C) 80xy3 - 105x2 y4 80) Find fxx when f(x, y) = x4y4 -
D) -14
D) 24x2 79)
B) 160xy3 - 105x2 y4 D) 160xy3 - 21x2 y4
2 x7 y8 + 17x - y.
80)
A) 12x2y3 - 42 2 x5 y7 C) 12x2y4 - 42 2 x5 y8
B) 12x3y4 - 42 2 x6 y8 D) 3x2 y4 - 6 2 x5 y8
81) Find fyy when f(x, y) = x3 y4 - 2 x7 y8 + 17x - y.
81)
A) 12x3y3 - 56 2 x7 y7 C) 2x3 y2 - 6 2 x7 y6
B) 12x3y2 - 56 2 x7 y6 D) 2x3 y3 - 6 2 x7 y7
82) Find fyx when f(x, y) = x4 y4 - 2 x8 y7 + 17x - y.
82)
A) 12x2y4 - 56 2 x6 y7 C) 16x3y3 - 56 2 x7 y6
B) 16x3y2 - 56 2 x7 y5 D) 12x2y2 - 42 2 x7 y5
83) Find fxy when f(x, y) = x5 y5 - 2 x7 y8 + 17x - y.
83)
A) 20x3y3 - 56 2 x6 y6 C) 25x4y3 - 56 2 x6 y6
B) 20x3y5 - 42 2 x5 y8 D) 25x4y4 - 56 2 x6 y7
84) Find fxx, where f(x, y) = x4 y5 + x5y.
84)
A) 12x2y5 + 20x3 C) 12x2y5 + 20x3 y
B) 4x3 y5 + 5x4 y D) 20y3
85) Find gyy when g(x,y) = x .
85)
y
A) 3x
B) 2x
y2
C) - 2x
y3
y3
D) - 3x
y2
86) Find gyx when g(x, y) = x .
86)
y
A) - 1
y2
B) 1
C) 0
y
20
D) x
2
y2
87) Find fxx when f(x, y) = 8xexy.
87)
A) 8(xexy + x2 yexy) C) 8(xy2 exy + yexy + exy)
B) 8xy2exy + 16yexy D) 8x2 y2exy
88) Find fxy when f(x, y) = 8xexy.
88)
A) 8x2 y2exy C) 8(2xexy + x2yexy)
B) 8(2xexy + xy2 exy) D) 8(2xyexy + exy)
89) Find fyx when f(x, y) = ln 2x + 9y . A)
9
B)
(2x + 9y)2
89) -9
C)
(2x + 9y)2
-18
(2x + 9y)2
D)
18
(2x + 9y)2
90) Find fxx when f(x, y) = ln 2x + 9y . A) -
91) Find
4
2z
81
B) -
2x + 9y 2
90) 18
C) -
2x + 9y 2
2x + 9y 2
D)
18 2x + 9y
when z = ex ln y .
y2
x
91) x
A) - e
x
B) e
y
x
C) e
D) - e
y2
y
y2
92) Find fyy when f(x, y) = x ln (y - x). A) -
x
(y - x)2
B)
92) x
C)
(y - x)2
x y-x
D) -
1
(y - x)2
93) Find fxy when f(x, y) = xy2 + yex2 + 5. A) 2xex2
93)
B) 2y + 2xex2
C) y + xex2
D) 2yex2
94) Find fxy when f(x, y) = x .
94)
x+y
A) y - x
B)
(x + y)3
C)
2y (x + y)3 (x + y)3
x
(x + y)3
D) x - y
(x + y)3
Find values of x and y such that both fx (x, y) = 0 and fy(x, y) = 0.
95) f(x,y) = x3 - 4xy + 8y A) x = 4 , y = 2 3
95) B) x = 2, y = 3
C) x = 2, y = 2
21
D) x = 0, y = 0
96) f(x, y) = x2 + xy + y2 - 3x + 2
96)
A) x = 0, y = 0
B) x = 1, y = - 1
C) x = -2, y = 1
D) x = 2, y = -1
97) f(x,y) = x3 + y3 - 9xy A) x = -3, y = -3
B) x = 1, y = 1
C) x = 0, y = 0
D) x = 3, y = 3
2
97)
Find the indicated derivative.
98) Find fx for f(x, y, z) = 7x10y7 + 7x3z 9 + 3y5. A) 70x9y7 + 21x2 z 9 C) 70x9 + 21x2
98) B) 490x9 y6 + 189x2z 8 D) 49x10y6 + 63x3z 8
99) Find fy for f(x, y, z) = 6x8 y5 + 10x3z 6 + 4y3. A) 240x7 y4 + 180x2z 5 + 12y2 C) 30x8y4 + 12y2
99) B) 48x7y5 + 30x2 z 6 D) 30y4 + 12y2
2
100) Find fz for f(x, y, z) = 10x - y .
100)
4z + 10x
2
2
(4z + 10x)2
4z + 10x
(4z + 10x)2
z
101) Find fy for f(x, y, z) =
x + y2
A) - yz
101) z
C) - z(2y + 1)
3/2 2(x + y2 )
yz
3/2 (x + y2 )
102)
8 3 A) 45x - 40xy
8 4 7 B) 45x - 10xy + z
5x9 - 10xy4 + z 7
5x9 - 10xy4 + z 7
45x8 - 10y4 5x9 - 10xy4 + z 7
D) ln 45x8 - 10y4 8
6
103) Find fx for f(x, y, z) = 10x4e(3y + 2z ). 7
D) -
3/2 2(x + y2 )
102) Find fx for f(x, y, z) = ln 5x9 - 10xy4 + z7 .
C)
(4z + 10x)2
.
B) -
x + y2
2 D) 10x - y
C) -40x + 4y
B) -40x + 4y
A) 40z + 10x
103)
5
8
A) 40x3e(24y + 12z ) 8
6
B) 40x3e(3y + 2z ) 6
C) (40x3 + 10x4)e(3y + 2z )
D) 40x3
22
8
4
104) Find fz for f(x, y, z) = 6x7 e(8y + 2z ). 8
104)
4
8
4
A) 8z 3 e(8y + 2z )
B) 48x7z 3 e(8y + 2z )
C) 8z 3 e2z 4
D) 48x7z 3 e8 z 3
105) Find fx for f(x, y, z) = 4x3 y6z 8 ln 3x9 . A) 12x2y6 z 8 ln 27x8 C) 36x2y6 z 8 + 12x2y6 z 8 ln 3x9
105) B) 324x2 y6 z8 ln 27x8 D) 36x3y6 z 8 + 12x2y6 z 8 ln 27x8
106) Find fzy for f(x, y, z) = ln 8xy - xz - y2 . A)
106)
8x - 2y
B)
2 (8xy - xz - y2 ) 2
2
C) 8x - x y + xy
D)
2 (8xy - xz - y2 )
107) Find fxz for f(x, y, z) = 2x6y5 + 4x6z 10. A) 12x5y5 + 240x5 z9 C) 240x5 z 9
8x2 - 2xy
2 (8xy - xz - y2 ) 8x 2
2 (8xy - xz - y2 )
107) B) 10z 9 D) 40x6z 9
Solve the problem. 108) A company has the following production function for a certain product: p(x, y) = 23x0.2y0.8 .
108)
Find the marginal productivity with fixed capital, px .
A) 4.6 y
1.2
x
B) 4.6 y
0.8
C) 4.6 x
x
0.8
y
D) 4.6xy0.8
109) A company has the following production function for a certain product:
109)
p(x, y) = 31x0.2y0.8 . Find the marginal productivity with fixed labor, py .
A) 24.8yx0.2
B) 24.8 x
0.2
C) 24.8 y
y
x
0.8
D) 24.8 y
0.2
x
110) The production function z for an industrial country was estimated as z = x6 y7, where x is the
110)
amount of labor and y the amount of capital. Find the marginal productivity of labor. A) 7x6 y6 B) 14x6y6 C) 12x5y7 D) 6x5 y7
111) Suppose that the manufacturing cost of a precision instrument is approximated by
M(x,y) = 15x2 + 25y2 - 2xy, where x is the cost of materials and y is the cost of labor. Find Mx(3, 8). A) 1,552
B) 394
C) 74
23
D) 87
111)
112) Under certain conditions the wind speed S, in miles per hour, of a tornado at a distance d from its center can be approximated by the function S =
aV
0.51d5
112)
, where a is an atmospheric constant, and V
is the approximate volume of the tornado, in cubic feet. Assume that a is 0.78, and find SV.
A) 1.53 d5
B) 1
C)
d5
a
D) 0.78
1.02d5
d5
113) Under certain conditions the wind speed S, in miles per hour, of a tornado at a distance d from its center can be approximated by the function S =
aV
0.51d3
113)
, where a is an atmospheric constant, and V
is the approximate volume of the tornado, in cubic feet. Assume that a is 0.78, and find Sd.
A) 1.5
B) - 4.59V
d3
D) 1.5V
C) 1.5V
d4
d
d2
114) Under certain conditions the wind speed S, in miles per hour, of a tornado at a distance d from its center can be approximated by the function S =
aV
0.51d5
114)
, where a is an atmospheric constant, and V
is the approximate volume of the tornado, in cubic feet. Interpret
S . V
A) The rate of change in speed per unit change in volume while distance is held constant. B) The rate of change in speed per unit change in distance while volume is held constant. C) The rate of change in volume per unit change in speed while distance is held constant.. D) The rate of change in speed per unit change in volume and distance. 115) The intelligence quotient in psychology is given by Q(m, c) = 100 m , where m is an individual's c
mental age and c is the individual's chronological, or actual, age. Find
A) - 100m c2
B) - 100m
C) 100m
c
Q . c
D) 100m c
116) A company's monthly sales, in thousands, is given by S(x, y) = 5x0.7y0.5, where x is the amount
spent on newspaper advertising per month in thousands of dollars and y is the amount spent on radio advertising per month in thousands of dollars. Suppose the company currently spends $5,000 on newspaper advertising per month and $4,000 on radio advertising per month. What would be the effect on sales if the company increases the amount spent on newspaper advertising to $6,000, while the amount spent on radio advertising remains constant? A) Sales would decrease by $408.93. B) Sales would increase by $3,856.46.
C) Sales would increase by $16,719.50.
D) Sales would increase by $4,089.33.
24
115)
116)
117) The surface area of a certain mammal, in square meters, is approximated by
117)
A(W, H) = 0.22W0.49H0.64, where W is the weight of the animal in kilograms and H is the height in meters. Find
A . H
A) 0.11W0.49H-0.36 C) 0.11W-0.51H0.64
B) 0.14W0.49H-0.36 D) 0.14W-0.51H0.64
Find all points where the function has any relative extrema or saddle points and identify the type of relative extremum. 118) f(x,y) = x3 - 12xy + 8y3 118)
A) Relative maximum at (1, 2) B) Relative minimum at (2,1) and relative maximum at (0, 0) C) Relative minimum at (2,1) and saddle point at (0, 0) D) Saddle point at (2,1) 119) f(x,y) = x2 + xy + y2 - 3x + 2 A) Relative minimum at (2, -1) and saddle point at (0, 0) B) Relative maximum at (-2, 1) C) Relative maximum at (2, -1) and saddle point at (0, 0) D) Relative minimum at (2, -1)
119)
120) f(x,y) = x3 - 12x + y2 A) Relative minimum at (2, 0) and relative maximum at (-2, 0) B) Relative maximum at (2, 0) C) Relative minimum at (2, 0) D) Relative minimum at (2, 0) and saddle point at (-2, 0)
120)
121) f(x,y) = 4xy - x2 y - xy2
121)
A) Relative minimum at 2 , 2 and saddle point at (0, 0) 3 3
B) Relative maximum at 4 , 4 and saddle point at 2 , 2 3 3
3 3
C) Relative minimum at 4 , 4 and saddle point at (0, 0) 3 3
D) Relative maximum at 4 , 4 and saddle point at (0, 0) 3 3
122) f(x,y) = x2 - y2 A) Saddle point at (0, 0) B) Saddle point at (0, 0) and relative maximum at (1, -1) C) Relative maximum at (0, 0) D) Relative minimum at (0, 0)
25
122)
123) f(x, y) = x2 + y2 - 11x - 3y
123)
A) Relative minimum at 11 , 3 2
2
B) Relative minimum at 11 , 3 and saddle point at (0, 0) 2
2
C) Saddle point at 11 , 3 2
2
D) No relative extrema or saddle points 124) f(x, y) = 6xy A) Relative minimum at (-1, -1), saddle point at (0, 0) B) Relative maximum at (0, 0) C) Saddle point at (0, 0) D) No relative extrema or saddle points
124)
125) f(x,y) = x2 + 2y2 - xy2 A) Saddle point at (0, 0) and relative minima at (2, 2) and (2, -2) B) Relative minimum at (0, 0) and relative maxima at (2, 2) and (2, -2) C) Relative maximum at (0, 0) and saddle points at (2, 2) and (2, -2) D) Relative minimum at (0, 0) and saddle points at (2, 2) and (2, -2)
125)
126) x2 - y2 + 8xy A) Relative minimum at (0, 0) C) Relative maximum at (0, 0)
126) B) Saddle point at (0, 0) D) There are no relative extrema.
127) x3 + y3 + 6xy A) Relative maximum at (0, 0) and relative minimum at 2, 2 B) Saddle point at (0, 0) and relative minimum at - 2, - 2 C) Saddle point at (0, 0) and relative maximum at 2, 2 D) Saddle point at (0, 0) and relative maximum at - 2, - 2
127)
128) f(x,y) = 1 + xy - 8 x
128)
y
A) Relative maximum at 1 , -4
B) Relative maximum at - 1 , 4
C) Relative minimum at 1 , -4
D) Relative minimum at - 1 , 4
2
2
2
2
129) f(x,y) = e(x2 + y2) A) Relative maximum at (0, 0) C) Relative maximum at (0, 1)
129) B) Relative minimum at (0, 0) D) Relative minimum at (0, 1)
26
130) f(x, y) = e-(x2 + y2 - 4y) A) Relative maximum at at (0, 2) and relative minimum at at (0, -2) B) Saddle point at (0, 2) C) Relative maximum at (0, 2) D) No relative extremum or saddle points. 131) f(x, y) = exy A) Relative maximum at (0, 0) C) Relative minimum at (0, 0)
131) B) Saddle point at (0, 0) D) No relative extrema or saddle points.
132) f(x, y) = 1 + 1 x
132)
y
A) Relative minimum at (100, 100) C) Relative maximum at (1, 1)
B) Saddle point at (0, 0) D) No relative extrema
133) f(x, y) = ex+y A) Relative minimum at (0, 1) C) Saddle point at (0, 0)
133) B) Relative minimum at (0, 0) D) No relative extrema.
134) y + 4 + x x
130)
134)
y
A) Relative minimum at 41/3, 4 2/3 B) Saddle point at 4 1/3, 4 2/3 C) Relative maximum at 4 1/3, 42/3 D) There are no relative extrema or saddle points. 135) (x - 4) ln(xy)
135)
A) Relative minimum at 4, 1 . 4
B) Saddle point at 4, 1 4
C) Relative maximum at 4, 1 . 4
D) There are no relative extrema or saddle points Find all relative extrema for the function, and then match the equation to its graph.
27
136) z = y4 - 2y2 + x2 - 5
136)
4
A) Relative maxima of 1 at (0, 1) and at (0, -1); 4
saddle point at (0, 0)
B) Relative maxima of 9 at (0, 1) and at (0, -1); 4
saddle point at (0, 0)
28
C) Saddle points at (0, 0), (-1, 1), (1, -1), (1, 1), and (-1, -1)
D) Relative minima of - 9 at (0, 1) and 4
at (0, -1); saddle point at (0, 0)
29
137) z = -2x3 - 3y4 + 6xy2 + 1
137)
2
A) Relative maxima of 3 at (1, -1) and at (-1, 1); 2
saddle point at (0, 0)
B) Relative maxima of 3 at (1, 1) and at (1, -1) 2
30
C) Saddle points at (0, 0), (-1, 1), (1, -1), (1, 1), and (-1, -1)
D) Relative minima of - 1 at (1, 1) and at (1, -1) 2
31
138) z = -x4 + y4 + 2x2 - 2y2 + 1
138)
4
A) Relative maxima of 1 at (-1, 1) and 4
at (1, 1)
B) Saddle points at (0, 0), (-1, 1), (1, -1), (1, 1), and (-1, -1)
32
C) Relative maxima of 1 at (1, 1) and 4
at (-1, -1); saddle point at (0, 0)
D) Relative minima of - 3 at (0, 1) and 4
at (0, -1); saddle point at (0, 0)
33
139) z = -y4 + 4xy - 2x2 + 1
139)
8
A) Relative maxima of 3 at (1, -1) 8
and at (1, 1);
B) Relative minima of - 7 at (0, 1) and 8
at (0, -1); saddle point at (0, 0)
34
C) Saddle points at (0, 0), (-1, 1), (1, -1), (1, 1), and (-1, -1)
D) Relative maxima of 9 at (1, 1) 8
and at (-1, -1); saddle point at (0, 0)
Solve the problem. 140) Suppose that the labor cost for a building is approximated by C(x,y) = 10x2 + 2y2 - 400x - 480y + 14,000, where x is the number of days of skilled labor and y is
the number of days of semiskilled labor required. Find the x and y that minimize cost C. A) x = 20, y = 120 B) x = 120, y = 48 C) x = 48, y = 360 D) x = 40, y = 240
35
140)
141) A firm produces two kinds of tennis balls, one for recreational players which sells for $2.50 per
141)
can, and one for serious players which sells for $4.00 per can. The total revenue from the sale of x thousand cans of the first ball and y thousand cans of the second ball is given by R(x, y) = 2.5x + 4y. The company determines that the total cost, in thousands of dollars, of producing x thousand cans of the first ball and y thousand cans of the second ball is given by C(x, y) = x2 - 2xy + 2y2 . Find the number of each type of ball which must be produced and sold in order to maximize the profit. A) 2000 of the $2.50 cans and 5000 of the $4.00 cans
B) 3000 of the $2.50 cans and 4000 of the $4.00 cans C) 4500 of the $2.50 cans and 3250 of the $4.00 cans D) 5000 of the $2.50 cans and 3000 of the $4.00 cans 142) A computer firm markets two kinds of electronic calculator that compete with one another. The
142)
total revenue function is R p, q = 80p - 6p2 - 4pq + 68q - 2q2 , where p is the price of the first
calculator (in multiples of $10), and q is the price of the second calculator (in multiples of $10). What prices should be charged in order to maximize the total revenue? A) $5 and $90 B) $20 and $120 C) $15 and $155 D) $40 and $170
143) A closed rectangular box with a volume of 16 cubic feet is made from two kinds of materials. The
143)
top and bottom are made of a material costing $0.10 per square foot, and the sides are made of a material costing $0.05 per square foot. Find the dimensions of the box so that the cost of materials is minimal. A) 1 ft by 1 ft by 16 ft B) 2 ft by 2 ft by 4 ft
C) 2 ft by 2 ft by 8 ft
D) 2 ft by 4 ft by 2 ft
144) A rectangular metal tank with an open top is to hold 256 cubic feet of liquid. What are the
144)
dimensions of the tank that require the least material to build? A) 8 ft by 8 ft by 4 ft B) 2 ft by 2 ft by 64 ft
C) 2 ft by 16 ft by 4 ft
D) 16 ft by 4 ft by 4 ft
145) A flat plate is located on a coordinate plane. The temperature of the plate, in degrees Fahrenheit,
145)
at point (x, y) is given by T(x, y) = x 2 + y2 - 3x - 3y. Find the coordinates of the point on the plate where the temperature is minimal. A) 3 , 3 B) - 3 ,- 3 2 2 2 2
C) (6, 6)
D) (3, 6)
146) A flat plate is located on a coordinate plane. The temperature of the plate, in degrees Fahrenheit, at point (x, y) is given by T(x, y) = x 2 + y2 - 7x - 5y. What is the maximum temperature on the plate?
A) No maximum
B) 75°
C) 126°
36
D) 18°
146)
147) The profit (in thousands of dollars) that a company earns from producing x tons of
147)
brass and y tons of steel can be approximated by P(x, y) = 56xy - 8x3 - y3 . Find the amount of brass and steel that maximize profit and find the value of the maximum profit. A) 28 tons of brass and 14 tons of steel; maximum profit is $813,037 3 3
B) 4 tons of brass and 56 tons of steel; maximum profit is $594,636 5
C) 28 tons of brass and 7 tons of steel; maximum profit is $447,294 5
D) 14 tons of brass and 28 tons of steel; maximum profit is $813,037 3
3
Find the indicated relative minimum or maximum. 148) Minimum of f(x,y) = x2 + y2 , subject to x + y = 1 A) f 1 , 1 = 1 2 2
148)
B) f(0, 1) = 1
C) f 1 , 1 = 1
2
2 2
D) f(0, 1) = 1
2
149) Minimum of f(x,y) = x2 + 2y2 - xy, subject to x + y = 8 A) f(2, 6) = 25
149)
B) f(5, 3) = 28
C) f(6, 2) = 25
D) f(3, 5) = 28
150) Minimum of f(x, y, z) = x2 + y2 + z2 ,
150)
subject to x + 2y - z = 3 A) f - 1 , 2, 1 = 9 2 2 2
B) f 3 , 6 , - 3 = 54
C) f(0, 1, -1) = 2
D) f 1 , 1, - 1 = 3
7 7 2
7
2
49
2
151) Maximum of f(x,y) = 4xy, subject to x + y = 8 A) f(4, 4) = 64
151) B) f(0, 8) = 72
C) f(2, 6) = 72
D) f(3, 5) = 64
152) Minimum of f(x,y) = x2 - 14x + y2 - 16y, subject to 2x + 3y = 12 A) f(2, 0) = -24
152)
B) f(3, 2) = -61
C) f(1, 5) = -68
D) f(0, 1) = -15
153) Maximum of f(x,y) = xy, subject to x + y = 100 A) f(0, 100) = 0
153) B) f(50, 50) = 100
C) f(100, 0) = 0
D) f(50, 50) = 2500
154) Minimum of f(x,y) = x2 + y2 - xy, subject to x - y = 10 A) f(2, -1) = 7
154)
B) f(1, 2) = 3
C) f(5, 5) = 25
37
D) f(5, -5) = 75
155) Minimum of f(x,y) = x2 + 4y2 + 6, subject to 2x - 8y = 20 A) f(2, -2) = 26
155)
B) f(-2, -2) = 26
C) f(2, 2) = 26
D) f(-2, 2) = 26
156) Minimum of f(x, y) = x2 + y2 ,
156)
subject to x - 3y = 6 A) f - 9 , - 13 = 10 5 5
B) f 3 , - 9 = 18
C) f - 3 , - 11 = 26
D) f 9 , - 7 = 26
5
5
5
5
5
5 5
5 5
157) Maximum of f(x, y, z) = xy + z, subject to x2 + y2 + z2 = 1 A) f(1, 1, 0) = 1
157)
B) f(1, 1, 1) = 1
C) f(0, 1, 0) = 1
D) f(0, 0, 1) = 1
Solve the problem. 158) Find two positive numbers whose sum is 36 and whose product is a maximum. A) 18 and 18 B) 9 and 27 C) 27 and 27 D) 18 and 27
158)
159) Find two positive numbers x and y such that x + y = 60 and xy2 is maximized. A) x = 15 and y = 45 B) x = 20 and y = 40 C) x = 1 and y = 59 D) x = 30 and y = 30
159)
160) Find three positive numbers whose sum is 144 and whose product is a maximum. A) 48, 36, and 36 B) 72, 36, and 36 C) 72, 72, and 72 D) 48, 48, and 48
160)
161) The total cost to hand-produce x large dolls and y small dolls is given by
161)
C(x,y) = 2x2 + 7y2 + 4xy + 40. If a total of 40 dolls must be made, how should production be allocated so that the total cost is minimized? A) Make 20 large dolls and 20 small ones B) Make 39 large dolls and 1 small one
C) Make 40 large dolls and 0 small ones
D) Make 0 large dolls and 40 small ones
162) The production level P of a factory during one time period is modeled by P(x, y) = Kx1/2y1/2
where K is a positive integer, x is the number of units of labor scheduled and y is the number of units of capital invested. If labor costs $2,900/unit, capital costs $800/unit and the owner has $1,800,000 available for one time period, what amount of labor and capital would maximize production? A) 300.0 units of labor and 1,000.0 units of capital
B) 620.7 units of labor and 2,250.0 units of capital C) 310.3 units of labor and 1,125.0 units of capital D) 1,125.0 units of labor and 310.3 units of capital
38
162)
163) A farmer has 320 m of fencing. Find the dimensions of the rectangular field of maximum area that can be enclosed by this amount of fencing. A) 80 m by 80 m B) 70 m by 90 m
C) 32 m by 128 m
163)
D) 80 m by 240 m
164) A farmer has 360 m of fencing. Find the area of the largest rectangular field that he can enclose
164)
165) What are the dimensions of a rectangular box, open at the top, which has maximum volume when
165)
with his fencing. Assume that no fencing is needed along one edge of the field. A) 50,875 m2 B) 22,525 m2 C) 16,200 m2 D) 32,400 m2
the surface area is 48 in.2 ? A) x = 4 in., y = 4 in., z = 2 in.
B) x = 6 in., y = 6 in., z = 3 in. D) x = 4 in., y = 2 in., z = 2 in.
C) x = 8 in., y = 2 in., z = 6 in.
166) The material for the bottom of a rectangular box costs $3 per square foot while the material for the
166)
167) Find the dimensions of the right circular cylinder with maximum volume if its surface area is
167)
sides and top costs $1 per square foot. Find the greatest capacity such a box can have if the total amount available for material is $12. A) 4 ft3 B) 1 ft3 C) 2 ft3 D) 3 ft3
24 in.2 . A) r = 2 in., h = 6 in.
B) r = 2 in., h = 4 in. D) r = 3 in., h = 3 in.
C) r = 3 in., h = 8 in.
168) Find the dimensions of the right circular cylinder with maximum surface area, if its volume is
168)
64 ft3 .
A) r = 3 ft, h = 2 ft
B) r = 8 ft, h =
C) r = 6 ft, h = 1 ft
D) r = 2 ft, h = 32 ft
16
ft
169) What is the greatest area that a rectangle can have if the length of its diagonal is 2 m? A) 5 m 2 B) 2 m 2 C) 1 m 2 D) 2 2 m 2
169)
170) Assuming that a cylindrical container can be mailed only if the sum of its height and
170)
circumference do not exceed 300 centimeters, what are the dimensions of the cylinder with the largest volume that can be mailed? A) Height 100 centimeters and radius 300/ centimeters
B) Height 200 centimeters and radius 100 centimeters C) Height 300 centimeters and radius 100/ centimeters D) Height 100 centimeters and radius 100/ centimeters 171) A rectangular box with square base and no top is to have a volume of 32 ft3 . What is the least amount of material required? A) 36 ft2 B) 42 ft2
C) 48 ft2 39
D) 40 ft2
171)
Evaluate dz.
172) z = 8x2 + 4xy + 4y2 ;
172)
x = 2, y = 8, dx = 0.01, dy = -0.02 A) -0.56 B) 0.80
C) 0.56
D) -0.80
173) z = 2x + 5xy;
173)
y
x = 6, y = 16, dx = 0.03, dy = 0.02 A) -0.27 B) 0.27
C) 0.07
D) -0.07
174) z = ln (2x + 12y);
174)
x = 5, y = 6, dx = 0.02, dy = 0.03 A) 0.004 B) -0.004
C) 0.005
175) z = x3 - 9xy2 + 5y; x = 3, y = 3, dx = 0.01, dy = -0.03 A) 4.32 B) 0.93 C) 4.17
D) -0.005 175) D) 4.44
2
176) z = x + 3y ; x = -2, y = 3, dx = 0.01, dy = -0.02
176)
4x2 - y
A) 0.00734
B) -0.01396
C) -0.00260
D) 0.00308
Evaluate dw.
177) w = 7x4 5y ln (7z);
177)
x = 3, y = 2, z = 7, dx = 0.01, dy = -0.02, dz = 0.03 A) 100.73 B) 58.15
C) 65.83
D) -27.21
178) w = 11x3 y + 11x2z;
178)
x = 3, y = 9, z = 3, dx = 0.03, dy = 0.01, dz = 0.02 A) 85.14 B) 91.08
C) 88.11
D) 89.10
Use the total differential to approximate the quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference of the two results to four decimal places. 179) 1.022 + 3.982 179)
A) 4; 4.1086; 0.1086 C) 4.1086; 4.1086; 0
B) 3.150; 3.1590; 0.0090 D) 3.1086; 4.1086; 1.0000
180) (5.012 - 0.932 )1/3 A) 2.8941; 2; 0.8941 C) 1.8941; 2.8939; 1.0002
180) B) 2; 2.8939; 0.8939 D) 2.8941; 2.8939; 0.0002
40
181) 0.97e0.02 A) 1.01; 0.9896; 0.0204 C) 0.99; 1; 0.0100
181) B) 0.99; 0.9896; 0.0004 D) 1; 0.9896; 0.0104
182) 0.98ln(1.02) A) 0.02; 0.01; 0.0100 C) 1; .02; 0.9800
182) B) 0; 0.0194; 0.0194 D) 0.02; 0.0194; 0.0006
Solve the problem.
183) The cost to make a product is M(x,y) = 9x2 + 6y2 - 4xy + 60, where x is material cost, y is labor
183)
184) The production function for a certain country is z = x0.3y0.6, where x stands for units of labor and
184)
185) The width of a rectangle is measured as 17.5 cm, while the length is measured as 34.7 cm. The
185)
186) Approximate the amount of aluminum needed for a beverage can of radius 2.8 cm and height 16
186)
cost. The company spends $10 on materials and $9 on labor. Use the differential to estimate the change in cost if the company spends $14 on materials and $6 on labor. A) $372 B) -$408 C) -$1,152 D) $1,152
y for units of capital. At present, x is 38 and y is 47. Use differentials to estimate the change in z if x becomes 39 and y becomes 52. A) 2.15 B) 3.10 C) 0.62 D) 1.57
width measurement could be off by 0.9 cm, and the length could be off by 1.2 cm. Estimate the maximum possible error in calculating the area of the rectangle. A) 42.00 cm2 B) 62.46 cm2 C) 52.23 cm2 D) 57.39 cm2
cm. Assume the walls of the can are 0.1 cm thick. A) 330.75 cm3 B) 33.07 cm3
C) 4.93 cm3
D) 28.15 cm3
187) The number of liters of blood pumped through the lungs in one minute is given by
187)
b C= a-v
Suppose a = 150, b = 180, v = 120. Estimate the change in C if a becomes 140, b becomes 186, and v changes to 128. A) 0.6 liters B) 3.4 liters C) -0.2 liters D) 3.8 liters
188) The dimensions of a cardboard box were measured as 40 cm, 90 cm and 40 cm, with the
188)
189) The base radius and the height of a cylindrical can tank were measured at 35 cm and 65 cm.The
189)
percentage error in each dimension not exceeding 3 percent. Approximate the worst possible percentage error in the volume of the box. Assume that the cardboard is of negligible thickness. A) 7% B) 3% C) 6% D) 9%
percentage errors in these measurements do not exceed 0.2 for the base radius and 0.6 for the height. Approximate the worst possible percentage error in the volume of the can. Assume that the sides and ends of the can are of negligible thickness. A) 1.2% B) 1% C) 0.8% D) 1.6%
41
190) The storage tank has the form of a cylinder with one hemisphere end (see figure). The perimeter of
190)
a circular cross-section perpendicular to the axis of symmetry of the tank was measured at 37.7 ± 1 ft and the perimeter of a cross-section lying in the plane containing the axis of symmetry was measured at 50.8 ± 2.5 ft. Approximate the worst possible percentage error in the volume of the tank. Assume that the sides and ends of the tank are of negligible thickness.
A) 73.4
B) 0.4
C) 35.9
D) 440.6
191) The resistance R of a circuit containing two resistors in parallel is 1 = 1 + 1 , where the two R
R1
R2
191)
resistors have resistances R 1 and R 2 . For resistors with R1 = 220 ± 0.5 ohm and R2 = 120 ± 0.8 ohm, approximate the error in R.
A) 0.2
B) 0.4
Evaluate the integral. 2 192) xy(6x - 3) dx 0 A) 2x(6x - 3) 2
193) 2
3x+2 - e3x+2
5 3
D) 0.8
192) B) 21
C) 10y
D) 18y
e3x+y dx
A) e
194)
C) 0.6
3
193) 6+y - e6+y
C) e
B) 3(e6 +y - e6+y)
3
D) e3x+2 - e3x+2
5x + y dy
194)
A) 2 [(25 +y)3/2 - (15 + y)3/2 ]
B) 1 [(25 +y)3/2 - (15 + y)3/2 ]
C) 2 [(5x + 5)3/2 - (5x + 3)3/2 ]
D) 1 [(5x + 5)3/2 - (5x + 3)3/2 ]
3
5
3
5
42
2
195) 0
(x4 y + y) dx
195)
A) 42 y
B) 2x4
5
4
196) 1
196)
A) 1023 x2 + 3x - 9
B) 21y4 - 3
C) 26 y4 - 12
D) 1022 x2 + 3x - 9
2
3
1
5
e2x + 7y dx
197)
-1
A) 1 e(7y + 2) - 1 e(7y - 2)
B) 1 e(7y + 2) - 1 e(7y - 2)
C) 1 e(2x + 7) - 1 e(2x - 7)
D) 1 e(2x + 7) - 1 e(2x - 7)
7 2
11
198) 2
2
2
7
198)
C) 2 (4y + 33)3/2 - 2 (4y + 6)3/2
D) 1 (3x + 44)3/2 - 1 (3x + 8)3/2
6
9
27
18
9
18
2 yex + y dy
199)
A) 1 [e(x + 64) - e(x + 1)]
B) 8e(x + 64) - e(x + 1)
C) 1 [e(8 + y2) - e(1 + y2 ) ]
D) y[e(8 + y2 ) - e(1 + y2 ) ]
2 2
5 1
7
B) 2 (4y + 33)3/2 - 2 (4y + 6)3/2
8
200)
2
A) 1 (3x + 44)3/2 - 1 (3x + 8)3/2 27
1
7
3x + 4y dy
6
199)
5
(x + x2 y4 - 3) dx
5
197)
D) 32 y + 2
C) 2x4 + 2
x x2 + 5y dx
200)
A) 1 [(25 + 5y)3/2 - (1 + 5y)3/2 ]
B) 2 [(25 + 5y)3/2 - (1 + 5y)3/2 ]
C) 1 [(x2 + 5)3/2 - (x2 + 1)3/2 ]
D) 1 [5(25 + 5y)3/2 - (1 + 5y)3/2 ]
3
3
3
3
43
1
201) 0
y y2 + 7x dx
201)
A) 1 (7x + 1)3/2 - 1 (7x)3/2
B) 2 y(y2 + 7)3/2 - 2 (y2 + 1)3/2
C) 2 y(y2 + 7)3/2 - 2 y4 21 21
D) 1 (7x + 1)3/2 - 1 (7x + 7)3/2 6 6
3
3
Evaluate the iterated integral. 7 6 202) (8x + 9y) dx dy 0 0
A) 2331 2
203) 0
2
0
5
207)
8
D) 777 2
203) D) 10
204) B) 625
C) 625
2
D) 625 4
-7
4
0 A) 160 3
205) B) 2272
C) 4544
3
3
D) 18176 3
xy2 dx dy
3 A) 609 1
C) 8
(7x2 y + 5xy) dy dx
0 A) 568 3 4
1
C) 333
2
2xy dx dy
0
4
206)
B) 111
B) 12
A) 1,250
0
202)
0
5
205)
15
(1 + x + y) dx dy
A) 5
204)
15
206) B) 420
C) - 420
D) - 609
x 2 + y2 dx dy
207)
-1
C) 130
B) 22
3
44
D) 136 3
6
208) 0
4 0
1 dx dy (x + 1)(y + 1)
A) ln 35 2
209) 0
2
4
C) 1 ln 5
B) 7 ln 5
209)
B) 205
C) 211
6
6
x 4x dy dx + y 3
210)
1 21 A) ln 4 + 42 2
B) 21 ln 4 + 168
C) 45 + 15 ln 5
D) 45 + 15 ln 5 - 15 ln 2
2 0
D) 203
6
2
2
211)
D) ln 5 · ln 7
7
(x3 + 2x2 y - y3 + xy) dy dx
-3 A) 209 6 5
210)
2
208)
1
3
2
2
y y2 + 7x dy dx
211)
0 2 A) (155/2 - 145/2 - 1) 15
B) 2 (155/2 - 1)
C) 2 (155/2 - 145/2 - 1)
D) 2 (155/2 - 145/2 )
15
105
105
Find the double integral over the rectangular region R with the given boundaries.
212)
(1 + x + y) dx dy R 0 x 3, 0 y 3 A) 10
213)
B) 18
C) 27
D) 36
(3xy) dx dy R 0 x 5, 0 y 1 A) 25 4
214)
212)
213)
C) 75
B) 75
4
D) 75 2
(x2 + y2 ) dx dy R 0 x 4, -1 y 1 A) 130 3
214)
B) 136
C) 160
3
3
45
D) 22
215)
1 dx dy (x + 1)(y + 1)
215)
R 0 x 6, 0 y 8
A) ln 63
216)
B) ln7 · ln 9
B) 1
C) 1
2
217)
B) 126
C) 128
5
5
D) 189 5
(x + x2 y4 - 3) dy dx
218)
B) 11
C) 14
15
15
D) - 43 15
2 (x3 + 5y) 3x e dx dy
219)
R 0 x 1, 0 y 1 A) 1 (e6 - e5 - e - 1) 5
B) 1 (e6 - e5 - e + 1) 5
C) 1 (e5 - e4 - e - 1)
D) 1 (e5 - e4 - e + 1)
4
220)
D) 2
(x4 y + y) dx dy
R 0 x 1, 0 y 2 A) 13 15
219)
9
216)
R 0 x 2, 0 y 3 A) 144 5
218)
D) 1 ln 7
x dx dy y R 0 x 1, 1 y e A) 3 2
217)
C) 9 ln 7
8
(x + y)3
4
dx dy
R 0 x 1, 1 y 2 A) 4 5
220)
B) 9
C) 7
2
2
D) 4 3
Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. 221) z = 6x2 y; 0 x 4, 0 y 3
A) 676
B) 2256
C) 576 46
D) 1256
221)
222) z = x2 + y2 ; 0 x 1, 0 y 1 A) 8
222)
B) 4
3
C) 1
3
3
223) z = 8x + 4y + 7; 0 x 1, 1 y 3 A) 28 B) 38 224) z = x3; 0 x 2, 0 y 3 A) 18 225) z =
8
(x + y)3
D) 2 3
223) C) 36
D) 26 224)
B) 12
C) 4
D) 6
; 0 x 1, 1 y 2
A) 4 3
225)
B) 1 3
D) 8
C) 8 3
D) 2 3
3
226) z = (x + y)2 ; -1 x 1, -1 y 1 A) 4 3
C) 2
3
226)
B) 1 3
227) z = x ; 0 x 1, 1 y e
227)
y
A) 1 6
B) 1
C) 1
3
2
228) z = e2x + 3y; 0 x 1, 0 y 1
B) 1 (e5 - e3 - e2 - 1) 6
C) 1 (e5 - e3 - e2 + 1)
D) 1 (e5 - e3 - e2 + 1)
6
4
229) z = 4x2 + 9y2; 0 x 1, 0 y 1
229)
B) 5 3
230) z = x + y; 0 x 1, 0 y 1 A) 2 3
Evaluate the double integral. 1 4 231) y dy dx 0 4x A) 16 3
4
228)
A) 1 (e5 - e3 - e2 - 1) 4
A) 26 3
D) 1
C) 17 3
D) 13 3
C) 1 3
D) 4 3
230)
B) 8 3
231) B) 64
C) 32
3
47
D) 8
4
232) 1
ln x
0 A) 25 4
ey
3x
r8
0
x/2
1 0
33
D) 1
34
(x + y) dy dx
236) B) 54
x2
y
0 y2 A) 5 24
239)
C) 1
35
y
C) 63
D) 45
(x2 - y2 ) dy dx
0 A) 44 115 1
238)
7
0
2 0
D) 81
235) B) 1
A) 36
237)
5
s ds dr
17
0
C) 27
5
0
6
D) 8
234) B) 54
A) 1
236)
C) 4
x2 dy dx
0 A) 162 7
1
235)
2
233) B) 9
3 0
2
D) 9
ey dx dy
A) 18
234)
C) 25
4
5
0
232) B) 9
ln 5
233)
ey dy dx
237) B) 32
C) 44
115
105
D) 32
105
(xy + 1) dx dy
238) B) 9
C) 7
24
24
D) 25 24
ex + y dx dy
0 1 2 A) (e - e)2 2
239) B) 1 (e2 - e)2
C) 1 (e - 1)2
e
3
48
D) 1 (e - 1)2 2
Use the region R to evaluate the double integral. (2xy - y2 + 1) dy dx
240)
240)
R R bounded by x = 2y + 2, x = 3 - 3y, y = 0 A) 31 B) 33 14 50
C) 33
250
D) 125 33
2xy dx dy
241)
241)
R
R bounded by y = x2 + 4, y = 3x + 2, x = -2, x = 1 A) 243 B) - 243 4 4
C) - 241 4
D) 241 4
dy dx
242)
242)
R R bounded by y =
x + 4, y =
A) 31
3 x, y = 0 5
B) 27
6
C) - 31
2
6
D) - 27 2
x2 + y2 dx dy
243)
243)
R R bounded by x = 10, y = 0, and y = 4x A) 190,000 B) 1,900 3 3
C) 80,000
D) 19,000 3
x2 + y4 dx dy
244)
244)
R
R bounded by x = 0, y = 9, y = x2 A) 592,272 B) 1,773,252 55 55
245)
C) 198,612 55
D) 67,392 55
exey dx dy
245)
R R bounded by x = 0, x = ln y, y = ln 4 A) 6 - 2 ln 4 + e B) 2 ln 4 - 6 + e
C) 4 ln 4 - 8 + e
49
D) 8 - 4 ln 4 + e
(x3 - y3 ) dy dx
246)
246)
R R bounded by x = 0, y = 1, y = x A) - 3 B) 1 20 6
C) 1
D) - 1
3
3
(x + y) dx dy
247)
247)
R R bounded by xy = 4, x + y = 5 A) 9 B) 12
C) 8
D) 6
x2 y2 dx dy
248)
248)
R R bounded by 0 x y, 1 y 3 A) 370 B) 370 9 3
C) 364
D) 364
C) 6
D) 12
C) 13
D) 11 2
9
Find the average value of the function f over the region R. 249) f(x, y) = 4x + 8y; 0 x 1, 0 y 1 A) 8 B) 10
249)
250) f(x, y) = 4x + 3y; 0 x 2, 0 y 6 A) 7
3
250)
B) 4
251) f(x, y) = e2x; 0 x 1 , 0 y 1 2
251)
2
B) e - 1
A) e - 1
C) 2e - 1
2
D) 2e - 1 4
252) f(x, y) = 1 ; 1 x 3, 1 y 3
252)
xy
A) ln 3
2
2
253) f(x, y) = A) 1 5
1
(xy)2
B) ln 3
C) ln 3
9
4
D) ln 3
2
3
; 1 x 5, 1 y 5
253)
B) ln 5
C) ln 5
5
25
50
D) 1
25
Solve the problem.
254) A company's cost for operating two warehouses is C(x, y) = 6x2 + 4x + 12y2 + 8y + 9, where x is
254)
255) A production function is given by P(x, y) = 270x2 y, where x is the number of units of labor and y
255)
256) A company's cost for operating two warehouses is C(x, y) = 4x3 y dollars, where x is the number of
256)
257) A company's cost for operating two warehouses is C(x, y) = 6xy2 + 12x2 y + 4y dollars, where x is
257)
258) Determine if the graph of the equation f(x, y) = 2x2 + 3y2 + 9 is a sphere, a hemisphere, a
258)
the number of units in warehouse A and y is the number in B. Find the average cost to store a unit if A has 20 to 60 units, and B has 32 to 80 units. A) $98,650 B) $89,560 C) $47,034 D) $62,375
is the number of units of capital. Find the average production level if x varies from 10 to 80 and y from 50 to 60. A) 216,810,000 B) 252,945,000 C) 1,517,670,000 D) 36,135,000
units in warehouse A and y is the number in B. Find the average cost to store a unit if A has 1 to 2 units, and B has 3 to 5 units. A) $120 B) $90 C) $60 D) $150
number of units in warehouse A and y is the number in B. Find the average cost to store a unit if A has 3 to 5 units, and B has 1 to 2 units. A) $718 B) $359 C) $356 D) $712
paraboloid, or none of these. A) Hemisphere B) Paraboloid
C) Sphere
D) None of these
259) Determine the type of curve given by vertical slices of the graph of the equation f(x, y) = 5x2 + 2y2 + 7. A) Ellipse
B) Circle
C) Parabola
259)
D) None of these
260) Determine the type of curve given by horizontal slices of the graph of the equation
260)
f(x, y) = 3x2 + 4y2 + 2.
A) Circle
B) Paraboloid
C) Ellipse
D) None of these
261) If a, b, and c are all nonzero, then determine if the plane ax + by + cz = 1 always intersects all three
261)
262) Determine if the graph of the equation f(x, y) = 9 - x2 - y2 is a sphere, a hemisphere, a
262)
coordinate planes (i.e. the planes x = 0, y = 0, z = 0), sometimes intersects them, or never intersects them. A) Always B) Never C) Sometimes
paraboloid, or none of these. A) Hemisphere B) Paraboloid
C) Sphere
D) None of these
263) Determine if the graph of the equation x = 0 is a plane identical to, parallel to (but not identical to), or perpendicular to the yz-plane. A) Parallel B) Perpendicular
C) Identical 51
D) None of these
263)
264) Determine which coordinate plane (i.e. xy-plane, yz-plane, or xz-plane) the graph of the equation z = 2 is parallel to. A) xz-plane
B) yz-plane
264)
C) xy-plane
Answer the question.
265) Given f(x,y) = x2 + y2, the purpose of finding the critical points is which of the following? (i) find local minima, if they exist (ii) find local maxima, if they exist (iii) find saddle points, if they exist (iv) find interior points, if they exist A) Only (i) and (ii) are correct.
265)
B) All are correct. D) (i), (ii), and (iii) are all correct.
C) Only (iii) is correct.
266) If (a,b) is a critical point in the interior of the domain of f(x,y) and if
266)
H = fxx(a,b) fyy(a,b) - fxy(a,b) 2 , then f has a local minimum at (a,b) if (i) H > 0 and fxx(a,b) > 0.
(ii) H > 0 and fxx(a,b) < 0. (iii) H < 0. A) Only (iii) is correct.
B) Only (ii) is correct. D) Only (i) is correct.
C) (i), (ii), and (iii) are all incorrect.
267) If (a,b) is a critical point in the interior of the domain of f(x,y) and if
267)
D = fxx(a,b) fyy(a,b) - fxy(a,b) 2 , then f has a local maximum at (a,b) if (i) D > 0 and fxx(a,b) > 0.
(ii) D > 0 and fxx(a,b) = 0. (iii) D < 0. A) Only (i) is correct.
B) Only (ii) is correct. D) (i), (ii), and (iii) are all incorrect.
C) Only (iii) is correct.
268) If (a,b) is a critical point in the interior of the domain of f(x,y) and if
D = fxx(a,b) fyy(a,b) - fxy(a,b) 2 , then f has a saddle point at (a,b) if (i) D > 0 and fxx(a,b) > 0.
(ii) D > 0 and fxx(a,b) < 0. (iii) D < 0. A) Only (ii) is correct.
B) Only (iii) is correct. D) None of the above is correct.
C) Only (i) is correct.
52
268)
269) If we are looking for the critical points (a,b) on the surface of z = f(x,y), then we study the tangent
269)
lines to (a,b) along the slices parallel to the x-axis and y-axis. These tangent lines must have which of the following properties? (i) They must be perpendicular. (ii) They must be horizontal. (iii) They must be horizontal, but not perpendicular. (iv) They must be perpendicular, but not horizontal. A) Only (iv) is correct.
C) Only (ii) is correct.
B) Both (i) and (ii) are correct. D) Only (iii) is correct.
270) If we wish to locate the candidates for local extrema and saddle points in the interior of the
270)
domain of a function f(x,y), then how should we proceed?
(i) Set the partial derivatives equal to each other. (ii) Set the partial derivatives equal to zero. (iii) Set the partial derivatives equal to the critical values. A) Only (i) is correct. B) Only (ii) is correct.
C) Only (iii) is correct.
D) None of the above is correct.
271) If we wish to find the candidates for local extrema and saddle points for f(x,y) = x4 y + y3 x4 , then
271)
we must find which of the following? (i) fx and fxx (ii) fy and fyy (iii) fxy (iv) fyx (v) fxxx
A) (i) and (ii) C) All
B) (i), (ii), and (iii) D) (i), (ii), and (v)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
272) Write an "area" word problem for which finding the solution would involve evaluating
272)
the double integral x2
5 0
dy dx.
0
273) Write an "area" word problem for which finding the solution would involve evaluating the double integral 2 0
2x x2
dy dx.
53
273)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
274) True or false? Consider the double integral 2
6
274)
x 2 + y dx dy.
0 3 The first step in calculating this integral involves integrating with respect to x. A) True B) False
275) True or false? Consider the double integral 2
6
275)
x 2 + y dx dy.
0 3 The first step in calculating this integral involves integrating with respect to y. A) True B) False
276) True or false? Consider the double integral 2
6
276)
x 2 + y dy dx.
0 3 The first step in calculating this integral involves holding x constant. A) True B) False
277) True or false? Consider the double integral 2
6
277)
x 2 + y dx dy.
0 3 The first step in calculating this integral involves holding x constant. A) True B) False
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
278) Write a "volume" word problem for which finding the solution would involve evaluating the double integral 2 1
6
(y + x) dy dx.
4
54
278)
279) Write a "volume" word problem for which finding the solution would involve evaluating the double integral 2 1
6
(y + x) dx dy.
4
55
279)
Answer Key Testname: UNTITLED9
1) D 2) C 3) A 4) C 5) C 6) A 7) A 8) C 9)
10)
56
Answer Key Testname: UNTITLED9
11)
12) 13) C 14) D 15) A 16) B 17) C 18) C 19) A 20) C 21) A 22) C 23) A 24) C 25) C 26) D 27) B 28) B 29) B 30) B 31) C 57
Answer Key Testname: UNTITLED9
32) A 33) B 34) C 35) B 36) B 37) C 38) C 39) B 40) A 41) C 42) A 43) A 44) C 45) C 46) B 47) D 48) C 49) D 50) B 51) D 52) A 53) D 54) C 55) D 56) A 57) C 58) A 59) C 60) D 61) B 62) C 63) A 64) A 65) A 66) C 67) C 68) A 69) D 70) A 71) C 72) C 73) D 58
Answer Key Testname: UNTITLED9
74) A 75) A 76) A 77) A 78) B 79) C 80) C 81) B 82) C 83) D 84) C 85) B 86) A 87) B 88) C 89) C 90) A 91) D 92) A 93) B 94) D 95) B 96) D 97) D 98) A 99) C 100) B 101) D 102) C 103) B 104) B 105) C 106) B 107) C 108) B 109) B 110) D 111) C 112) A 113) B 114) A 115) A 59
Answer Key Testname: UNTITLED9
116) C 117) B 118) C 119) D 120) D 121) D 122) A 123) A 124) C 125) D 126) B 127) D 128) B 129) B 130) C 131) B 132) D 133) D 134) A 135) B 136) D 137) B 138) B 139) D 140) A 141) C 142) C 143) B 144) A 145) A 146) A 147) D 148) C 149) B 150) D 151) A 152) B 153) D 154) D 155) A 156) B 157) D 60
Answer Key Testname: UNTITLED9
158) A 159) B 160) D 161) C 162) C 163) A 164) C 165) A 166) C 167) B 168) D 169) B 170) D 171) C 172) D 173) C 174) C 175) C 176) D 177) C 178) B 179) C 180) D 181) B 182) D 183) A 184) A 185) C 186) B 187) D 188) D 189) B 190) A 191) B 192) C 193) C 194) C 195) A 196) B 197) B 198) A 199) A 61
Answer Key Testname: UNTITLED9
200) A 201) C 202) A 203) B 204) B 205) D 206) B 207) D 208) D 209) B 210) A 211) C 212) D 213) C 214) B 215) B 216) B 217) D 218) D 219) B 220) D 221) C 222) D 223) B 224) B 225) A 226) C 227) C 228) C 229) D 230) D 231) A 232) D 233) D 234) A 235) D 236) D 237) D 238) A 239) D 240) C 241) B 62
Answer Key Testname: UNTITLED9
242) A 243) A 244) B 245) C 246) A 247) A 248) C 249) C 250) C 251) A 252) A 253) D 254) C 255) D 256) C 257) C 258) B 259) C 260) C 261) A 262) A 263) C 264) C 265) D 266) D 267) D 268) B 269) B 270) B 271) B 272) Find the area of the region bounded by the curve y = x2 and the lines y = 0, x = 0, and x = 5. 273) Find the area of the region between the curves y = x2 and y = 2x for x between 0 and 2. 274) A 275) B 276) A 277) B 278) Find the volume bounded above by f(x,y) = y + x which lies over the region for which 1 x 2 and 4 y 6. 279) Find the volume bounded above by f(x,y) = y + x which lies over the region for which 4 x 6 and 1 y 2.
63
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the general solution for the differential equation. 1) dy = 6x2 dx
A) y = 6x3 + C
1)
3
B) y = x + C
C) y = x3 + C
3
D) y = 2x3 + C
2) dy = x - 14
2)
dx
2
A) y = 2x2 - 14 + C
B) y = x - x + C
C) y = x3 - 14x + C
D) y = x - 14x + C
2
2
2
3) dy = 30x2 - 18x
3)
dx
A) y = 10x3 - 18x2 + C C) y = 30x3 - 18x2 + C
B) y = 10x3 - 9x2 + C D) y = 30x3 - 9x2 + C
4) dy = 2e3x
4)
dx
A) y = 2e3x + C
B) y = 1 e3x + C
C) y = 2 e3x + C
3
3
5) dy - 9x2 = 8
D) y = 6e3x + C
5)
dx
A) y = 3x3 + 4x + C C) y = 3x3 + 8x + C
B) y = 3x3 - 4x + C D) y = -3x3 + 8x + C
6) 2x2 - 2 dy = 3
6)
dx
3
B) y = x - 3x + C
A) y = 3x3 + 3x + C 2
3
3
3
C) y = x + 3x + C
2
D) y = x + C
3
3
1
7) (y8 - y) dy = x
7)
dx
A) 9y9 - 2y2 = 2x2 + C C) 2y9 - 9y2 = 9x2 + C
B) y9 - 9y2 = 9x + C D) 9y9 - 2y = 2x2 + C
8) y dy = x2 - 7x
8)
dx
A) y = 1 x3 - 7x + C
B) y2 = 1 x3 - 7x2 + C
C) y2 = 2 x3 - 7x2 + C
D) y = 2 x3 - 7x2 + C
3
3
3
3
9) dy = y - 2
9)
dx
A) y = Mex + 2 C) y = 2xex + C
B) y = Me2x + C D) y = Me(x + 2) + C
10) dy = 10xy
10)
dx
A) y = e5x2 + C
B) y = Me5x2
C) y = ln 5x2 + C
D) y = Me10x
11) dy = 6xy - 12x2 y
11)
dx
A) y = ke(3x2 - 4x3 )
B) y = ln(3x2 - 4x3 ) + C
C) y = ke(6x - 3x2 )
D) y = 3x2 - 4x3 + C
12) dy = y dx
12)
x3 2
A) y = - x + C
B) y = - 1 ln 1 + C
C) y = Me-1/(4x4)
D) y = Me-1/(2x2)
2
2
x2
13) dy = y2e2x
13)
dx
A) y = - 1 + C e2x 2
B) y = - 1 + C
C) y = 1 + C e2x 2
2e2x
2
2x
D) y = - e
2
+C
14) dy = 5x4 y - 3x2 y
14)
dx
1 5 x - x3
A) y = - x5 - x3
B) y = -
C) y = e(x5 - x3 ) + C
D) y = Me(x5 - x3 )
15) dy = y2(6 - ex)
15)
dx
A) y = 6x - ex + C C) y =
B) y =
1 x e - 6x + C
k
6x - ex
D) y = 3
3 6x - ex + C
Find the particular solution for the initial value problem. 16) dy = 4x + 18; y(0) = -18 dx
16)
A) y = 2x2 + 18x - 9 C) y = 4x2 + 18x - 18
B) y = 2x2 + 18x - 18 D) y = 4x2 + 18x - 9
17) dy = 21 ; y(1) = 16 dx
17)
x
A) y = ln x + 16 C) y = 21 ln x + 10.5
B) y = ln x + 14 D) y = 21 ln x + 16
18) dy + 2x = 3x2 ; y(0) = 4
18)
dx
A) y = 3x3 + 2x2 + 4 C) y = 3x3 + x2 + 4
B) y = x3 - x2 + 4 D) y = x3 + 2x2 + 4
2
19) dy = x ; y(0) = 3 dx
19)
y
3
A) y2 = x + 3 3
3
3
B) y2 = 2x + 3
C) y2 = 2x + 9
3
3
20) (6x + 3)y = dy ; y(0) = 1
3
D) y2 = x + 9 3
20)
dx
A) y = e6x2 + 3x
B) y = e3x2 + 3x + 1
C) y = e3x2 + 3x
D) y = e6x2 + 3x + 1
3
21) x dy - 4y x = 0; y(0) = 1
21)
dx
B) y = e8x1/2
A) y = e4x-1/2 + 1
D) y = e8x1/2 + 1
C) y = e8x-1/2
22) dy = 6x2 - 4x + 22; y(1) = 5
22)
dx
A) y = 2x3 - 4x2 + 22x - 15 C) y = 2x3 - 2x2 + 22x - 17
B) y = 6x3 - 4x2 + 22x - 19 D) y = 2x3 - 2x2 + 22x + 17
23) dy = 4xe2x; y(0) = 15
23)
dx
A) y = 4xe2x - e2x + 16 C) y = 4xe2x - 2e2x + 17
B) y = 2xe2x - e2x + 16 D) y = 2xe2x + 15
24) x dy = 4x2 e2x; y(0) = 9
24)
dx
A) y = 2xe2x - e2x + 10 C) y = 4xe2x - 2e2x + 11
B) y = 4xe2x - e2x + 10 D) y = 2xe2x + 9
3
25) dy = 2 - x ; y(0) = 2 dx
25)
4y + 5
B) 2y2 + 5y = 2x - 1 x4 + 18
A) 2y2 + 5y = 2x - x4 + 18
4
C) 2y2 + 5y = 2x - 1 x4 + 39 3
D) 2y2 + 5y = 2x - 1 x4 + 2
2
4
26) dy = ex - y; y(0) = 5
26)
dx
B) y = ln(ex + e5 - 1) D) y = -ln(ex + e5 - 1)
A) y = ln x + 5 C) y = x + 5 2
27) dy = y ; y(e) = 8 dx
27)
x
B) y = -
A) y = 8 ln |x| C) y =
8 9 ln |x| - 8
D) y =
8 8 ln |x| - 9
8 ln |x|
28) x3 dy = y; y(1) = 1
28)
dx
A) y = e-1/(2x2 ) + 1/2
B) y = e-1/x2 + 1
C) y = e-1/(2x2 ) + 1
D) y = e-1/(4x4 ) + 1/4
4
29) dy = x1/2y2; y(4) = 9
29)
dx
A) y = -
9 3/2 6x - 42
B) y = -
9 1/2 6x - 49
C) y = -
9 3/2 6x - 49
D) y = -
1 3/2 6x - 42
30) dy = (x + 1)3ey; y(3) = 0
30)
dx
4
4
A) y = -ln 65 - (x + 1)
B) y = -ln 257 - (x + 1)
C) y = e65 - (x + 1)4/4
D) y = -ln 257 - (x + 1)
4
4
2
2
Solve the problem. 31) Sales (in thousands) of a certain product are declining at a rate proportional to the amount of sales, with a decay constant of 6% per year. Write a differential equation to express the rate of sales decline. A) dy/dt = -0.06t B) dy/dt = -0.94y C) dy/dt = -0.06y D) dy/dt = e-0.06t
31)
32) The amount of a radioactive substance decreases exponentially, with a decay constant of 4% per
32)
33) Sales (in thousands) of a certain product are declining at a rate proportional to the amount of
33)
34) The population of a country is predicted to increase from 11.3 million in 2000 to 23.7 million in
34)
month. Find a general solution to the differential equation which expresses the rate of change. A) y = -Me0.04t B) y = Me-0.96t C) y = -0.04t D) y = Me-0.04t
sales, with a decay constant of 12% per year. By writing and solving a differential equation, determine how much time will pass before sales become 40% of their original value. A) 7.6 years B) 6.7 years C) 5.1 years D) 5.9 years
2050. Assuming the unlimited growth model dy/dt = ky fits this population growth, express the population y as a function of the year t. Let 2000 correspond to t = 0.
A) y = 11.3e0.01681t
B) y = 11.3e0.01481t
23.7 -0.01681t 1+e
D) y = 23.7e0.01481t
C) y =
35) Assume that the rate of change of population of a certain city is given by dy = 6000e0.06t, where y dt
is the population at time t, in years. The population was 100,000 in 1980 (t = 0 in 1980). Predict the population in 2050. A) 7,368,633 B) 6,668,633 C) 6,968,633 D) 6,168,633
5
35)
36) A wild animal preserve can support no more than 160 elephants. 32 elephants were known to be
36)
in the preserve in 1980. Assume that the rate of growth of the population is dP = 0.0004P(160 - P) dt
where t is time in years. Find a formula for the elephant population in terms of t. Let 1980 correspond to t = 0. 160 160 A) P = B) P = -0.64t 1 + 4.00e 1 + 4.00e-0.064t
C) P =
160 1 - 4.00e-0.64t
D) P =
160 1 + 5.00e-0.064t
37) A wild animal preserve can support no more than 190 elephants. 34 elephants were known to be
37)
in the preserve in 1980. Assume that the rate of growth of the population is dP = 0.0007P(190 - P) dt
where t is time in years. How long will it take for the elephant population to increase from 34 to 110? [First find a formula for the elephant population in terms of t.] A) 14.4 years B) 13.8 years C) 12.0 years D) 15.0 years
38) When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical
38)
examiner to determine the time of death as closely as possible. If the temperature of the medium has been fairly constant and less than 48 hours have passed since death, Newton's law of cooling dT = -k(T - TM), where k is a constant, T is the can be used. Newton's law of cooling states, dt temperature of the object after t hours, and TM is the (constant) temperature of the surrounding medium. Assuming the temperature of a body at death is 98.6°F, the temperature of the surrounding air is 71°F, and at the end of one hour the body temperature is 89°F, what is the temperature of the body after 3 hours? Round to the nearest tenth of a degree. A) 78.7°F B) 89°F C) 7.7°F D) 72.4°F
39) When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical examiner to determine the time of death as closely as possible. If the temperature of the medium has been fairly constant and less than 48 hours have passed since death, Newton's law of cooling dT = -k(T - TM), where k is a constant, T is the can be used. Newton's law of cooling states, dt temperature of the object after t hours, and TM is the (constant) temperature of the surrounding medium. Assuming the temperature of a body at death is 98.6°F, the temperature of the surrounding air is 69°F, and at the end of one hour the body temperature is 91°F, when will the temperature of the body be 76°F? Round to the nearest tenth of an hour. A) 1.9 hr B) 0.3 hr C) 0.8 hr D) 4.9 hr
6
39)
A person's weight depends both on the daily rate of energy intake, say C calories per day, and the daily rate of energy consumption, typically between 12 and 20 calories per pound per day. Using an average value of 16 calories per pound per day, a person weighing w pounds uses 16w calories per day. If C = 16w, then weight remains constant, and weight gain or loss occurs according to whether C is greater or less than 16w. To determine how fast a change in weight will occur, a plausible assumption is that dw/dt is proportional to the net excess (or deficit) C - 16w in the number of calories per day. 40) Assume C is constant and write a differential equation to express this relationship. Use k to 40) represent the constant of proportionality. A) dw = k(C - 16w) B) dw = C(k - 16w) dt dt
C) dw = k(16w - C)
D) dw = k(C + 16w)
dt
dt
41) Assuming C is constant, a differential equation to express this relationship is dw/dt = k(C - 16w),
41)
where k is the constant of proportionality. The units of dw/dt are pounds per day, and the units of C - 16w are calories per day. What units must k have?
A) pounds/calorie C) calorie/pounds
B) pounds/day D) pounds calories/day2
42) Write a differential equation using the fact that 3500 calories is equivalent to one pound. A) dw/dt = 3500(C - 16w) B) dw/dt = (16w - C)/3500 C) dw/dt = C(3500 - 16w) D) dw/dt = (C - 16w)/3500
42)
43) Given that 3500 calories is equivalent to one pound, a differential equation to express this
43)
relationship is dw/dt = (C - 16w)/3500. Solve this differential equation. A) w = C/16 - e-0.0065M e-0.0065t /16 B) w = C - e-0.0046M e-0.0046t
C) w = C - e-0.0065M e-0.0065t
D) w = C/16 - e-0.0046M e-0.0046t /16
Solve the problem. 44) The table shows the population of a certain city for selected years between 1950 and 2003. Years after 1950 Population 0 7891 20 103,087 30 191,064 40 241,552 50 265,058 55 269,116 Use the logistic regression function on your calculator to determine the logistic equation that best fits the data. 266,076.8 278,715.3 A) P = B) P = -0.1215t 1 + 23.128e 1 + 25.311e-0.1374t
C) P =
254,180.3 1 + 26.118e-0.1402t
D) P =
7
271,976.2 1 + 24.361e-0.1345t
44)
45) The table shows the population of a certain city for selected years between 1950 and 2003.
45)
46) The table shows the population of a certain city for selected years between 1950 and 2003.
46)
47) The table shows the population of a certain city for selected years between 1950 and 2003.
47)
Years after 1950 Population 0 7891 20 103,087 30 191,064 40 241,552 50 265,058 55 269,116 By using your calculator to find the logistic regression equation that best fits the data, determine the limiting size of the population. A) 271,976 B) 266,078 C) 254,180 D) 278, 715
Years after 1950 Population 0 7891 20 103,087 30 191,064 40 241,552 50 265,058 55 269,116 By using your calculator to find the logistic regression equation that best fits the data, determine when the population of the city will first exceed 271,000. A) In 2012 B) In 2010 C) In 2015 D) In 2018
Years after 1950 Population 0 7891 20 103,087 30 191,064 40 241,552 50 265,058 55 269,116
Write a logistic differential equation in the form
dP = kP(M - P) that models the growth of the dt
population. [You will first need to use your calculator find the logistic regression equation that best fits the data.] A) dP = 0.1345 P(271,976.2 - P) B) dP = (4.945 × 10-7 )P(271,976.2 - P) dt dt
C) dP = 0.1207 P(278,903.2 - P)
D) dP = (4.741 × 10-7 )P(278,903.2 - P)
dt
dt
8
48) The table shows the population of a certain city for selected years between 1950 and 2003.
48)
Years after 1950 Population 0 12,421 20 143,112 30 290,089 40 375,297 50 445,052 55 471,126 Use the logistic regression function on your calculator to determine the logistic equation that best fits the data. 494,193.8 482,549.6 A) P = B) P = -0.1084t 1 + 24.126e 1 + 22.095e-0.1132t
C) P =
478,549.6 1 + 21.095e-0.1209t
D) P =
499,107.3 1 + 23.521e-0.0981t
49) The table shows the population of a certain city for selected years between 1950 and 2003.
49)
50) The table shows the population of a certain city for selected years between 1950 and 2003.
50)
Years after 1950 Population 0 12,421 20 143,112 30 290,089 40 375,297 50 445, 052 55 471,126 By using your calculator to find the logistic regression equation that best fits the data, determine the limiting size of the population. A) 478,550 B) 494,194 C) 499,107 D) 482,550
Years after 1950 Population 0 12,421 20 143,112 30 290,089 40 375,297 50 445, 052 55 471,126 By using your calculator to find the logistic regression equation that best fits the data, determine when the population of the city will first exceed 480,000. A) In 2018 B) In 2015 C) In 2023 D) In 2020
9
51) The table shows the population of a certain city for selected years between 1950 and 2003. Years after 1950 Population 0 12,421 20 143,112 30 290,089 40 375,297 50 445, 052 55 471,126
Write a logistic differential equation in the form
51)
dP = kP(M - P) that models the growth of the dt
population. [You will first need to use your calculator find the logistic regression equation that best fits the data.] A) dP = (2.193 × 10-7 )P(494,193.8 - P) B) dP = (2.346 × 10-7 )P(482,549.6 - P) dt dt
C) dP = (1.623 × 10-7 )P(499,107.3 - P)
D) dP = (2.526 × 10-7 )P(478,549.6 - P)
dt
dt
Find the general solution for the differential equation. 52) dy + 2y = 17 dx
52)
A) y = 17 + Ce2x
B) y = 17 + Ce2x
C) y = 17 + Ce-2x
D) y = 17 + e2x + Ce-2x
5 2
2
53) dy + 3y = 12
53)
dx
A) y = 12 + Ce-3x
C) y = 1 + Ce3x
B) y = 4 + Ce-3x
4
D) y = 4 + Ce-12x
54) dy + 2xy = 19x
54)
dx
A) y = 19 + Cex2
B) y = 19 + 2x + Ce-x2
C) y = 19 + Cex2
D) y = 19 + Ce-x2
2
5
2
55) x dy - 2y - 2x = 0
55)
dx
A) y = -2x + Cx2
B) y = - 2 + C
C) y = -2x2 + Cx3
x
10
D) y = -2 + Cx
56) x dy + 3xy - x2 = 0
56)
dx
A) y = x - 1 + Ce3x
B) y = x + 1 + Ce3x
C) y = x - 1 + Ce-3x
D) y = x - 1 + Ce-3x
3
9
3
3
9
3
3
3
57) 5 dy - 10xy - x = 0
57)
dx
A) y = - 1 + Cex2
B) y = - 1 + Cex2 /2
C) y = - 1 + Ce-x2
D) y = 1 + Ce-x2/2
10
10
10
10
58) x2 dy + xy = 6x6 + 5x5 , x > 0
58)
dx
A) y = x6 + x5 + C
B) y = k x5 + x4 + 1
C) y = x6 + x5 + C
D) y = x5 + x4 + C
x
x
x
59) x2 + 5xy = 5x dy
59)
dx
A) y = - x + 1 + Cex
B) y = - x + Cex
3
D) y = x - 1 + Cex
5
5
C) y = x + C
5
20
60) y - x dy = 10x3 , x > 0
60)
dx
A) y = -10x3 + Cx
B) y = -5x3 + Cx
C) y = 5x2 + Cx
D) y = -5x2 + C
61) x dy + 2y = 5x2 + 12x
61)
dx
A) y = 5x3 + 12x2 + C
B) y = 5x2 + 12x + C
4
D) y = 5x + 4x + C
x2
x
2
C) y = 5x + 4x3 + C 4
4
11
x2
Solve the differential equation subject to the initial condition. 62) dy - xy - x = 0; y(1) = 10 dx
62)
A) y = -1 + 11e(x2 -1)/2
B) y = - 1 + 11 ex2 - 1
C) y = -1 + 11e-x2
D) y = - 1 + 11e-x2
2
2
2
63) dy + y = 2ex; y(0) = 5
63)
dx
A) y = 2ex + 2e-x C) y = 4e2 + 20e-x
B) y = 5ex D) y = ex + 4e-x
64) dy + 7y = 3; y(0) = 1
64)
dx
A) y = 4 e-7x + 3 7
7
B) y = 4 e7x + 3 7
C) y = 3 e-7x + 4
7
7
7
D) y = 3 e7x + 4 7
7
65) x dy + 4y = x2 ; y(2) = 9
65)
dx
A) y = x + 400 6
3x4
2
2
B) y = x + 380 6
C) y = x + 400
3x4
6
3x4
66) dy + xy = 3x; y(0) = -6
2
D) y = x + 380 6
3x3
66)
dx
A) y = 3ex2 /2 - 9
B) y = -9ex2/2 + 3
C) y = -9e-x2 /2 + 3
D) y = 3e-x2/2 - 9
67) 2 dy - 4xy = 8x; y(0) = 19
67)
dx
A) y = -1 + 20ex2
B) y = -2 + 21ex2
C) y = 2 + 19ex2
D) y = -2 + 21e-x2
68) x dy + (1 + x)y = 5; y(5) = 5
68)
dx
5-x
5-x
A) y = 5 + 25 e
B) y = 5 + 20e
x
x
-5 - x
7-x
C) y = 5 + 20e
D) y = 5 + 22e
x
x
12
69) dy + 6xy - e-3x2 = 0; y(0) = 5
69)
dx
A) y = (x + 5)e-3x
B) y = (x + 5)e-3x2
C) y = xe-3x2 + 5
D) y = xe-3x + 5
70) dy + 4x3 y - 4xe-x4 = 0; y(0) = 250
70)
dx
A) y = 2xe-x4 + 250e-x4
B) y = 2x2 e-x4 + 250e-x4
C) y = 4x2 ex4 + 250ex4
D) y = 2x2 e-x3 + 250e-x3
Solve the problem. 71) The rate of change in the concentration of a drug with respect to time in a user's blood is given by dC = -kC + D(t), dt
71)
where D(t) is dosage at time t and k is the rate at which the drug leaves the bloodstream. If D(t) is a constant D, and C(0) = C0 , solve the differential equation to find a formula for C(t), the concentration at time t.
A) C(t) = C) C(t) =
D + (kC0 - 1)e-kt
-kt)
B) C(t) = D(1 - e
k
k
D + (kC0 - D)e-kt
D) C(t) =
k
D + (kC0 - D)ekt k
72) If a population is changed by either immigration or emigration, a model for the population is
72)
dy = ky + f(t), dt
where y is the population at time t and f(t) is some function of t that describes the net effect of the emigration/immigration. Assume that k = 0.02 and y(0) = 10,000. Solve this differential equation for y, given that f(t) = 12t. A) y = -600t - 30,000 + 40,000e0.02t B) y = 600t + 30,000 + 40,000e-0.02t
C) y = 600t - 30,000 + 40,000e-0.02t
D) y = -600t - 30,000 + 40,000e-0.02t
73) If a population is changed by either immigration or emigration, a model for the population is dy = ky + f(t), dt
where y is the population at time t and f(t) is some function of t that describes the net effect of the emigration/immigration. Assume that k = 0.02 and y(0) = 10,000. Solve this differential equation for y, given that f(t) = -4t. A) y = -200t + 10,000 + 0e-0.02t B) y = 200t + 10,000 + 0e-0.02t
C) y = 200t + 10,000 + 0e0.02t
D) y = -200t - 10,000 + 0e0.02t
13
73)
74) Newton's Law of Cooling states that the rate of change of temperature of an object is proportional
74)
to the difference in temperature between the object and the surrounding medium. Thus, if T is the temperature of the object after t hours and T0 is the (constant) temperature of the surrounding medium, then dT = -k(T - T0 ), dt where k is a constant. Assume that the temperature of a body at death is 98.6°F, the temperature of the surrounding air is 57°F, and at the end of one hour the body temperature is 87°F. Use Newton's Law of Cooling to determine the number of hours that have elapsed when the temperature of the body is 67°F. A) 5.2 hours B) 5.6 hours C) 4.4 hours D) 6.7 hours
75) Newton's Law of Cooling states that the rate of change of temperature of an object is proportional
75)
to the difference in temperature between the object and the surrounding medium. Thus, if T is the temperature of the object after t hours and T0 is the (constant) temperature of the surrounding medium, then dT = -k(T - T0 ), dt where k is a constant. Suppose the air temperature surrounding a body remains at a constant 12° F, k = 0.24, and the temperature of the body at death is 98.6°F. By solving the differential equation, determine a formula for the temperature at any time t. A) T = 98.6e-0.24t + 12 B) T = 98.6e-0.24t - 12
D) T = 86.6e0.24t + 12
C) T = 86.6e-0.24t + 12
76) Newton's Law of Cooling states that the rate of change of temperature of an object is proportional
to the difference in temperature between the object and the surrounding medium. Thus, if T is the temperature of the object after t hours and T0 is the (constant) temperature of the surrounding medium, then dT = -k(T - T0 ), dt where k is a constant. A cup of coffee with a temperature of 102°F is placed in a freezer with a temperature of 0°F. After 6 minutes, the temperature of the coffee is 61.6°F. What will its temperature be 15 minutes after it is placed in the freezer? Round your answer to the nearest degree. A) 24°F B) 29°F C) 27°F D) 23°F
14
76)
77) Newton's Law of Cooling states that the rate of change of temperature of an object is proportional
77)
to the difference in temperature between the object and the surrounding medium. Thus, if T is the temperature of the object after t hours and T0 is the (constant) temperature of the surrounding medium, then dT = -k(T - T0 ), dt where k is a constant. A cup of coffee with a temperature of 105°F is placed in a freezer with a temperature of 0°F. After 8 minutes, the temperature of the coffee is 55.4°F. When will its temperature be 44°F? Round your answer to the nearest minute. A) 24 minutes after being placed in the freezer
B) 14 minutes after being placed in the freezer C) 11 minutes after being placed in the freezer D) 21 minutes after being placed in the freezer 78) Newton's Law of Cooling states that the rate of change of temperature of an object is proportional
78)
to the difference in temperature between the object and the surrounding medium. Thus, if T is the temperature of the object after t hours and T0 is the (constant) temperature of the surrounding medium, then dT = -k(T - T0 ), dt where k is a constant. A dish of lasagna baked at 400°F is taken out of the oven into a kitchen that is 73°F. After 8 minutes, the temperature of the lasagna is 324.1°F. What will its temperature be 16 minutes after it was taken out of the oven? Round your answer to the nearest degree. A) 274°F B) 266°F C) 260°F D) 252°F
Use Euler's method to approximate the indicated function value to three decimal places using h = 0.1. 79) dy = x2 + y2 ; y(0) = 1; find y(0.4) dx
A) 1.689
B) 1.569
C) 1.462
D) 1.573
80) dy = xy + 2; y(0) = 0; find y(0.4)
80)
dx
A) 0.809
B) 0.837
C) 0.812
D) 0.828
81) dy = 1 - e-x; y(0) = 0; find y(0.4)
81)
dx
A) 0.042
B) 0.072
C) 0.054
D) 0.051
82) dy = e-y + x; y(0) = 0; find y(0.4)
82)
dx
A) 0.502
79)
B) 0.408
C) 0.342
15
D) 0.406
83) dy = 1 + y ; y(1) = 0; find y(1.4) dx
83)
x
A) 0.428
B) 0.486
C) 0.529
D) 0.452
84) dy = x + y; y(0) = 1; find y(0.4)
84)
dx
A) 1.603
B) 1.501
C) 1.491
D) 1.402
85) dy = x + y2 ; y(0) = 0; find y(0.4)
85)
dx
A) 0.101
B) 0.039
C) 0.060
D) 0.058
86) dy = x + y2 ; y(0) = 0; find y(0.5)
86)
dx
A) 0.061
B) 0.098
C) 0.100
D) 0.141
87) dy = 1 + y; y(0) = 2; find y(0.5)
87)
dx
A) 4.561
B) 3.427
C) 3.832
D) 3.864
88) dy = ey + e-x; y(1) = 2; find y(1.3)
88)
dx
A) 12.278
B) 12.704
C) 12.908
D) 11.907
89) dy = x2 ; y(0) = 2; find y(0.4)
89)
dx
A) 2.020
B) 2.005
C) 2.026
D) 2.014
90) dy = 4x + 3; y(1) = 2; find y(1.4)
90)
dx
A) 2.770
B) 5.040
C) 3.630
D) 3.080
91) dy = 2xy; y(1) = 1; find y(1.4)
91)
dx
A) 2.982
B) 2.928
C) 2.591
D) 2.287
92) dy = 1 ; y(1) = 1; find y(1.4) dx
92)
x
A) 1.247
B) 1.410
C) 1.351
D) 1.274
93) dy = -4 + x; y(0) = 1; find y(0.5)
93)
dx
A) -0.540
B) -0.940
C) -0.900
16
D) -0.880
94) dy = x2 y; y(0) = 1; find y(0.5)
94)
dx
A) 1.014
B) 1.042
C) 1.030
D) 1.003
95) dy = 4x + 3; y(1) = 0; find y(1.5)
95)
dx
A) 3.990
B) 2.770
C) 3.900
D) 4.181
96) dy = x ; y(0) = 2; find y(0.3) dx
96)
y
A) 2.005
B) 2.024
C) 2.015
D) 2.030
97) dy = x2 - x; y(0) = 0; find y(0.3)
97)
dx
A) -0.034
B) -0.025
C) -0.055
D) -0.046
98) dy - y = 3ex; y(0) = 80; find y(0.4)
98)
dx
A) 119.718
B) 120.018
C) 118.736
D) 119.140
Solve the problem. 99) A rumor spreads through a community of 800 people at the rate dN = 0.1(800 - N)N1/2 dt
99)
where N is the number of people who have heard the rumor at time t (in hours). Use Euler's method with h = 0.5 to find the number who have heard the rumor after 1.5 hours, if 1 person heard it initially. A) about 702 people B) about 719 people
C) about 711 people
D) about 681 people
100) A population of algae consists of 4000 algae at time t = 0. Conditions will support at most 600,000
algae. Assume that the rate of growth of algae is proportional both to the number present (in thousands) and to the difference between 600,000 and the number present (in thousands). Write a differential equation using 0.03 for the constant of proportionality. A) dy/dt = 0.03y(y - 600) B) dy/dt = 4000y(600 - 0.03y)
C) dy/dt = 0.03y(600 - y)
D) dy/dt = 0.03(600 - y)
17
100)
101) A population of algae consists of 4000 algae at time t = 0. Conditions will support at most 300,000
101)
algae. The rate of growth of algae is proportional both to the number present (in thousands) and to the difference between 300,000 and the number present (in thousands). Given that the constant of proportionality is 0.01, a differential equation is dy = 0.01y(300 - y). dt Approximate the number of algae present when t = 2, using h = 0.5. A) about 141 thousand B) about 125 thousand
C) about 133 thousand
D) about 128 thousand
102) A population of ants, y, living in a colony grows at a rate
102)
dy = 0.05y - 0.1y1/2, dt
where t is time in weeks. The initial population is 1000 insects. Using Euler's method with h = 1, what is the number of ants after 7 weeks? A) 1380 ants B) 1383 ants C) 1381 ants D) 1377 ants
103) Richard deposits $2000 in an IRA at 10% interest compounded continuously for his retirement in
103)
104) Richard deposits $2000 in an IRA at 10% interest compounded continuously for his retirement in
104)
105) Sally plans to make continuous deposits to a savings account. She wants to accumulate $40,000 in
105)
25 years. He intends to make continuous deposits at the rate of $2500 a year. How much will he have accumulated in 20 years? Round your answer to the nearest dollar. A) $177,505 B) $170,505 C) $174,505 D) $180,505
25 years. He intends to make continuous deposits at the rate of $2500 a year. How long will it take for Richard to accumulate $110,000? A) 17.1 years B) 16.7 years C) 15.1 years D) 16.1 years
10 years. If the account earns 9% interest compounded continuously and she makes no initial deposit, what yearly deposit will be required? Round your answer to the nearest dollar. A) $3066 B) $2665 C) $2866 D) $2466
Find the requested equation. 106) The system of equations dy = y - 2xy dt
106)
dx = -x + 3xy dt describes the influence of the populations (in thousands) of two competing species on their growth rates. Find an equation relating x and y, assuming y = 2 when x = 1. A) 3y - ln y = ln x - ln 2 + 11 B) 3y - ln y = ln x - 2x - ln 3 + 11
C) 3y - ln y = ln x - 2x - ln 2 + 8
D) 3y - ln y = ln x - 2x + 8
18
107) The system of equations
107)
dy = y - 3xy dt
dx = -x + 5xy dt describes the influence of the populations (in thousands) of two competing species on their growth rates. Find an equation relating x and y, assuming y = 3 when x = 1. A) 5y - ln y = ln x - 3x - ln 3 + 12 B) 5y - ln y = ln x - ln 3 + 18
C) 5y - ln y = -3x - ln 3 + 12
D) 5y - ln y = ln x - 3x - ln 3 + 18
Solve the problem. 108) Suppose an isolated island has a native population of 7000 and a person from a visiting ship introduces a disease which has an infection rate of 0.00004. Assume that the rate of spread of the disease satisfies the following logistic equation: dy y =k 1y, dt N
108)
where N is the size of the population and y is the number infected at time t. Write an equation for the number of infected natives after t days. 6999 6999 A) y = B) y = -0.28t 1 + 7000e 1 + 7000e-0.00004t
C) y =
7000 1 + 6999e-0.28t
D) y =
7000 1 + 6999e-0.00004t
109) Suppose an isolated island has a native population of 8000 and a person from a visiting ship
109)
introduces a disease which has an infection rate of 0.00005. Assume that the rate of spread of the disease satisfies the following logistic equation: dy y =k 1y, dt N where N is the size of the population and y is the number infected at time t. Write an equation for the number of natives who remain uninfected after t days. 8000 A) y = B) y = 63,992,000 0.4t 7999 + e 7999 + e0.00005t
C) y = 63,992,000
D) y = 63,992,000
7999 + e0.4t
7999 + e-0.4t
110) Suppose an isolated island has a native population of 10,000 and a person from a visiting ship
introduces a disease which has an infection rate of 0.00005. Assume that the rate of spread of the disease satisfies the following logistic equation: dy y =k 1y, dt N where N is the size of the population and y is the number infected at time t. How many individuals are infected after 23 days? A) 9230 B) 9190 C) 9080
19
D) 8980
110)
111) Suppose an isolated island has a native population of 10,000 and a person from a visiting ship
111)
introduces a disease which has an infection rate of 0.00005. Assume that the rate of spread of the disease satisfies the following logistic equation: dy y =k 1y, dt N where N is the size of the population and y is the number infected at time t. How many individuals remain uninfected after 10 days? A) 9754 B) 9854 C) 9554
D) 146
112) An influenza epidemic spreads at a rate proportional to the product of the number of people
112)
infected and the number not yet infected. Assume that 50 are infected at the beginning of the epidemic in a community of 8000 people and 200 are infected 10 days later. Write an equation for the number of people infected, y, after t days. 8000 8000 A) y = B) y = -0.14t 1 + 159e 1 + 7999e-0.12t
C) y =
8000 1 + 7999e-0.14t
D) y =
8000 1 + 159e-0.12t
113) Suppose a rumor starts among 4 people in an office building. That is, y0 = 4. Suppose 300 people
113)
work in the building and 60 people have heard the rumor in 4 days. Write an equation for the number who have heard the rumor in t days. Assume that the rate of spread of the rumor satisfies the logistic equation: dy y =k 1y, dt N where N is the size of the population and y is the number who have heard the rumor after t days. 300 300 A) y = B) y = 1 + 299e-0.729t 1 + 74e-0.729t
C) y =
300 1 + 299e-0.618t
D) y =
300 1 + 74e-0.618t
114) Suppose a tank contains 100 gallons of a solution of 10 lb of salt dissolved in water, which is kept
114)
uniform by stirring. If pure water is allowed to flow into the tank at a rate of 3 gallons per minute, and the mixture flows out at a rate of 2 gallons per minute, find an expression for the amount of salt in the tank after t minutes. 5 5 A) y = 10 B) y = 10 (3t + 50)2 (t + 100)2
C) y = 100 (t - 100)2
D) y =
103 (t - 100)
115) Suppose a tank contains 100 gallons of a solution of 10 lb of salt dissolved in water, which is kept
uniform by stirring. Pure water is allowed to flow into the tank at a rate of 3 gallons per minute, and the mixture flows out at a rate of 2 gallons per minute. How much salt is present in the tank after 30 minutes? A) Approx. 6.9 lb B) Approx. 5.9 lb C) Approx. 5.1 lb D) Approx. 8.3 lb
20
115)
116) Suppose a tank contains 100 gallons of a solution of 10 lb of salt dissolved in water, which is kept uniform by stirring. Pure water is allowed to flow into the tank at a rate of 3 gallons per minute, and the mixture flows out at a rate of 2 gallons per minute. How long will it take for the amount of salt in the mixture to be reduced to 5.5 lb? A) Approx. 35 min B) Approx. 49 min
C) Approx. 89 min
D) Approx. 69 min
21
116)
Answer Key Testname: UNTITLED10
1) D 2) D 3) B 4) C 5) C 6) B 7) C 8) C 9) A 10) B 11) A 12) D 13) A 14) D 15) C 16) B 17) D 18) B 19) C 20) C 21) B 22) C 23) B 24) A 25) B 26) B 27) B 28) A 29) C 30) A 31) C 32) D 33) A 34) B 35) B 36) B 37) B 38) A 39) D 40) A 41) A 42) D 22
Answer Key Testname: UNTITLED10
43) D 44) D 45) A 46) C 47) B 48) B 49) D 50) C 51) B 52) C 53) B 54) D 55) A 56) C 57) A 58) D 59) A 60) B 61) D 62) A 63) D 64) A 65) C 66) C 67) B 68) B 69) B 70) B 71) C 72) A 73) C 74) C 75) C 76) B 77) C 78) B 79) D 80) D 81) C 82) D 83) D 84) C 23
Answer Key Testname: UNTITLED10
85) C 86) C 87) C 88) B 89) D 90) B 91) D 92) C 93) C 94) C 95) C 96) C 97) B 98) C 99) B 100) C 101) D 102) A 103) C 104) D 105) D 106) C 107) D 108) C 109) D 110) C 111) B 112) A 113) B 114) B 115) B 116) A
24
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Decide whether or not the function is a probability density function on the indicated interval. 1) f(x) = 3 x2 ; [3, 5] 98
A) No
B) Yes
2) f(x) = 4 x3 ; [0, 3]
2)
81
A) No
B) Yes
3) f(x) = 1 x - 1 ; [3, 4] 3
3)
6
A) No
B) Yes
4) f(x) = 1 x - 1 ; [2, 5] 9
1)
4)
18
A) Yes
B) No
5) f(x) = 3 x2 ; [1, 4]
5)
63
A) Yes
B) No
6) f(x) = 4 x3 ; [0, 3]
6)
74
A) Yes
B) No
7) f(x) = 3x2 ; [-2, 2] A) Yes
B) No
8) f(x) = 2x; [-2, 5] A) No
B) Yes
7)
8)
2
9) f(x) = 3x ; [-2, 2]
9)
16
A) Yes
B) No
1
10) f(x) = 12 x2 - 12 x + 1 ; [0, 5] 355
355
10)
355
A) No
B) Yes
Find a value of k that will make f a probability density function on the indicated interval. 11) f(x) = kx; [0, 5] A) 2 B) 2 C) 1 D) 1 5 25 25 5
12) f(x) = kx2; [0, 3] A) 3 26
13) f(x) = kx2; [-1, 4] A) 3 64
14) f(x) = kx1/2; [1, 9] A) 3 54
15) f(x) = kx2; [1, 2] A) 1 3
16) f(x) = kx; [2, 4] A) 1 8
17) f(x) = kx; [0, 6] A) 1 9
18) f(x) = kx2; [1, 3] A) 1 9
19) f(x) = kx3; [0, 2] A) 1 8
20) f(x) = kx3; [1, 2] A) 4 17
11)
12) B) 1 27
C) 2 9
D) 1 9
B) 1 16
C) 1 21
D) 3 65
B) 1 56
C) 3 17
D) 3 52
13)
14)
15) B) 3
C) 1
D) 3
B) 1 16
C) 1 6
D) 1 12
B) 1 35
C) 1 18
D) 1 36
7
2
8
16)
17)
18) B) 3
C) 1
D) 3
B) 1 4
C) 1 15
D) 4 15
B) 4 15
C) 1 4
D) 3 16
26
13
28
19)
20)
2
Find the cumulative distribution function for the given probability density function. 21) f(x) = 1 x - 1 ; [2, 4] 5 10 2
21)
2
A) F(x) = x - x - 2 , 2 x 4
B) F(x) = x - x - 2 , 2 x 4
5
10
2
2
C) F(x) = x - x + 2 , 2 x 4
D) F(x) = x - x - 4 , 2 x 4
10
10
22) f(x) = 3 x2 ; [2, 4]
22)
56
3
3
A) F(x) = x , 2 x 4
B) F(x) = x - 2 , 2 x 4
56
C) F(x) =
x3 + 8 56
56
, 2 x 4
D) F(x) =
x3 - 8 56
, 2 x 4
23) f(x) = 3 x1/2; [1, 9]
23)
52
A) F(x) = x
1/2 - 1
C) F(x) = x
3/2
26
26
, 1 x 9
, 1 x 9
B) F(x) = x
3/2 - 1
D) F(x) = x
3/2 + 1
26 26
, 1 x 9 , 1 x 9
Find the indicated probability. 24) f(x) = 1 (1 + x)-3/2; [0, ), P(x 3) 2
A) - 1 2
24) C) 1
B) - 1
2
3
D) 1
3
25) f(x) = e-x; [0, ), P(x 9) A) 0 B) 0.0001 C) 00.9999 D) The function f(x) is not a probability density function.
25)
26) f(x) = 1 e-x/4 ; [0, ), P(1 x 5)
26)
4
A) 0.0615
B) 0.1,231
C) 0.4,923
3
D) 0.0820
27) f(x) =
5
(x + 5)2
; [0, ), P(2 x 6)
27)
A) 0.5,195 B) 0.1,732 C) 0.2,597 D) The function f(x) is not a probability density function.
28) f(x) =
x3 12
if 0 x 2
16 3x3
if x > 2
A) 313 432
, P(0 x 3)
28)
B) 67
C) 19
108
27
D) 143 216
Solve the problem. 29) The life (in months) of an automobile battery has a probability density function defined by 1 f(x) = e-x/4 for x in [0, ). Find the probability that the life of a randomly selected battery is 4 greater than 5 years. A) 0.0716
B) 0.2,865
C) 0.1,784
D) 0.7,135
30) The time between major earthquakes in the Alaska panhandle region is a random variable with probability density function f(x) =
29)
30)
1 -x/660 e for x in [0, ), where t is measured in days. Find the 660
probability that the time between a major earthquake and the next one is less than 300 days. A) 0.3,653 B) 0.0010 C) 0.6,347 D) 0.0006
31) The time of a telephone call (in minutes) to a certain town is a continuous random variable with a
31)
probability density function defined by f(x) = 6x-7 for x 1. Find the probability that the call lasts between 1 and 2 minutes. A) 0.9844 B) 1.0156 C) 0.0156 D) 0.8594
32) The time of a telephone call (in minutes) to a certain town is a continuous random variable with a
32)
probability density function defined by f(x) = 2x-3 for x 1. Find the probability that the call lasts more than 2 minutes. A) 0.5000 B) 0.2500 C) 0.7500 D) 0.6250
33) The time to failure t, in hours, of a certain machine can often be assumed to be exponentially distributed with probability density function 1 -t/83 f(t) = e ,0 t< , 83
What is the probability that a failure will occur in 46 hours or less? A) 0.4255 B) 0.5106 C) 0.3404
4
D) 0.3191
33)
Find the expected value of the probability density function to the nearest hundredth. 34) f(x) = 1 ; [3, 6] 3
A) 4.00
B) 4.17
C) 5.00
34) D) 4.50
35) f(x) = 1 ; [0, 5]
35)
5
A) 5.00
B) 2.40
C) 2.50
D) 12.50
36) f(x) = x - 1 ; [2, 6] 8
36)
4
A) 4.67
B) 4.00
C) 4.33
D) 5.00
37) f(x) = 2(1 - x); [0, 1] A) 1.00
B) 0.67
C) 0.33
D) 0.50
37)
38) f(x) = 1 - 1 ; [1, 4]
38)
x
A) 2.50
B) 2.67
39) f(x) = 3x-4 ; [1, ) A) 1.50 40) f(x) = 4x-5 ; [1, ) A) 1.00
C) 3.00
D) 2.83 39)
B) 1.75
C) 1.25
D) 1.00 40)
B) 1.67
C) 0.80
D) 1.33
41) f(x) = 1 ; [2, 8]
41)
6
A) 5.00
B) 5.33
C) 6.50
D) 4.50
42) f(x) = x - 1 ; [3, 4] 3
42)
6
A) 3.75
B) 8.28
C) 3.50
D) 3.53
2
43) f(x) = 3x ; [3, 5]
43)
98
A) 4.20
B) 4.16
C) 1.31
D) 4.78
Find the variance of the probability density function to the nearest hundredth. 44) f(x) = 1 ; [3, 6] 3
A) 1.89
B) 0.38
C) 0
5
44) D) 0.75
45) f(x) = 1 ; [0, 5]
45)
5
A) 0
B) 2.09
C) 2.08
D) 1.04
46) f(x) = x - 1 ; [2, 6] 8
46)
4
A) 0.16
B) 0.84
C) 0.91
D) 0.88
47) f(x) = 2(1 - x); [0, 1] A) 1.00
B) 1.50
C) 0.06
D) 0.60
47)
48) f(x) = 1 - 1 ; [1, 4]
48)
x
A) 0.52
B) 0.57
C) 0.61
D) 0.53
49) f(x) = 3x-4 ; [1, ) A) 0.73
B) 0.69
C) 0.50
D) 0.75
50) f(x) = 4x-5 ; [1, ) A) 0.20
B) 0.25
C) 0.22
D) 0.33
49)
50)
51) f(x) = 1 ; [2, 8]
51)
6
A) 2.86
B) 2.75
C) 3.00
D) 2.98
52) f(x) = x - 1 ; [3, 4] 3
52)
6
A) 0.08
B) 0.05
C) 0.10
D) 0.06
2
53) f(x) = 3x ; [3, 5]
53)
98
A) 0.25
B) 0.27
C) 0.31
D) 0.33
Find the standard deviation of the probability density function to the nearest hundredth. 54) f(x) = 1 ; [3, 6] 3
A) 0.61
B) 0.94
C) 0.87
D) 0.86
55) f(x) = 1 ; [0, 5]
55)
5
A) 1.40
54)
B) 1.02
C) 1.45
6
D) 1.44
56) f(x) = x - 1 ; [2, 6] 8
56)
4
A) 0.93
B) 0.92
57) f(x) = 2(1 - x); [0, 1] A) 0.25
C) 0.95
D) 0.94 57)
B) 0.24
C) 0.33
D) 0.31
58) f(x) = 1 - 1 ; [1, 4]
58)
x
A) 0.76
B) 0.75
59) f(x) = 3x-4 ; [1, ] A) 0.87 60) f(x) = 4x-5 ; [1, ] A) 0.57
C) 0.73
D) 0.71 59)
B) 0.85
C) 0.71
D) 0.83 60)
B) 0.47
C) 0.44
D) 0.50
61) f(x) = 1 ; [2, 8]
61)
6
A) 1.72
B) 1.69
C) 1.73
D) 1.66
62) f(x) = x - 1 ; [3, 4] 3
62)
6
A) 0.32
B) 0.28
C) 0.25
D) 0.24
2
63) f(x) = 3x ; [3, 5]
63)
98
A) 0.61
B) 0.50
C) 0.54
D) 0.56
Find the probability to the nearest hundredth that the random variable of the probability density function has a value greater than the mean. 64) f(x) = 1 ; [3, 6] 64) 3
A) 0.50
B) 0.41
C) 0.53
D) 0.48
65) f(x) = 1 ; [0, 5]
65)
5
A) 0.53
B) 0.49
C) 0.50
D) 0.45
66) f(x) = x - 1 ; [2, 6] 8
A) 0.54
66)
4
B) 0.50
C) 0.55
7
D) 0.46
67) f(x) = 2(1 - x); [0, 1] A) 0.29
67) B) 0.33
C) 0.50
D) 0.44
68) f(x) = 1 - 1 ; [1, 4]
68)
x
A) 0.48
B) 0.55
C) 0.53
D) 0.50
69) f(x) = 3x-4 ; [1, ) A) 0.50
B) 0.30
C) 0.32
D) 0.26
70) f(x) = 4x-5 ; [1, ) A) 0.23
B) 0.29
C) 0.50
D) 0.32
69)
70)
71) f(x) = 1 ; [2, 8]
71)
6
A) 0.53
B) 0.50
C) 0.45
D) 0.48
72) f(x) = x - 1 ; [3, 4] 3
72)
6
A) 0.49
B) 0.57
C) 0.51
D) 0.54
2
73) f(x) = 3x ; [3, 5]
73)
98
A) 0.62
B) 0.56
C) 0.50
D) 0.54
Find the probability to the nearest hundredth that the value of the random variable is within one standard deviation of the mean. 74) f(x) = 1 ; [0, 6] 74) 6
A) 0.67 75) f(x) = 2(1 - x); [0, 1] A) 0.75
B) 0.75
C) 0.58
D) 0.50 75)
B) 0.50
C) 0.82
D) 0.64
76) f(x) = 1 x; [0, 2]
76)
2
A) 0.69
B) 0.75
C) 0.50
D) 0.63
Find the median of the random variable for the given probability density function. 77) f(x) = 5x-6 ; [1, )
A) 3.35
B) 1.08
C) 0.93
8
77) D) 1.15
78) f(x) = 1 ; [4, 9]
78)
5
B) 17
A) 4
C) 31
3
4
D) 13 2
79) f(x) = x - 1 ; [1, 5] 8
79)
8
A) 3.83
B) 3
C) 0.50
D) 2.41
Find the expected value, the variance, and the standard deviation, when they exist, for the probability density function. Give an exact answer for the expected value and round the variance and standard deviation to four decimal places when appropriate. x3 if 0 x 2 12 80) f(x) = 19 80) if x > 2 3x3
A) 37 ; does not exist; does not exist
B) 37 ; 8.2611; does not exist
C) 37 ; 8.2611; 2.8742
D) 37 ; 2.8742; does not exist
10
10
10
10
Solve the problem. 81) The annual rainfall in Maine is a random variable with probability density function defined by 1 1 f(x) = x+ for x in [0, 5]. Find the mean annual rainfall. 15 2
A) 3.000
B) 2.994
C) 3.194
81)
D) 3.360
82) The annual rainfall in Maine is a random variable with probability density function defined by
82)
1 1 f(x) = x+ for x in [0, 5]. Find the variance. 15 2
A) 2.550
B) 1.601
C) 0.815
D) 1.643
83) The annual rainfall in Maine is a random variable with probability density function defined by
83)
1 1 f(x) = x+ for x in [0, 5]. Find the standard deviation. 15 2
A) 1.265
B) 1.270
C) 1.597
D) 0.903
84) The annual rainfall in Maine is a random variable with probability density function defined by
1 1 f(x) = x+ for x in [0, 5]. Find the probability of rainfall within one standard deviation from 15 2 the mean. A) 0.594
B) 0.365
C) 0.612
9
D) 0.622
84)
85) The annual rainfall in Maine is a random variable with probability density function defined by
85)
1 1 f(x) = x+ for x in [0, 5]. Find the probability that the rainfall is greater than the mean. 15 2
A) 0.512
B) 0.553
C) 0.500
D) 0.549
86) The annual rainfall in Maine is a random variable with probability density function defined by
86)
1 1 f(x) = x+ for x in [0, 5]. Find the probability that the rainfall is less than the mean. 15 2
A) 0.447
B) 0.445
C) 0.489
D) 0.500
87) The annual rainfall in Maine is a random variable with probability density function defined by
87)
1 1 f(x) = x+ for x in [0, 5]. Find the probability of rainfall greater than one standard deviation 15 2 above the mean. A) 0.346
B) 0.188
C) 0.159
D) 0.086
88) The annual rainfall in Maine is a random variable with probability density function defined by f(x) =
88)
1 1 x+ for x in [0, 5]. Find the probability of rainfall less than two standard deviations 15 2
below the mean. A) 0.031
B) 0.037
C) 0
D) 0.026
89) The life (in years) of a certain species of whale is a random variable with probability density function defined by f(x) =
A) 6.4 years
89)
1 4 3+ for x in [4, 9]. Find the mean life of this species of whale. 23 x
B) 8.1 years
C) 9.6 years
D) 7.5 years
90) The time between major earthquakes in a particular region of the Mediterranean is a random
90)
1 variable with probability density function f(x) = e-x/1,500 for x in [0, ), where t is 1,500
measured in days. Find the expected value and the standard deviation of this probability density function. A) µ = 1,500 days; = 2,121 days B) µ = 1,500 days; = 1,500 days
C) µ = 1,500 days; = 3,000 days
D) µ = 3,000 days; = 2,121 days
Find the mean and standard deviation of the specified probability density function. 91) f(x) = 1 for [12, 20] 8
A) µ = 16, = 2.31 C) µ = 1.6, = 2.29
91)
B) µ = 15, = 2.31 D) µ = 15.5, = 2.28
92) f(x) = 4 for [2.25, 2.50] A) µ = 2.500, = 0.069 C) µ = 2.375, = 0.072
92) B) µ = 2.375, = 0.144 D) µ = 2.500, = 0.070
10
93) f(x) = 5 for [5.2, 5.4] A) µ = 5.32, = 0.058 C) µ = 5.39, = 0.056
93) B) µ = 5.30, = 0.058 D) µ = 5.30, = 0.056
94) f(x) = 1 for [6, 13]
94)
7
A) µ = 3.50, = 2.02 C) µ = 9.50, = 2.02
B) µ = 3.45, = 2.02 D) µ = 9.50, = 1.73
95) f(x) = 0.04e-0.04x for [0, ) A) µ = 25, = 25 B) µ = 25, = 15 96) f(x) = e-x for [0, ) A) µ = 1, = 1
95) C) µ = 2.5, = 2.5
96) B) µ = 0.5, = 1
C) µ = 10, = 10
97) f(x) = 0.07e-0.07x for [0, ) A) µ = 1.429, = 1.249 C) µ = 14.29, = 15.1
B) µ = 14.29, = 14.29 D) µ = 14.1, = 14.29
98) f(x) = 0.3e-0.3x for [0, ) A) µ = 33.33, = 33.33 C) µ = 3.33, = 3.50
B) µ = 3.75, = 3.75 D) µ = 3.33, = 3.33
99) f(x) = 0.6e-0.6x for [0, ) A) µ = 16.67, = 16.67 C) µ = 1.5, = 1.5
B) µ = 1.71, = 1.67 D) µ = 1.67, = 1.67
100) f(x) = 0.2e-0.2x for [0, ) A) µ = 5, = 5
D) µ = 20, = 20
D) µ = 0.5, = 0.5 97)
98)
99)
100) B) µ = 0.5, = 0.6
C) µ = 6, = 6
D) µ = 0.5, = 0.5
Find the proportion of observations of a standard normal distribution that are between the given z-scores. 101) 0 and 0.35 A) 0.1406 B) 0.1368 C) 0.3632 D) 0.6368
102) 0 and 0.94 A) 0.1736 103) 0 and 0.75 A) 0.4591 104) 0 and 2.5 A) 0.9938
101)
102) B) 0.3289
C) 0.8264
D) 0.3264 103)
B) 0.2734
C) 0.9599
D) 0.4599 104)
B) 0.9940
C) 0.4062
11
D) 0.4938
105) 1.26 and 2.15 A) 0.5880
B) 0.0898
C) 0.0862
D) 0.0881
106) -1.2 and 0.5 A) 0.5784
B) 0.5601
C) 0.1934
D) 0.5764
107) -2.34 and -1.1 A) 0.1263
B) 0.6261
C) 0.1260
D) 0.1249
108) -3.0 and 2.65 A) 0.0037
B) 0.9961
C) 0.4956
D) 0.9946
109) 0.11 and 2.11 A) 0.9388 110) -0.75 and 0.75 A) 0.5528
105)
106)
107)
108)
109) B) 0.4383
C) 0.4423
D) 0.4388 110)
B) 0.7734
C) 0.5467
Use the standard normal curve table to find the z-score for the given condition. 111) 4.01% of the total area is to the left of z. A) 1.70 B) -1.76 C) -1.74
112) 4.01% of the total area is to the right of z. A) 1.74 B) 1.76
D) 0.2734
111) D) -1.75 112)
C) -1.75
D) 1.75
113) 20.05% of the total area is to the right of z. A) 0.82 B) 0.83
C) 0.84
D) -0.84
114) 74.86% of the total area is to the left of z. A) 0.67 B) -0.67
C) 0.68
D) 0.66
115) 25.14% of the total area is to the right of z. A) 0.67 B) -0.68
C) 0.33
D) -0.67
116) 3.01% of the total area is to the right of z. A) 1.89 B) 1.88
C) -1.88
D) -1.89
113)
114)
115)
116)
117) 82.89% of the total area is to the left of z. A) -0.95 B) -0.96
117) C) 0.96
118) 33% of the total area is to the right of z. A) 0.45 B) 0.44
D) 0.95 118)
C) -0.44
12
D) 0.74
119) 30.15% of the total area is to the right of z. A) -0.52 B) 0.52
C) 0.88
D) 0.53
120) 30.15% of the total area is to the left of z. A) -0.88 B) -0.52
C) -0.53
D) 0.52
119)
120)
Solve the problem. 121) The mean clotting time of blood is 7.35 seconds, with a standard deviation of 0.35 seconds. What is the probability that blood clotting time will be less than 7 seconds? Assume the distribution is normal. A) 15% B) 14% C) 16% D) 84%
122) The life span of a certain insect in days is uniformly distributed over the interval [20, 36]. What is the expected life of this insect? A) 29 days B) 28 days
C) 26 days
B) 10.392
C) 4.619
122)
D) 30 days
123) The life span of a certain insect in days is uniformly distributed over the interval [20, 36]. What is the standard deviation? A) 4.33
121)
123)
D) 5.774
124) A company installs 5000 light bulbs. Each bulb has an average life of 500 hours with a standard
124)
125) The life span of a certain insect in days is uniformly distributed over the interval [20, 36]. What is
125)
126) The life span of a certain insect in days is uniformly distributed over the interval [20, 36]. What is
126)
deviation of 100 hours. The life of each bulb is approximated by a normal curve. Find the number of bulbs that can be expected to last less than 500 hours. A) 2500 B) 2400 C) 1000 D) 3000
the probability an insect will live between the mean and one standard deviation above the mean? A) 0.25 B) 0.29 C) 0.28 D) 0.79
the probability the insect will live 24 days or less? A) 0.20 B) 0.25
C) 0.75
D) 0.24
127) A company installs 5000 light bulbs. Each bulb has an average life of 500 hours with a standard
127)
128) A machine fills quart soda bottles with an average of 32.3 oz per bottle, with a standard deviation
128)
deviation of 100 hours. The life of each bulb is approximated by a normal curve. Find the number of bulbs that can be expected to last between 290 hours and 540 hours. A) 1641 B) 3188 C) 1639 D) 3190
of 1.2 oz. What is the probability that a filled bottle will contain less than 32 oz? Assume the distribution is normal. A) 38% B) 41% C) 60% D) 40%
13
129) A machine produces bolts with an average diameter of 0.30 inches and a standard deviation of
129)
130) The number of new mini-vans sold by a particular salesperson during the month of March is
130)
0.01 inches. What is the probability that a bolt will have a diameter greater than 0.32 inches? Assume the distribution is normal. A) 98% B) 3% C) 2% D) 1%
exponentially distributed with a mean of 8. What is the probability that the salesperson will sell between 3 and 7 mini-vans in March? A) 0.183 B) 0.262 C) 0.270 D) 0.299
14
Answer Key Testname: UNTITLED1
1) B 2) B 3) B 4) A 5) A 6) B 7) B 8) B 9) A 10) B 11) B 12) D 13) D 14) D 15) B 16) C 17) C 18) B 19) B 20) B 21) B 22) D 23) B 24) C 25) B 26) C 27) C 28) C 29) B 30) A 31) A 32) B 33) A 34) D 35) C 36) A 37) C 38) D 39) A 40) D 41) A 42) D 15
Answer Key Testname: UNTITLED1
43) B 44) D 45) C 46) D 47) C 48) B 49) D 50) C 51) C 52) A 53) C 54) C 55) D 56) D 57) B 58) B 59) A 60) B 61) C 62) B 63) D 64) A 65) C 66) C 67) D 68) C 69) B 70) D 71) B 72) C 73) D 74) C 75) D 76) D 77) D 78) D 79) A 80) A 81) C 82) B 83) A 84) D 16
Answer Key Testname: UNTITLED1
85) B 86) A 87) B 88) B 89) A 90) B 91) A 92) C 93) B 94) C 95) A 96) A 97) B 98) D 99) D 100) A 101) B 102) D 103) B 104) D 105) D 106) D 107) C 108) D 109) D 110) C 111) D 112) D 113) C 114) A 115) A 116) B 117) D 118) B 119) B 120) B 121) C 122) B 123) C 124) A 125) B 126) B 17
Answer Key Testname: UNTITLED1
127) B 128) D 129) C 130) C
18
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. List the first n terms of the geometric sequence satisfying the given conditions. 1) a1 = 8, r = 5, n = 4
A) 8, 13, 18, 23 C) 8, 40, 200, 5,000
1)
B) 5, 40, 320, 2,560 D) 8, 40, 200, 1,000
2) a1 = 2, r = 2 , n = 3
2)
3
A) 2, 4 , 16
B) 2, 4 , 8
3 27
C) 2, 8 , 16
3 9
D) 4 , 8 , 16
9 27
3 9 27
3) a1 = 4 , r = 4, n = 4
3)
3
A) 4 , 16 , 64 , 256
B) 4, 16 , 16 , 32
C) 2 , 8 , 16 , 32
D) 16 , 64 , 256 , 1024
3
3
3
3
3
3 9 27 81
3
4) a1 = -7, r = 2, n = 4 A) -14, -28, -56, -112 C) -7, -14, -28, -56
9
3
27
3
3
4) B) -7, 14, -28, 56 D) -7, -5, -3, -1
5) a1 = 3, r = 1 , n = 4
5)
2
A) 3, 7 , 4, 9 2
B) 3 , 3 , 3 , 3
2
2 4 8 16
6) a1 = -3, r = -5, n = 4 A) -3, 15, 45, 135
D) 3, 3 , 3 , 3
C) 3, 6, 12, 24
2 4 8
6) B) -3, 15, 75, 375
C) -3, 15, -75, 375
D) -3, 15, -45, 135
7) a1 = 3 , r = -7, n = 4
7)
4
A) 3 , - 21 , 63 , - 1029 4
16 64
B) 3 , - 189 , - 189 , - 7203
256
4
C) 3 , - 21 , - 147 , - 1029 4
4
4
8) a2 = 10, a 3 = 20, n = 4 A) 10, 20, 40, 80
16
64
256
D) 3 , - 21 , 147 , - 1029
4
4
4
4
4
8) B) 5, 10, 20, 80
C) 1, 10, 20, 40 1
D) 5, 10, 20, 40
9) a3 = 150, a 4 = 75, n = 5
9)
A) 600, 300, 150, 75, 75
B) 300, 150, 75, 75 , 75
C) 500, 300, 150, 75, 75
D) 75 , 75 , 75, 150, 300
2
2
2
4
4
2
Find the indicated term of the geometric sequence. 10) a1 = 6, r = 4; Find a 6 .
A) a6 = 1,024
10) C) a6 = 3
B) a6 = 6,144
512
D) a6 = 7,962,624
11) a1 = 3, r = -4; Find a 3. A) a3 = 48
11) B) a3 = -192
12) a1 = 4, r = -4; Find a 4. A) a4 = -4,096
C) a3 = -48
D) a3 = 192 12)
B) a4 = - 204
C) a4 = -256
D) a4 = 1,024
13) a1 = 1,536, r = 1 ; Find a3 .
13)
4
A) a3 = 96
C) a3 = - 1
B) a3 = -96
96
D) a3 = 1
96
14) a1 = 1 , r = 1 ; Find a 5 . 4
14)
2
A) a5 = 1
16
B) a5 = 1
C) a5 = 1
64
512
D) a5 = 1
128
15) a1 = 7, r = - 8 ; Find a 5 .
15)
13
A) a5 = 28,672
B) a5 = 9,834,496
C) a5 = - 229,376
D) a5 = 199,927
28,561
28,561
371,293
4,096
16) a2 = 2, r = 1 ; Find a 4 .
16)
3
A) 18
B) 2
C) 2
27
9
17) a2 = -12, r = -3; Find a 5 . A) -108 B) 324
D) 7
17) C) -324
2
D) 36
Find a n for the given geometric sequence.
18) a1 = 7, r = 4
18)
A) an = 4(7)n
B) an = 7(4)n
C) an = 4(7)n-1
D) an = 7(4)n-1
19) a1 = 2, r = 1
19)
5
B) an = 2 1
A) an = 1 (2)n 5
n-1
C) an = 2 1
5
n
D) an = 1 (2)n-1 5
5
20) a1 = 3, r = -5
20)
A) an = -5(3)n
B) an = 3(-5)n
C) an = 3(-5)n-1
D) an = -5(3)n-1
21) a2 = -14, r = -2
21)
A) an = 7(-2n)
B) an = -14(-2)n - 1
C) an = 7(-2)n - 1
D) an = 7(-2)n
22) a5 = 7 , r = 1 256
22)
4
n-1 A) an = 7 · 1
B) an = 7 · 1
n-1
n C) an = 7 · 1
D) an = 7 · 1
n
256
4
1024
4
4
4
Find the common ratio for the geometric sequence. 23) 3 , 6 , 12 , 24 , 48 , . . . A) r = 1/ 2 B) r = 6
23) C) r = 16
D) r = 2
C) r = 6
D) r = 1 3
24) 6, 18, 54, 162, 486, . . . A) r = 3
24) B) r = 18
25) -8, -24, -72, -216, -648, . . . A) r = -24 B) r = 3 n 26) 2, -6, 18, -54, 162, . . . A) r = -6
25) C) r = -3
D) r = 3
C) r = -3
D) r = (-3)n-1
26) B) r = 2
27) 4, 1, 1 , 1 , 1 , . . .
27)
4 16 64
A) r = 1 4
C) r = 1
B) r = 4
16
3
D) r = 4
28) 3, - 3 , 3 , - 3 , 3 , . . . 2 4
28)
8 16
A) r = - 1
C) r = 1
B) r = 3
4
D) r = - 1
2
2
29) 2 , 8 , 32 , 128 , 512 , ... 3
3
3
3
29)
3
A) r = 1
B) r = 2
3
C) r = 1
3
D) r = 4
4
30) 1, 1 , 1 , 1 , 1 , ...
30)
2 4 8 16
A) r = 2
B) r = 3
D) r = 1
C) r = 1
2
Find a n for the given geometric sequence.
31) 4, 16, 64, 256, 1,024, . . . A) an = 4 + 12(n - 1)
B) an = 12n
C) an = 4 n-1 + 3
D) an = 4 n
32) 5, -10, 20, -40, 80, . . . A) an = 5(-2)n - 1
31)
32) B) an = a 1 - 2n
D) an = 5(-2)n
C) an = 5(-2n)
33) 3, - 3 , 3 , - 3 , 3 , . . . 4 16
33)
64 256
A) an = 3 - 1
n-1
16
C) an = 3 1
B) an = 3 - 1
n
D) an = 3 - 1
n-1
4
n-1
4
4
34) 1 , 1 , 1 , 1 , . . .
34)
3 21 147 1,029
A) an = 1 1
n
C) an = 1 1
n-1
B) an = 1 - 2 (n - 1) 3
3 7
7
D) an = 1 + 1 (n - 1) 3
3 7
7
35) 1 , - 1 , 1 , - 1 , . . . 4
8 16
35)
32
A) an = 1 - 1 4
n-1
B) an = 1
2
4
D) an = 1 - 1
C) an = 1 - 1 (n - 1) 4
n-1
2
2
4
4
-
3 8
n-1
36) - 1 , 1 , - 1 , 1 , . . . 4 28
36)
196 1372
A) an = - 1
n-1
4
C) an = - 1 - 1 4
+
2 7
B) an = - 1 - 1
n-1
D) an = - 1 - 1
n-1
4
n
7
7
Find the sum of the first five terms of the indicated geometric series. 37) 3, 12, 48, 192, . . . A) 1,023 B) 1,025 C) 1,033
38) 1, -2, 4, -8, . . . A) -11
7 4
37) D) 1,019 38)
C) 11
B) -31
D) 31
39) 4 , 8 , 16 , 32 , . . . 3 3
3
39)
3
A) 124 3
B) 41
C) 41
5
D) 124 15
40) 3, 3 , 3 , 3 , . . .
40)
2 4 8
A) 3 8
41) a1 = 8, r = -4 A) 1,640 42) a1 = 2.578, r = 3.245 A) 88.78
C) 93
B) 93
16
D) 3 4
41) B) 408
C) 1,820
D) 14,564 42)
B) 412.03
C) 232.11
D) 126.18
Use the formula for the sum of the first n terms of a geometric sequence to evaluate the sum. 4 43) 4(2)i i=0 A) 124 B) 60 C) 4 D) 15
44)
4
5(-2)i
44)
i=0
A) 55
43)
B) -5
C) -25
5
D) -135
45)
3 i=0
11 1 i 3 2
A) 11
B) 11
8
46)
4
3
A) 31
i=0
5
5 i=0
D) 55 8
46) C) 1
16
D) 31 8
2 (-3)i 5
A) - 182
48)
8
B) 2
16
5
C) 165
1 i 2
2
i=0
47)
45)
32
47) C) 122
B) 364
5
D) - 364 5
3 i 4
A) 3367 16
48) B) 3367
C) 6735
32
32
D) 6735 16
Solve the problem. 49) Eloise contracts to work for 13 days, receiving $0.04 the first day, $0.12 the second day, $0.36 the third day, and so on, with each day's pay triple that of the previous day. How much will she earn on the last day of the contract? A) $21,257.64 B) $5,314.41 C) $63,772.92 D) $163.84
50) Assuming two parents, four grandparents, etc., what is the total number of ancestors a person has going back 8 generations? A) 254 ancestors
B) 256 ancestors
C) 128 ancestors
49)
50)
D) 510 ancestors
51) Ms. Patterson proposes to give her daughter Claire an allowance of $0.15 on the first day of her
51)
52) A pendulum bob swings through an arc 40 inches long on its first swing. Each swing thereafter, it
52)
15-day vacation, $0.30 on the second day, $0.60 on the third day, and so on. Find the allowance Claire would receive on the last day of her vacation. A) $16,384.15 B) $2.25 C) $2,457.60 D) $4,915.20
swings only 87% as far as on the previous swing. What is the length of the arc after 6 swings? Round to two decimal places. A) 19.94 in. B) 15.09 in. C) 17.35 in. D) 174.00 in.
6
53) The population of a small town in 1988 was 11,000 people. Due to decline in industrial growth the
53)
54) While bungee jumping, Gregory falls from a height of 48 feet. He continues to bounce one-half the
54)
population has since been decreasing at a rate of 4% every year. What was the population of this town at the end of 1998? A) 8,112 B) 7,313 C) 5,925 D) 6,586
height from which he last fell. Write out the first five terms of this geometric sequence and find the general term. A) 48, 24, 12, 6, 3; a n = 48 B) 48, 24, 12, 6, 3; a n = 48 n 1 2 2n
C) 24, 12, 6, 3, 3 ; a n = 2
48 n 2 -1
D) 48, 46, 44, 42, 40; a n = 48 - 2(n - 1)
55) A particular substance decays in such a way that it loses half its weight each day. How much of the substance is left after 7 days if it starts out at 16 grams? A) 1 g B) 4 g C) 1 g 4 8
55)
D) 8 g
56) On a gambling trip to Las Vegas, Anthony tripled his bet each time he won. If his first winning bet
56)
57) When students at State University held a food drive for the needy, 1,458 cans of food were
57)
58) A job pays a salary of 32,000 the first year. During the next 8 years, the salary increases by 4% each
58)
was $2 and he won six consecutive bets, find how much he won on the sixth bet. Find the total amount he won on these six bets. A) $1,458; $728 B) $486; $728 C) $1,458; $2,186 D) $162; $242
collected on the first day of the drive, 486 the second day, 162 the third day, and so on. Find the total number of cans collected the first five days. A) 2,178 cans B) 3,690 cans C) 2,160 cans D) 2,184 cans
year. What is the salary for the 9th year? What is the total salary over the 9-year period? (Round to the nearest cent.) A) $45,545.98; $294,880.24 B) $45,545.98; $384,195.43
C) $43,794.21; $338,649.45
D) $43,794.21; $294,855.24
59) As Sunee improves her algebra skills, she takes 0.8 times as long to complete each homework
59)
assignment as she took to complete the preceeding assignment. If it took her 55 minutes to complete her first assignment, find how long it took her to complete the fifth assignment. Find the total time she took to complete her first five homework assignments. Round to the nearest minute. A) 23 min; 185 min B) 18 min; 162 min
C) 23 min; 162 min
D) 18 min; 185 min
Find the amount of the ordinary annuity. Round to the nearest cent. 60) R = $420, i = 0.04, n = 14 (Interest is compounded annually.) A) $1,748.33 B) $7,682.60 C) $6,983.27
7
60) D) $18,182.60
61) R = $4,200, i = 0.045, n = 8 (Interest is compounded annually.) A) $33,680.44 B) $12,701.38 C) $39,396.06
D) $132,729.39
62) R = $82,000, i = 0.065, n = 14 (Interest is compounded annually.) A) $286,052.27 B) $1,598,984.23 C) $1,784,918.20
D) $3,046,456.66
63) R = $13,100, i = 0.045, n = 8 (Interest is compounded annually.) A) $413,989.29 B) $122,878.18 C) $105,050.89
D) $39,616.20
64) R = $11,400, 10% interest compounded semiannually for 7 years A) $201,957.95 B) $201,928.00 C) $451,424.40
D) $223,424.40
65) R = $3,800, 4% interest compounded semiannually for 13 years A) $121,715.14 B) $317,949.44 C) $152,098.06 66) R = $2,300, 7% interest compounded quarterly for 11 years A) $145,692.63 B) $64,634.97 C) $281,970.82 67) R = $5,700, 5% interest compounded quarterly for 8 years A) $222,587.51 B) $54,429.92 C) $678,587.51
61)
62)
63)
64)
65) D) $127,949.44 66) D) $150,542.25 67) D) $214,209.89
Find the periodic payment that will amount to the given sum under the given conditions. Round to the nearest cent. 68) S = $24,000; interest is 8% compounded annually; payments are made at the end of each year for 68) 12 years. A) $1,264.68 B) $1,441.83 C) $1,951.27 D) $2,256.35
69) S = $77,000; interest is 6% compounded semiannually; payments are made at the end of each semiannual period for 8 years. A) $8,659.14 B) $3,820.04
C) $7,779.77
D) $2,999.32
70) S = $47,000; interest is 12% compounded quarterly; payments are made at the end of each quarter for 5 years. A) $1,749.14
B) $2,527.03
C) $1,578.35
69)
70)
D) $2,347.23
Find the present value of the ordinary annuity. Round to the nearest cent. 71) Payments of $1,480 are made annually for 13 years at 6% compounded annually. A) $13,097.26 B) $12,408.02 C) $13,756.60 D) $13,101.97
71)
72) Payments of $5,000 are made annually for 9 years at 7% compounded annually. A) $50,295.44 B) $32,576.16 C) $38,038.43 D) $25,164.77
72)
73) Payments of $730 are made semiannually for 16 years at 4% compounded semiannually. A) $13,047.69 B) $9,911.73 C) $17,131.88 D) $26,222.63
73)
8
74) Payments of $1,800 are made semiannually for 8 years at 5% compounded semiannually. A) $11,633.78 B) $19,507.99 C) $12,906.25 D) $23,499.01
74)
75) Payments of $9,800 are made quarterly for 10 years at 8% compounded quarterly. A) $160,244.04 B) $193,969.19 C) $307,951.34 D) $440,559.38
75)
76) Payments of $24,000 are made quarterly for 15 years at 4% compounded quarterly. A) $1,078,920.92 B) $1,262,089.22 C) $834,261.29 D) $619,384.99
76)
Find the lump sum deposited today that will yield the same total amount as the payments described. Interest is compounded annually. 77) Payments of $8,500 at the end of each year for 10 years at an interest rate of 4%. 77) A) $76,351.97 B) $102,051.91 C) $93,072.63 D) $68,942.61
78) Payments of $11,500 at the end of each year for 15 years at an interest rate of 7%. A) $221,900.33 B) $288,983.75 C) $104,741.01 D) $132,450.23 Find the payment necessary to amortize the loan. Round to the nearest cent. 79) $1,000, 7% compounded annually, 9 annual payments A) $153.49 B) $142.38 C) $167.47
79) D) $45.34
80) $90,000, 6% compounded annually, 9 annual payments A) $3,392.18 B) $13,232.00 C) $14,493.23
D) $12,228.12
81) $31,000, 10% compounded semiannually, 11 semiannual payments A) $3,732.06 B) $3,497.59 C) $4,014.64
D) $978.12
82) $2,400, 14% compounded quarterly, 10 quarterly payments A) $315.47 B) $288.58 C) $49.15 83) $6,600, 10% compounded monthly, 36 monthly payments A) $218.19 B) $208.03 C) $212.96 84) $55,000, 6% compounded monthly, 36 monthly payments A) $149.81 B) $1,631.93 C) $1,673.21
80)
81)
82) D) $266.62 83) D) $31.58 84) D) $1,716.85
Solve the problem. 85) To save for retirement, you decide to deposit $1,250 into an IRA at the end of each year for the next 30 years. If the interest rate is 12% per year compounded annually, find the value of the IRA after 30 years. Round to the nearest dollar. A) $54,021,054 B) $268,228 C) $36,200 D) $301,666
9
78)
85)
86) Looking ahead to retirement, you sign up for automatic savings in a fixed-income 401K plan that
86)
87) Scott deposits $100 each month into a savings account paying annual interest of 5.75%
87)
88) Lynn invests $300 each quarter in a fixed-interest mutual fund paying annual interest of 6%
88)
89) In her will the late Mrs Barbaroni said that each child in her family could have an annuity of
89)
90) In a lottery, a winner is paid $60,000 per year for 20 years. Assume that these payments form an
90)
91) Find the amount of each payment into a sinking fund if $10,000 must be accumulated. Payments
91)
92) Green Thumb Landscaping wants to build a $143,000 greenhouse in 2 years. The company sets up
92)
93) How much should be deposited semiannually into a sinking fund over 5 years to accumulate
93)
pays 6% per year compounded annually. You plan to invest $3,000 at the end of each year for the next 30 years. How much will your account have in it at the end of 30 years? Round to the nearest dollar. A) $235,632 B) $237,175 C) $238,945 D) $238,473
compounded monthly. How much will his account have in it at the end of 7 years? Round to the nearest dollar. A) $10,441 B) $10,158 C) $833 D) $10,312
compounded quarterly. How much will her account have in it at the end of 8 years? Round to the nearest dollar. A) $36,695 B) $12,335 C) $2,969 D) $12,206
$3,600 at the end of each year for 9 years, or the equivalent present value. If money can be deposited at 7% compounded annually, what is the present value? A) $27,387.67 B) $23,454.84 C) $36,212.71 D) $18,118.63
ordinary annuity and that the lottery managers can invest money at 6% compounded annually. Find the lump sum that the management must put away to pay off the winner. A) $902,777.82 B) $1,386,886.32 C) $2,207,135.46 D) $688,195.26
are made at the end of each quarter for 3 years, with interest of 12% compounded quarterly. A) $527.00 B) $592.80 C) $984.30 D) $704.62
a sinking fund with payments made quarterly. Find the payment into this fund if the money earns 12% compounded quarterly. A) $14,075.49 B) $16,081.26 C) $10,075.78 D) $7,536.10
$202,000 if the money earns 8% compounded semiannually? A) $12,126.06 B) $16,824.77 C) $34,432.92
D) $17,620.46
94) How much should be deposited monthly into a sinking fund over 4 years to accumulate $130,000 if the money earns 12% compounded monthly? A) $2,558.40 B) $3,244.80
C) $1,622.40
10
D) $2,123.40
94)
95) Chuck wants to start an IRA that will have $250,000 in it when he retires in 25 years. How much
95)
96) Anita wants to start an IRA that will have $250,000 in it when she retires in 25 years. How much
96)
should he invest semiannually in his IRA to do this if the interest is 6% compounded semiannually? A) $4556.68 B) $2303.29 C) $2216.37 D) $2216.91
should she invest semiannually in her IRA to do this if the interest is 11% compounded semiannually? A) $1075.58 B) $2185.06 C) $1015.36 D) $1015.14
Find the monthly house payment necessary to amortize the loan. Assume that interest is compounded monthly. 97) $193,121 at 8.05% for 30 years 97) A) $1,295.52 B) $1,496.94 C) $1,423.79 D) $43.18
98) $268,471 at 8.12% for 25 years A) $11,708.97 B) $2,093.49
98) C) $1,816.65
99) $353,387 at 7.84% for 25 years A) $15,367.31 B) $2,690.15
D) $2,987.21 99)
C) $3,875.39
D) $2,308.80
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Prepare an amortization schedule for the loan. 100) A construction company pays $70,000 for a truck. It amortizes the loan in semiannual payments for 5 years at 8.5% compounded semiannually. Prepare an amortization schedule showing the first three payments.
101) An insurance firm pays $30,000 for a new copy machine. It amortizes the loan in 8 annual
100)
101)
payments at 7% compounded annually. Prepare an amortization schedule showing the first three payments.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the Taylor polynomial of degree 3 at 0. 102) f(x) = 1 x+5
102)
A) 1 + 1 x + 1 x2 + 1 x3
B) 1 x + 1 x2 + 1 x3 + 1 x4
C) 1 - 1 x + 1 x2 - 1 x3
D) 1 x - 1 x2 + 1 x3 - 1 x4
5 5
25 25
125 125
625
5
625
5
11
25 25
125 125
625 625
103) f(x) = 1
103)
2-x
A) 1 x + 1 x2 + 1 x3 + 1 x4
B) 1 + 1 x + 1 x2 + 1 x3
C) 1 x - 1 x2 + 1 x3 - 1 x4
D) 1 - 1 x + 1 x2 - 1 x3
2
4
2
8
4
16
8
2
16
2
104) f(x) = e-6x A) 1 + 6x + 18x2 + 36x3 C) 1 - 6x + 18x2 - 36x3
4
8
4
16
8
16
104) B) 1 - 36x + 648x2 - 3888x3 D) 1 - 6x + 18x2 - 72x3
105) f(x) = e2x
105)
A) 1 + 2x + 2x2 + 4 x3 9
B) 1 + 2x + 2x2 + 4 x3 3
C) 1 +4x + 8x2 + 16 x3
D) 1 + 2x + 2x2 + 8 x3
3
3
106) f(x) = x + 4
106)
A) 2 + 1 x - 1 x2 + 1 x3 4 64 512
B) 2 - 1 x + 1 x2 - 1 x3 4 64 256
C) 2 + 1 x - 1 x2 + 1 x3
D) 2 - 1 x + 1 x2 - 1 x3
4
107) f(x) =
64
256
4
64
512
3
x-8 A) -2 + 1 x - 1 x2 + 5 x3 12 288 10,368
B) -2 + 1 x + 1 x2 +
5 x3 20,736
C) -2 + 1 x - 1 x2 +
D) -2 + 1 x + 1 x2 +
5 x3 10,368
12
288
107) 12
5 x3 20,736
12
108) f(x) = (36 - x)3/2
288 288
108)
A) 216 + 9x - 3 x2 + 1 x3 48 1728
B) 216 - 9x + 3 x2 + 1 x3 48 1728
C) 216 + 9x - 3 x2 + 1 x3
D) 216 - 9x + 3 x2 + 1 x3
48
3456
48
109) f(x) = ln(1 - 3x)
3456
109)
A) -3x + 9 x2 - 9x3 2
B) -3x - 9x2 - 27x3
C) -3x - 9 x2 - 9x3
D) -3x - 9 x2 - 9 x3
2
2
12
2
Use the appropriate Taylor polynomial of degree 3 at x = 0 to approximate the quantity. Round the answer to four decimal places. 110) ln 0.91 110) A) -0.3023 B) 0.9057 C) -0.0943 D) -0.1473
111) ln 1.285 A) 1.2521
B) 0.1951
C) 0.0471
D) 0.2521
112) e-0.03 A) 0.9691
B) 0.9701
C) 0.9695
D) 0.9704
113) e1.90 A) 0.9050
B) 6.7876
C) 6.0575
D) 6.6871
114) 4.1 A) 1.4505
B) 2.0248
C) 2.0252
D) 1.4858
111)
112)
113)
114)
115) 15.92 A) 3.9900 116)
3
115) B) 3.9911
C) 3.9905
D) 3.9891
7.6
116)
A) 0.9661
B) 1.9661
C) 1.9322
D) 1.9660
Solve the problem. 117) If P dollars are invested at an annual interest rate of r compounded n times a year, then the accumulated amount after t years is given by r nt A=P 1+ . n Find the Taylor polynomial of degree 2 at r = 0 for r nt f(r) = P 1 + . n
A) P + 1 Ptr + nt - 1 Ptr2
B) 1 + Ptr - nt - 1 Ptr2
C) P + 1 Ptr + nt - 1 Ptr2
D) P + Ptr + nt - 1 Ptr2
2 2
3n
2n
6n
2n
13
117)
118) If P dollars are invested at an annual interest rate of r compounded n times a year, then the
118)
accumulated amount after t years is given by r nt A=P 1+ . n
If $6,000 is invested for 9 years at a rate of 7% compounded monthly, then what is the difference, in thousands of dollars, between the actual accumulated amount and the amount given by the Taylor polynomial of degree 2 at r = 0 for r nt f(r) = P 1 + ? n
A) 0.29 thousand dollars C) 0.11 thousand dollars
B) 0.68 thousand dollars D) 0.89 thousand dollars
119) The strain S on a small experimental I-beam under a central point load of x pounds is given by S(x) = ln
x2 9
119)
+ 1 (see figure below).
Use a Taylor polynomial of degree 2 at x = 0 to estimate the strain on the beam when loaded with a 6 lb point load. Round to the nearest hundredth. A) 2.49 B) 1.61 C) -4.00 D) 4.00
120) The "normal curve" plays a central role in statistics. The curve is given by f(x) =
1 2 2 e-x /2 , where 2
and
are constants. Find a Taylor polynomial of degree 2 that
approximates the normal curve for x-values near zero. 2 A) 1 1 - x B) 2 2 2
C)
1 2
120)
- x2
D)
1 2
+ x2
1 2
1-
x2 4
Identify which geometric series converge. Give the sum of a convergent series. 121) 24 + 32 + 128 + 512 + . . . 3 27 243
121)
A) Converges to 216
B) Converges to 9
C) Converges to 96
D) Diverges
5
5
5
14
122) 1 + 0.3 + 0.09 + 0.027 + . . .
122)
A) Converges to 11 7
B) Converges to 10 7
C) Converges to 2
D) Diverges
123) 12 + 6 + 3 + 3 + . . .
123)
2
A) Converges to 30 C) Converges to 6
B) Converges to 24 D) Diverges
124) 2 + 8 + 32 + 128 + . . . A) Converges to 1024 C) Converges to 512
124) B) Converges to 1536 D) Diverges
125) 125 + 25 + 5 + 1 + . . .
125)
A) Converges to 100
B) Converges to 158
C) Converges to 625
D) Diverges
4
126) 2 + 2 + 2 + . . . 3
15
126)
75
A) Converges to 1
B) Converges to 5
C) Converges to 7
D) Diverges
6
6
127) 42 - 112 + 896 - 7168 + . . . 3
127)
9
A) Converges to 336
B) Converges to - 336
C) Converges to - 126
D) Diverges
11
11
11
128) 1 + 1 + 1.1
1
(1.1)2
+...
128)
A) Converges to 10 C) Converges to 12
B) Converges to 9 D) Converges to 11
15
129) e + 1 + 1 + 1 + . . . e
129)
e2
A) Converges to
e (e - 1)
B) Converges to
C) Converges to
e2 (e + 1)
D) Diverges
e2 (e - 1)
The nth term of a sequence is given. Calculate the fifth partial sum. 130) an = 1 7n
A) 137 315
B) 137
130)
C) 137
378
420
D) 137 210
131) an = 1
131)
n+2
A) 153 140
B) 451
C) 1,369
420
1,260
D) 1,361 1,260
132) an = 1
132)
n-7
A) 1357 720
133) an =
C) - 689
20
144
D) 1879 360
1 5n + 1
A) 132,305 576,576
134) an =
B) - 29
133) B) 6,505
C) 27,295
16,016
144,144
D) 5,365
18,018
1 (n + 2)(n + 5)
A) 161
1080
135) an = (-1)
134) B) 161
C) 191
1180
1180
D) 191
1080
n
135)
9n - 1
A) - 67,247
680,680
B) - 877
C) - 104,963
29,120
2,722,720
D) - 229,609
5,445,440
136) an = 5n + 1
136)
5n + 2
A) 654,905 141,372
B) 2,277,215
C) 1,652,215
424,116
424,116
16
D) 1,339,715 424,116
137) an = (-1)n+1
1 n(6n + 1)
A) 2,162,893
16,079,700
137) B) 1,644,493
C) 1,903,693
16,079,700
16,079,700
D) 15,223
176,700
n 2n 138) an = (-1) + (-1)
138)
(6n + 1)(6n + 2)
A) 32
2275
139) an = (-1)
B) 2
C) 2
325
175
D) 4
455
n(n+1)/2
139)
n+4
A) - 529
2,520
B) - 73
C) - 49
280
216
D) - 1,843 7,560
Solve the problem. 140) The repeating decimal 0.44444. . . can be expressed as infinite geometric series 1 1 2 1 3 + 0.4 + 0.4 +.... 0.4 + 0.4 10 10 10
140)
By finding the sum of the series, determine the rational number whose decimal expansion is 0.44444. . . . A) 4 B) 4 C) 4 D) 4 13 11 17 9
141) The repeating decimal 0.36363636. . . can be expressed as infinite geometric series
141)
1 1 2 1 3 + 0.36 + 0.36 +.... 0.36 + 0.36 100 100 100
By finding the sum of the series, determine the rational number whose decimal expansion is 0.36363636. . . . A) 4 B) 4 C) 4 D) 4 11 9 17 13
142) A ball is dropped from a height of 14 meters and returns to about 6/7 of its previous height on each bounce. About how far will the ball travel before it comes to rest? A) 98 m B) 266 m C) 350 m
D) 182 m
143) A pendulum bob swings through an arc 40 centimeters long on its first swing. For each swing
thereafter, it swings only 89% as far as on the previous swing. How far will it swing altogether before coming to a complete stop? A) 4000 cm B) 445 cm C) 445 cm D) 4000 cm 11 7 3 189
17
142)
143)
144) The recycling process for aluminum cans is not perfectly efficient, and only a portion of the
144)
145) A company makes a very durable product. The company sells 20,000 products in the first year,
145)
material of a particular can ends up in the next generation of cans. The "effective mass" of a can is its initial mass plus the cumulative mass of aluminum that it passes on to subsequent recycling generations. If the recycling process is 80% efficient, find the effective mass of a 14.2 gram can. Round to the nearest gram. A) 73 g B) 5 g C) 71 g D) 72 g
but will have diminishing sales due to the product's durability, so that each year it can expect to sell only seventy-five percent of the quantity it will have sold the year before. How many of the product can the company expect to eventually sell? A) 26,667 products B) 40,000 products
C) 80,000 products
D) 35,000 products
146) An object is rolling with a driving force that suddenly ceases. The object then rolls 10 meters in the
146)
first second, and in each subsequent interval of time it rolls 80% of the distance it had rolled the second before. This slowing is due to friction. How far will the object eventually roll? A) 50 m B) 20 m
C) 12.2 m
D) It will roll an infinite distance.
147) Acetone is a solvent frequently used to clean lubricants from machine components. Once used, the
147)
dirty solvent need not be discarded. It may be distilled to recover pure acetone from the soiled mixture. During a distillation cycle, 55% of the acetone can be recovered. Calculate the effective volume of 1 liter of acetone. [The effective volume is the original 1 liter plus the accumulated amount recovered from an infinite number of distillation cycles]. A) 1.818 L B) 2.222 L
C) 1.550 L
D) An infinite number of liters
148) After being struck with a hammer, a gong vibrates 54 vibrations in the first second and in each second thereafter makes
148)
6 as many vibrations as in the previous second. Find how many 7
vibrations the gong makes before it stops vibrating. A) 388 vibrations B) 58 vibrations
C) 63 vibrations
Find the Taylor series for the given function. Give the interval of convergence. 149) f(x) = 2 1 + 7x
A) 2x - 14x2 + . . . + (-1)n 2 · 7 nxn+1 + . . . ; - 1 1 2, 2
B) 2 - 14x + . . . + (-1)n 2 · 7 nxn + . . . ; - 1 1 7, 7
C) 2x + 14x2 + . . . + (-1)n 2 · 7n xn+1 + . . . ; - 1 1 2, 2
D) 2 + 14x + . . . + (-1)n 2 · 7 n xn + . . . ; - 1 1 7, 7
18
D) 378 vibrations
149)
150) f(x) = 3
150)
4-x
A) 3 + 3 x + 3 x2 + . . . + 4
16
3 xn + . . . ; (-3, 3) n+1 4
64
B) 3 + 3 x + 3 x2 + . . . + 3 xn + . . . ; (-3, 3) 16
64
4n
C) 3 + 3 x + 3 x2 + . . . + 4
16
64
3 xn + . . . ; (-4, 4) n+1 4
D) 3 x + 3 x2 + 3 x3 + . . . + 3 xn + . . . ; (-4, 4) 4
151) f(x) = 3x
16
64
4n
2
151)
5-x
A) 3 x2 + 3 x3 + 3 x4 + . . . + 3 xn+1 + . . . ; (-3, 3) 5
25
125
5n
B) 3 x + 3 x2 + 3 x3 + . . . + 3 xn + . . . ; (-5, 5) 5
25
125
5n
C) 3 x + 3 x2 + 3 x3 + . . . + 3 xn + . . . ; (-3, 3) 5
25
125
5n
D) 3 x2 + 3 x3 + 3 x4 + . . . + 5
25
125
152) f(x) = ln(1 + 4x)
3 x n+2 + . . . ; (-5, 5) n+1 5
152)
2
3
4
n n
2!
3!
4!
(n + 1)!
A) -4x - 4 x2 + 4 x3 - 4 x4 + . . . + (-1) 4 xn + . . . ; (-4, 4) 2
3
4
n n+1
2
3
4
n+1
2
3
4
n n+1
2!
3!
4!
2
3
4
n n+1
2
3
4
n+1
B) 4x - 4 x2 + 4 x3 - 4 x4 + . . . + (-1) 4 C) 4x - 4 x2 + 4 x3 - 4 x4 + . . . + (-1) 4
(n + 1)!
D) 4x - 4 x2 + 4 x3 - 4 x4 + . . . + (-1) 4 153) f(x) = e2x2 A) 1 + 2x +
xn+1 + . . . ; -
1 1 , 4 4
xn+1 + . . . ; -
1 1 , 4 4
xn+1 + . . . ; (-4, 4)
153) 22 2!
x2 +
23 3!
x3 + . . . +
2n n!
xn + . . . ; ( , )
2
3
n n
2!
3!
n!
2
3
n
2!
3!
n!
2
3
n
2!
3!
n!
B) 1 - 2x2 + 2 x4 - 2 x6 + . . . + (-1) 2 x2n + . . . ; ( , ) C) 1 + 2x2 + 2 x4 + 2 x6 + . . . + 2 x2n + . . . ; (-2, 2) D) 1 + 2x2 + 2 x4 + 2 x6 + . . . + 2 x2n + . . . ; ( , )
19
154) f(x) =
8
154)
1 + x2
A) 8 - 8x2 + 8x4 - 8x6 + . . . + (-1)n 8x2n + . . . ; ( , ) n
B) 8 - 8 x2 + 8 x4 - 8 x6 + . . . + (-1) 8 x2n + . . . ; (-1, 1) 2!
4!
6!
(2n)!
C) 8 - 8x + 8x2 - 8x3 + . . . + (-1)n 8xn + . . . ; (-1, 1) D) 8 - 8x2 + 8x4 - 8x6 + . . . + (-1)n 8x2n + . . . ; (-1, 1) 155) f(x) = x6 e-x
155)
A) x6 - x7 + 1 x8 - 1 x9 + . . . + 2
6
(-1)n n!
x6+n + . . . ; -
1 1 , 6 6
n
B) 1 + x6 - x7 + 1 x8 - 1 x9 + . . . + (-1) x6+n + . . . ; (- , ) 2
6
n!
n
C) 1 - x6 + x7 - 1 x8 + 1 x9 - . . . + (-1) x6+n + . . . ; - 1 , 1 2
6
n!
6 6
n
D) x6 - x7 + 1 x8 - 1 x9 + . . . + (-1) x6+n + . . . ; (- , ) 2
6
n!
156) f(x) = ln (1 - 7x2 )
156)
A) -7x2 - 49 x4 - . . . - 7 2
n+1
n+1
x2n+2 - . . . ; -
B) -7x2 + 49 x4 - . . . + (-1)
n+1 7 n+1
C) -7x2 + 49 x4 - . . . + (-1)
n+1 7 n+1
2
2
D) -7x2 - 49 x4 - . . . - 7 2
n+1
n+1
n+1
n+1
1 1 , 7 7
x2n+2 - . . . ; -
1 1 , 7 7
x2n+2 - . . . ; -
1 1 , 7 7
x2n+2 - . . . ; -
1 1 , 7 7
157) f(x) = 4x
157)
1 + 3x
A) 4x - 4 · 3x2 + . . . + (-1)n4 · 3nxn+1 + . . . ; (-3, 3) B) 4x + 4 · 3x2 + . . . + 4 · 3 nxn+1 + . . . ; - 1 , 1 3 3
C) 4x + 4 · 3x2 + . . . + 4 · 3 nxn+1 + . . . ; (-3, 3) D) 4x - 4 · 3x2 + . . . + (-1)n4 · 3nxn+1 + . . . ; - 1 , 1 3 3
20
5x + e-5x
158) f(x) = e
158)
2
2
4
6
2n
2!
4!
6!
(2n)!
A) 1 + 5 x2 + 5 x4 + 5 x6 + . . . + 5 2
3
n
2!
3!
n!
x2n + . . . ; ( , )
B) 1 + 5x + 5 x2 + 5 x3 + . . . + 5 xn + . . . ; ( , ) 2
4
6
2n
2!
4!
6!
(2n)!
C) 1 + 5 x2 + 5 x4 + 5 x6 + . . . + 5
x2n + . . . ; (-5, 5)
D) 1 + 52 x2 + 5 4x4 + 5 6 x6 + . . . + 52nx2n + . . . ; ( , ) Solve the problem. 159) Use the first five terms of the Taylor series to approximate the area of the region bounded by e-x f(x) = , x = 1, x = 2.7, and the x-axis. x
A) -0.770309
C) 0.370975
B) -0.622277
159)
D) -0.756304
160) Use the first five terms of the Taylor series to approximate the area of the region bounded by the
160)
standard normal curve, x = 0, x = 2.5, and the x-axis. The equation of the standard normal curve 1 -x2 /2 is f(x) = e .
A) 1.624507
B) 0.656337
C) 0.648084
D) 1.645192
161) Use the first five terms of the Taylor series to approximate the area of the region bounded by f(x) = ln(1 + 7x), x = 0, x = 1, and the x-axis. A) 464.1 B) 23.9167
C) 0.4
D) 720.5333
162) Use the first five terms of the Taylor series to approximate the area of the region bounded by f(x) = xe-7x , x = 0, x = 1, and the x-axis. A) 9.532 B) 37.0653
C) -11.9597
161)
162)
D) 0.4
163) In a certain country, the infant mortality rate is 8.5 per 1000 live births. Assuming that this is the
163)
164) At a certain university, 1 in 9 students is an engineering major. Suppose we randomly select
164)
expected value for a Poisson distribution, find the probability that in a random sample of 1000 live births, there were fewer than 5 cases of infant mortality. A) 0.059491 B) 0.063741 C) 0.074364 D) 0.089237
students until we find one who is an engineering major. What is the probability that we will find an engineering major within the first four students we select? (Assume a geometric distribution.) A) 0.176802 B) 0.574608 C) 0.375705 D) 0.773510
Use Newton's method to find a solution for the equation in the given interval. Round your answer to the nearest hundredth. 165) 3x2 + 8x - 7 = 0; [0, 1] 165)
A) 0.68
B) 0.67
C) 0.66 21
D) 0.69
166) 2x3 - x2 + 5x + 6 = 0; [-1, 0] A) -0.82 B) -0.81
C) -0.83
D) -0.84
167) x3 - x - 1 = 0; [1, 2] A) 1.33
B) 1.31
C) 1.34
D) 1.32
168) 2x3 - 3x2 - 7x + 1 = 0; [-2, -1] A) -1.33 B) -1.34
C) -1.36
D) -1.35
166)
167)
168)
169) 2x4 - 3x2 - 7x + 1 = 0; [1, 2] A) 1.80 B) 1.82
169) C) 1.79
D) 1.81
170) 2x4 - 3x2 - 7x + 1 = 0; [0, 1] A) 0.13 B) 0.15
C) 0.14
D) 0.12
171) 2x4 - 5x3 - 9x2 + 3x - 5 = 0; [3, 4] A) 3.64 B) 3.68
C) 3.67
D) 3.65
172) 2x5 - 3x2 - 7x + 1 = 0; [-2, -1] A) -1.21 B) -1.22
C) -1.20
D) -1.19
173) ex + 3x - 5 = 0; [0, 1] A) 0.85
170)
171)
172)
173) B) 0.84
C) 0.86
174) 5x1/3 - 3x2 + 9 = 0; [-2, -1] A) -1.14 B) -1.15 175) 3 ln x - 2x + 4 = 0; [4, 5] A) 4.13
D) 0.87 174)
C) -1.13
D) -1.12 175)
B) 4.15
C) 4.09
D) 4.11
176) x3 e-x + x - 4 = 0; [2, 3] A) 2.76
B) 2.54
C) 2.66
D) 2.68
177) x1/3 - x + 8; [10, 11] A) 10.18
B) 10.21
C) 10.19
D) 10.17
Use Newton's method to find the given root to the nearest thousandth. 178) 8 A) 2.828 B) 8.000 C) 2.833
D) 2.825
176)
177)
22
178)
179) 62 A) 7.871
B) 62.000
C) 7.874
D) 7.879
180) 176 A) 13.262
B) 13.269
C) 13.266
D) 13.268
181)
3
3
3
181) B) 2.565
C) 2.571
D) 2.568
B) 4.021
C) 4.018
D) 4.024
65
A) 4.027 183)
180)
17
A) 2.574 182)
179)
182)
185
A) 5.696
183) B) 5.701
C) 5.698
D) 5.694
Solve the problem.
184) Use Newton's method to find the critical point of the function f(x) = x3 - 6x2 + 11x - 5 that
184)
185) Use Newton's method to find the critical point of the function f(x) = x3 - 6x2 + 11x - 5 that
185)
186) Use Newton's method to find the critical point of the function f(x) = x4 - 4x3 + 2x2 - 6 that
186)
187) For a particular product, the revenue and cost functions are R(x) = 361 - x2 and C(x) = 4x + 7.
187)
corresponds to a relative maximum. Round your answer to the nearest hundredth. A) 1.43 B) 1.44 C) 1.45 D) 1.42
corresponds to a relative minimum. Round your answer to the nearest hundredth. A) 2.58 B) 2.61 C) 2.57 D) 2.59
corresponds to a relative minimum within the interval [2, 4]. Round your answer to the nearest hundredth. A) 2.66 B) 3.53 C) 3.57 D) 2.62
Approximate the break-even point to the nearest hundredth. A) 2.91 B) 2.85 C) 2.94
D) 2.89
188) Suppose that P dollars are loaned, with the money to be repaid in n monthly payments of M
dollars each. Then the true annual interest rate is found by solving the equation 1 - (1 + i)-n P = 0 for i, the monthly interest rate, and then multiplying i by 12 to get the true i M annual rate. This equation can best be solved by Newton's method, letting 1 - (1 + i)-n P f(i) = . Suppose that P = $5000, n = 24, and M = $243. Let the initial guess for i i M be i1 = 0.01. Use Newton's method to find i2 . Give your answer to five decimal places.
A) 0.01244
B) 0.01288
C) 0.01264
23
D) 0.01162
188)
189) The volume V (in liters) of 1 mole of a gas is related to its temperature T (in Kelvin) and pressure
189)
a P (in atmospheres) by van der Waals' equation P + (V - b) = RT, where the constant R is V2
0.08207. For krypton, a = 2.318 and b = 0.03978. Use Newton's method to find the volume V of 1 mole of krypton if P = 2.0 atmospheres and T = 330 K. Round your answer to two decimal places. A) 17.61 L B) 12.53 L C) 15.49 L D) 13.50 L
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
190) Marcus Tool and Die Company produces a specialized milling tool designed specifically
190)
for machining ceramic components. Each milling tool sells for $4, so the company's revenue in dollars for x units sold is R(x) = 4x. The company's cost in dollars to produce x tools can be modeled as C(x) = 304 + 30x5/8. Use Newton's method to find the break-even point for the company (that is, find x such that C(x) = R(x)). Use x = 370 as your initial guess and show all of your work to find x3 as your approximation.
191) A team of engineers is testing an experimental high-voltage fuel cell with a potential
191)
application as an emergency back-up power supply in cell phone transmission towers. Unfortunately, the voltage of the prototype cell drops with time according to the equation V(t) = -0.0306t3 + 0.373t2 - 2.16t + 15.1, where V is in volts and t is the time of operation in hours. The cell provides useful power as long as the voltage remains above 6.6 volts. Use Newton's method to find the useful working time of the cell to the nearest tenth of an hour (that is, solve V(t) = 6.6 volts). Use t = 7 hours as your initial guess and show all of your work to find x 3 as your approximation.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use l'Hospital's rule, if applicable, to find the limit. 2 192) lim x - 49 x 7 x-7
A) 7
192)
B) 14
C) -14
D) -7
2
193) lim x - 5x + 4
193)
x-4
x 4
A) 3 3
B) 8
C) 13
D) -1
2
194) lim x - 9x + 8 x 1
A) 12
194)
x-1
B) 18
C) 21
24
D) -15
8
4
195) lim 9x - 8x + 3x
2
195)
x 0 9x3 + 6x2 - 9x
A) - 1
B) 9
9
C) 0
D) Does not exist
x2 - 4x + 4 x 2 x3 - 11x2 + 32x - 28
196) lim
A) 1
196) B) - 1
9
C) 0
5
D) - 1
10
x
197) lim e - 1 x 0
197)
x7
A) 1
B) 1
5040
C) 0
7
D) Does not exist
xex x 0 e6x - 1
198) lim
198) B) 1
A) 0
C) 1
6
D) Does not exist
2x 199) lim e - 1
199)
x 0 3x2 + 7x
B) 2
A) 0
C) 2
3
7
D) Does not exist
ex
200) lim
200)
x 0 15x3 - 7x2 + 3x
A) 1
15
B) 0
C) 1
D) Does not exist
x2 + 6x + 5 x-7
201) lim
x 7
A) 1
201) B) - 1
C) 0
7
D) Does not exist
202) lim
x-8 x 64 x - 64
A) 1
16
202) C) 1
B) 0
8
25
D) Does not exist
203) lim ln(x - 7) x 8
203)
x-8
A) 0
C) 1
B) 1
8
D) Does not exist
6+x-3 x-3
204) lim
x 3
204)
A) 1
B) 1
18
C) 1
6
3
D) Does not exist
x2 + 13 - 7 x2 - 36
205) lim
x 6
A) 1
205) B) 1
12
C) 1
7
14
D) Does not exist
3
x-4 x x 64 - 64
206) lim
206)
A) 1
B) 1
16
207) lim
10 - x x
x 0
48
D) 16 3
10 + x
A) 2 10
207) 10 10
B) -
C)
10 10
D) Does not exist
9x + e-9x - 2
208) lim e x 0
C) 1
12
208)
x
A) 0
B) 18
C) 9
D) Does not exist
209) lim (6 + x)ln(x + 1) B) 1
A) 6
210) lim
x 1
209)
ex - 1
x 0
C) 0
6
D) Does not exist
x - x4 ln x
A) -1
210) B) - 7
C) 0
2
26
D) -4
1+
211) lim
x - (1 + x)1/2 2
211)
x2
x 0
A) 1
B) 1
8 1 + 5x x
212) lim
x 0
C) 1
16
12
D) Does not exist
1 - 5x
A) 1
212) B) 1
10
D) 1
C) 5
5
213) lim 1 - 1 x 0 x
213)
x4
B) 1
A) 0
C) -1
4
D) Does not exist
214) lim 1 + 1 x 0 x
4
214) x
B) 1
A) -1
C) 0
4
D) Does not exist
215) lim x3 (ln x)3
215)
x 0+
A) 3
C) - 1
B) 0
3
D) Does not exist
x ln(ex - 1)
216) lim
x 0+
216) C) 1
B) 0
A) -1
2
D)
2 217) lim ln (2x + 12x)
217)
ln x
x 0+
A) 1
B) 2
C)
D) 0
x
218) lim e
218)
x5
x
A)
B) 1
C) 0
5
27
D) 1
120
219) lim (ln x)
4
219)
x
x
B) 1
A)
C) 4
4
D) 0
2
220) lim 4x - 19
220)
x ln x
x
A) 0
B) 1
C) 4
D)
221) lim ln(ln x)
221)
x
x
A) 1
B) -1
2
222) lim x
C)
D) 0
6x ln(ex + 1)
A) 1
222) B) 6
6
223) lim ln(10e
C) 0
D)
x - 1)
223)
14 x
x
A) 1
14
B) 0
C)
D) 5 7
2 224) lim 6x - 3x + 3
224)
11x 2 + 3x + 3
x
A)
B) 0
C) 1
28
D) 6
11
Answer Key Testname: UNTITLED2
1) D 2) B 3) A 4) C 5) D 6) C 7) D 8) D 9) A 10) B 11) A 12) C 13) A 14) B 15) A 16) C 17) B 18) D 19) B 20) C 21) C 22) B 23) D 24) A 25) D 26) C 27) A 28) D 29) D 30) D 31) D 32) A 33) D 34) C 35) A 36) B 37) A 38) C 39) A 40) C 41) A 42) B 29
Answer Key Testname: UNTITLED2
43) A 44) A 45) D 46) D 47) D 48) B 49) A 50) D 51) C 52) A 53) B 54) A 55) C 56) B 57) A 58) C 59) A 60) B 61) C 62) C 63) B 64) D 65) D 66) D 67) A 68) A 69) B 70) A 71) D 72) B 73) C 74) D 75) D 76) A 77) D 78) C 79) A 80) B 81) A 82) B 83) C 84) C 30
Answer Key Testname: UNTITLED2
85) D 86) B 87) D 88) D 89) B 90) D 91) D 92) B 93) B 94) D 95) C 96) C 97) C 98) B 99) B 100) Payment Amount of Interest Number Payment for Period 0 1 $8738.11 $2975.00 2 $8738.11 $2730.07 3 $8738.11 $2474.73
Portion to Principal $5763.11 $6008.04 $6263.38
Principal at End of Period $70,000 $64,236.89 $58,228.85 $51,965.47
Payment Amount of Interest Number Payment for Period 0 1 $5024.03 $2100.00 2 $5024.03 $1895.32 3 $5024.03 $1676.31
Portion to Principal $2924.03 $3128.71 $3347.72
Principal at End of Period $30,000 $27,075.97 $23,947.26 $20,599.54
101)
102) C 103) B 104) C 105) B 106) A 107) B 108) D 109) C 110) C 111) D 112) D 113) D 114) B 115) A 116) B 31
Answer Key Testname: UNTITLED2
117) D 118) A 119) D 120) A 121) A 122) B 123) B 124) D 125) C 126) B 127) D 128) D 129) B 130) C 131) A 132) B 133) B 134) A 135) A 136) A 137) C 138) A 139) A 140) D 141) A 142) D 143) A 144) C 145) C 146) A 147) B 148) D 149) B 150) C 151) D 152) B 153) D 154) D 155) D 156) D 157) D 158) A 32
Answer Key Testname: UNTITLED2
159) C 160) C 161) A 162) A 163) C 164) C 165) D 166) C 167) D 168) C 169) D 170) C 171) C 172) C 173) D 174) C 175) A 176) D 177) D 178) A 179) C 180) C 181) C 182) B 183) C 184) D 185) A 186) D 187) C 188) C 189) D
33
Answer Key Testname: UNTITLED2
190) Find the root of f(x) = C(x) - R(x) = 304 + 30x5/8 - 4x. f (x) =
75 -3/8 -4 x 4
x1 = 370 x2 = 370 -
f(370) 304 + 30 · 3705/8 - 4(370) = 370 = 386.59 f (370) 75 · 370-3/8 - 4 4
x3 = 386.59 -
f(386.59) 304 + 30· 386.595/8 - 4(386.59) = 386.59 = 386.45 f (386.59) 75 -3/8 · 386.59 -4 4
The break-even point is x = 386.45 tools. 191) Find the root of f(x) = -0.0306t3 + 0.373t2 - 2.16t + 15.1 - 6.6.
f (x) = -0.0918t2 + 0.746t - 2.16 x1 = 7 x2 = 7 -
f(7) -0.0306(7)3 + 0.373(7)2 - 2.16(7) + 15.1 - 6.6 =7= 7.81 f (7) -0.0918(7)2 + 0.746(7) - 2.16
x3 = 7.81 -
-0.0306(7.81)3 + 0.373(7.81)2 - 2.16(7.81) + 15.1 - 6.6 f(7.81) = 7.81 = 7.71 f (7.81) -0.0918(7.81)2 + 0.746(7.81) - 2.16
The useful working time is t = 7.71 hours.
192) B 193) A 194) D 195) C 196) B 197) D 198) B 199) C 200) D 201) D 202) A 203) B 204) B 205) C 206) C 207) B 208) A 34
Answer Key Testname: UNTITLED2
209) A 210) B 211) A 212) C 213) D 214) D 215) B 216) B 217) A 218) A 219) D 220) D 221) D 222) B 223) A 224) D
35
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the degree measure to radians. Leave the answer as a multiple of . 1) 45°
A)
B)
6
C)
3
1) D)
4
5
2) -36°
2)
A) -
7
3) 330°
4) 450°
5) 162°
D) -
6
B) 11 12
C) 11 6
D) 11 5
B) - 5 2
C) - 5 4
D) 5 2
B) 10 11
C) 8 9
D) 9 10
5)
A) 4 5
6) 288°
A)
5
4)
A) 5
7) 670°
C) -
4
3)
A) 11 3
A)
B) -
6) 5
B) 16
C)
5
5
D)
5
7) 67 18
8) 700°
A) 70 9
B)
67 9
C)
31 18
D)
67 36
8) B) 35 9
C) 17 9
D) 35 18
Convert the radian measure to degrees.
9)
9)
4
A) 90°
B) 22.5°
C) 32.5°
1
D) 45°
10) -
10)
3
A) -120°
B) -30°
C) -60°
D) -40°
11) 7
11)
8
A) 157.5°
B) 88.75°
C) 315°
D) 78.75°
12) - 5
12)
2
A) -225°
B) -450°
Find the indicated trigonometric function for , given that defined by the given point. 13) (18, 24); find sin A) 4 B) 4 3 5
14) (18, 24); find cos A) 4 3
15) (-15, 36); find sin A) - 12 13
16) (-20, -48); find cos A) - 5
13
17) (-15, 36); find sec A) - 13 12
18) (9, 12); find csc A) 4 3
19) (9, -12); find csc A) 4 5
C) -235°
D) -900°
is an angle in standard position with the terminal side
13) C) 3 4
D) 3 5
C) 3 5
D) 4 5
14) B) 3 4
15) B) 12
C) - 5
13
13
D) 5
13
16) B) - 13
C) - 12
D) 5
B) - 13 5
C) - 5 13
D) 13 5
B) 5 4
C) 3 4
D) 5 3
B) 5 4
C) - 5 3
D) - 5 4
5
13
13
17)
18)
19)
2
20) (-4, -3); find tan A) - 4 5
21) (-8, 4); find cot A) - 8 9
If
20) B) 3 4
C) 4 3
D) - 3 5 21)
B) 4
C) - 2
9
D) - 1 2
is an angle in the indicated quadrant, determine whether the given function is positive or negative. 22) II, sec A) Positive B) Negative
23) III, cot A) Negative
23) B) Positive
24) IV, cot A) Positive
24) B) Negative
25) II, sin A) Negative
25) B) Positive
26) III, cos A) Negative
26) B) Positive
27) IV, sin A) Positive
B) Negative
28) II, tan A) Negative
B) Positive
29) III, csc A) Positive
B) Negative
30) IV, sec A) Positive
B) Negative
27)
28)
29)
30)
31) I, csc A) Positive
31) B) Negative
Give the exact value. 32) sec 30°
A) 2
22)
32) B) 2
C)
3
3 2
D) 2 3 3
33) cos 60°
33)
A) 2 3 3
B) 1 2
C)
2 2
C)
2 2
D)
3 2
34) cos 45° A)
34) 3 2
B) 1 2
D) 2
35) cos 150°
35) 2 2
A) -
B)
3 2
C)
2 2
D) -
3 2
36) tan 120° A) - 3
36) B) 3
C)
3 3
D) -
3 3
37) cot 300°
37)
A) -1
B)
3 3
C) - 3
D) -
3 3
38) sec 120° A) 2
38) B) -2
C) 2 3
D) - 2 3
C) - 2
D) - 2 3 3
3
3
39) sec 210° A) 2 3 3
39) B) 2
40) csc 300°
40)
A) 2
C) 2 3
B) -2
3
D) - 2 3 3
41) csc 210°
41)
A) -2
C) - 2 3
B) 2
3
D) 2 3 3
Find the exact value of the following expression without using a calculator.
42) sec
42)
6
A) 2 3 3
B) 2
C) 2
4
D)
3 2
43) tan
43)
3
A) 3
44) cos
3 2
B)
C)
3 3
D) 2
44)
4
A)
3 2
B) 2
C)
2 2
D) 1 2
45) sin 11
45)
6
A)
3 2
B) 1
C) - 1
2
2
D) -
3 2
46) cos 7
46)
4
A)
2 2
B) 1
C)
2
3 2
D) -
2 2
47) tan 4
47)
3
A) - 3
C) 3
B) 1
D) -
3 3
48) csc -2
48)
3
A) - 1 2
B) - 2
C) - 3
D) - 2 3 3
49) sec -3
49)
4
A) - 2
B) -2
C)
2 2
D) - 2 3 3
50) cot -5
50)
6
A)
51) sec( ) A) 1
3 3
B) -
3 3
C) - 3
D) 3
C) -1
D) Undefined
51) B) 0
5
Find all values of x between 0 and 2 that satisfy the equation. 52) cos x = - 3 2
A)
,
3
,
6
11 6
C)
4
,
4
D)
,
6
6
3 2
53) sin x = A)
3
B)
52)
,
6
6
53) B)
,
4
C)
4
3
,
3
D)
,
3
3
54) tan x = 1
54)
3
A)
6
,
6
B)
3
,
C)
3
4
,
4
D)
55) csc x = -2 A)
6
,
3
,
11 6
55) 6
B)
56) sec x = -2 A)
,
6
4
,
C)
4
3
,
3
D)
11 , 6 6
56) 3
B)
11 , 6 6
C)
6
,
6
Use a calculator to find the function value to four decimal places. 57) sin 28.8° A) 0.4818 B) 0.5498 C) 0.8763
58) cos 52.0° A) 0.7880 59) cot 42.0° A) 1.1106
D)
4
,
4
57) D) -0.5018 58)
B) 1.2799
C) -0.1630
D) 0.6157 59)
B) 2.2914
C) 0.9004
D) 0.6691
60) tan 76.4° A) 0.2419
B) 4.1335
C) 0.9720
D) 1.5635
61) csc 42.2° A) -0.9777
B) 1.4887
C) 0.7408
D) 0.6717
62) tan 472° A) 7.1153
B) -2.4751
C) -0.3746
D) -0.7813
60)
61)
62)
6
63) sin 0.2831 A) 0.9602
B) 0.2793
C) 1.0415
D) 0.2909
64) sec 0.26 A) 0.2571
B) 0.2660
C) 0.9664
D) 1.0348
65) tan 3.23 A) -0.9961
B) 0.0886
C) -1.0039
D) -0.0883
63)
64)
65)
Give the amplitude or period as requested. 66) Amplitude of f(x) = -4 sin x
A)
66) C) 2
B) -4
4
67) Amplitude of f(x) = -3 sin 2x A) 3
B)
D) 4
67) C) 3 2
3
D)
2
68) Period of f(x) = sin 3x
68)
A) 2
B) 1
3
C) 2
D) 3
69) Amplitude of f(x) = 1 cos 5x
69)
3
A) 5
B) 1
3
C)
3
5
D) 5
70) Period of f(x) = cos 5x
70)
A) 2
B) 2
5
C) 5
D) 1
71) Period of f(x) = -5 cos 1 x
71)
4
B) 5
A) 8
C)
4
4
D) -5
72) Period of f(x) = 5 cos x
72)
A) 2
73) Amplitude of f(t) = -6 sin A) 6
C) 5
B)
5
D)
5
t- 3
73)
B) 10
C) -6
7
D) -3
74) Period of f(t) = 7 cos A) 20
10
t+ 5
B)
74) C) 20
10
D) 10
75) Period of f(x) = -7 sin(13 x + 1) A) 2 13
75)
B) 2 13
C) 13
Graph the function. 76) y = 3 cos x
D) 7
76)
A)
B)
C)
D)
8
77) y = -1.5 sin x
77)
A)
B)
C)
D)
78) y = 3 tan x
78)
9
A)
B)
C)
D)
79) y = cos 1 x
79)
4
10
A)
B)
C)
D)
80) y = -cos( x)
80)
11
A)
B)
C)
D)
81) y = 1 sin 4 x 2
81)
3
12
A)
B)
C)
D)
82) y = -2 cos x +
82)
2
13
A)
B)
C)
D)
83) y = 1 sin(x + )
83)
2
14
A)
B)
C)
D)
84) y = 3 tan 1 x + 5
2
84)
4
15
A)
B)
C)
D)
85) y = 3 sin(x - ) + 4
85)
16
A)
B)
C)
D)
Solve the problem. 86) Sales of snow shovels are seasonal. Suppose the sale of snow shovels in Maine is approximated by s(t) = 10,000 + 10,000 cos
6
86)
t , where t is time in months and t = 0 is October. What are the sales in
December? A) 17,071 snow shovels
B) 18,660 snow shovels D) 13,900 snow shovels
C) 15,000 snow shovels
87) The temperature in Fairbanks is approximated by T(x) = 37 sin 2 (x - 101) + 25, where T(x) is 365
the temperature on day x, with x = 1 corresponding to Jan 1 and x = 365 corresponding to Dec 31. Estimate the temperature, to the nearest degree, on day 26. A) -25° B) -11° C) 15° D) -36°
17
87)
88) A scientist studying ocean tides places an 8 ft high marker in the water at 6 am on a Monday
88)
morning. At that time the water is about 5.5 ft high and receding. The scientist observes that the water reaches its lowest level, 0.1 ft, at 9:18 am and then begins to rise. Assume that the water level, in feet, is given by 2 h(t) = 4.9 sin t + 5, 12.4 where t represents the number of hours after midnight. (In other words, the marker was placed in the water when t = 6.) Find the first time interval during which the marker is completely underwater. A) Approximately from 1:18 am to 4:54 am Tuesday
B) Approximately from 2:06 pm to 4:24 pm Monday C) Approximately from 11:24 pm Monday to 2:00 am Tuesday D) Approximately from 1:42 pm to 5:18 pm Monday 89) The voltage E in an electrical circuit is given by E = 5.6 cos(60 t), where t is time measured in seconds. Find the period.
A)
B) 30
30
C) 30
D) 1
30
90) The total sales in dollars of some small businesses fluctuates according to the equation S = A + B sin
6
89)
90)
x , where x is the time in months, with x = 1 corresponding to January, A = 8,100,
and B = 2,800. Determine the month with the greatest total sales and give the sales in that month. A) June; $8,100 B) December; $10,900
C) September; $5,300
D) March; $10,900
91) The total sales in dollars of some small businesses fluctuates according to the equation S = A + B sin
6
91)
x , where x is the time in months, with x = 1 corresponding to January, A = 6,400,
and B = 3,100. Determine the month with the least sales and give the sales in that month. A) June; $3,100 B) December; $6,400
C) March; $9,500
D) September; $3,300
92) The motion of a spring-mass system is described by the equation y = 7 sin t -
3
, where y is the
92)
distance in feet from the equilibrium position and t is time in seconds. If the weight is 23 feet from the ceiling in a state of equilibrium, find the time at which the weight first passes the equilibrium position. A) 1 sec B) 6 sec C) 1 sec D) 1 sec 6 12
93) The motion of a spring-mass system is described by the equation y = 8 sin t -
2
, where y is the
distance in feet from the equilibrium position and t is time in seconds. If the weight is 22 feet from the ceiling in a state of equilibrium, find the closest the weight will ever be to the ceiling. A) 14 ft B) 8 ft C) 22 ft D) 30 ft
18
93)
94) The motion of a spring-mass system is described by the equation y = 5 sin t -
3
, where y is the
94)
distance in feet from the equilibrium position and t is time in seconds. If the weight is 19 feet from the ceiling in a state of equilibrium, find the distance from the ceiling at time t = 4. A) 13 ft B) 17 ft C) 19 ft D) 15 ft
95) The position of a weight attached to a spring is s(t) = -8 cos(12 t) inches after t seconds. What is
95)
the maximum height that the weight reaches above the equilibrium position and when does it first reach the maximum height? A) The maximum height of 16 inches is first reached after 3 seconds.
B) The maximum height of 16 inches is first reached after 6 seconds. C) The maximum height of 8 inches is first reached after 6 seconds. D) The maximum height of 8 inches is first reached after 0.08 seconds. 96) The index of refraction for air, Ia, is 1.0003. The index of refraction for water, Iw, is 1.3. If
96)
Iw sin A , and A = 31.5°, find W to the nearest tenth. = Ia sin W
A) 23.7°
B) 21.7°
97) Snell's Law states that
c1 c2
=
C) 22.7°
D) 20.7°
sin 1 . Use this law to find the requested value. If c1 = 9 × 107 , sin 2
2 = 34°, find c2 . A) c2 = 7.2 × 107 B) c2 = 6.67 × 107
97)
1 = 49°, and
98) Snell's Law states that
c1 c2
=
C) c2 = 6.67 × 106
D) c2 = 6.43 × 109
sin 1 . Use this law to find the requested value. If c2 = 1.55 × 109 , sin 2
2 = 34°, find c1 . A) c1 = 2.12 × 109 C) c1 = 2 × 109
98)
1 = 46° and
99) Snell's Law states that c2 = 4.23 × 106 ,
A) 2 = 33°
c1 c2
=
B) c1 = 1.55 × 1011 D) c1 = 1.43 × 108
sin 1 . Use this law to find the requested value. If c1 = 6 × 106 , sin 2
1 = 47°, find
2 . Round your answer to the nearest degree.
B) 2 = 30°
C) 2 = 34°
19
D) 2 = 31°
99)
100) Snell's Law states that c2 = 5.76 × 109 ,
c1 c2
=
sin 1 . Use this law to find the requested value. If c1 = 8 × 109 , sin 2
100)
2 = 30°, find 1 . Round your answer to the nearest degree.
A) 1 = 45°
B) 1 = 47°
C) 1 = 42°
D) 1 = 44°
101) From a boat on the lake, the angle of elevation to the top of a cliff is 34°30'. If the base of the cliff is 2,703 feet from the boat, how high is the cliff (to the nearest foot)? A) 1,871 ft B) 1,868 ft C) 1,861 ft
101)
D) 1,858 ft
102) From a boat on the river below a dam, the angle of elevation to the top of the dam is 20°59'. If the
102)
103) From a balloon 810 feet high, the angle of depression to the ranger headquarters is 73°6'. How far
103)
104) When sitting atop a tree and looking down at his pal Joey, the angle of depression of Mack's line
104)
105) The air speed of an airplane is 550 km/hr and its angle of climb is 4.05°. What is its ground speed
105)
dam is 456 feet above the level of the river, how far is the boat from the base of the dam (to the nearest foot)? A) 1,179 ft B) 1,189 ft C) 1,169 ft D) 1,159 ft
is the headquarters from a point on the ground directly below the balloon (to the nearest foot)? A) 241 ft B) 236 ft C) 251 ft D) 246 ft
of sight is 55°54'. If Joey is known to be standing 18 feet from the base of the tree, how tall is the tree (to the nearest foot)? A) 33 ft B) 31 ft C) 27 ft D) 29 ft
(to the nearest km/hr)? A) 549 km/hr
B) 544 km/hr
C) 534 km/hr
D) 539 km/hr
106) At an altitude of 3,300 ft, the engine on a small plane fails. What angle of glide is needed to reach
106)
107) The chairlift at a ski resort has a vertical rise of 3,500 feet. If the length of the ride is 2.4 miles,
107)
108) A 31-foot ladder is leaning against the side of a building. If the ladder makes an angle of 22° 38
108)
an airport runway that is 5 miles away by land? (Round your answer to the nearest tenth of a degree.) A) 7.1° B) 88.9° C) 8.1° D) 89.9°
what is the average angle of inclination of the lift (to the nearest tenth of a degree)? A) 10.0° B) 16° C) 19.0° D) 13.0°
with the side of the building, how far is the bottom of the ladder from the base of the building? Round your answer to the hundredths place. A) 11.93 ft B) 2.56 ft C) 17.63 ft D) 13.23 ft
20
109) A contractor needs to know the height of a building to estimate the cost of a job. From a point 100
109)
110) A conservation officer needs to know the width of a river in order to set instruments
110)
feet away from the base of the building, the angle of elevation to the top of the building is found to be 50° 48 . Find the height of the building. Round your answer to the hundredths place. A) 122.61ft B) 121.08 ft C) 125.51 ft D) 126.84 ft
correctly for a study of pollutants in the river. From point A, the conservation officer walks 105 feet downstream and sights point B on the opposite bank to determine that = 50° (see figure). How wide is the river?
= 50°
105 ft. B) 80 ft
A) 125 ft
C) 88 ft
D) 163 ft
111) A weight attached to a spring is pulled down 5 inches below the equilibrium position. Assuming that the period of the system is
1 second, determine a trigonometric model that gives the position 6
of the weight at time t seconds.
A) y = 5 cos 12 t
B) y = -5 cos 12 t
C) y = -5 cos 6 t
D) y = 5 cos 1 t 6
112) A weight attached to a spring is pulled down 2 inches below the equilibrium position. Assuming that the frequency of the system is
5
112)
cycles per second, determine a trigonometric model that
gives the position of the weight at time t seconds. A) y = -2 cos 10t B) y = 2 cos 5t
C) y = 2 cos 10t
D) y = -2 cos 5t
113) Tides go up and down in a 14-hour period. The average depth of a certain river is 14 m and
ranges from 11 to 17 m. The depth of the river can be approximated by a sine curve. Write an equation that gives the depth x hours after midnight given that high tide occurs at 7:00 am. A) d = 3 sin x + 14 B) d = 3 sin x - + 1 7 14 2
C) d = 3 sin
111)
x + 14 7 2
D) d = 7 sin
21
x 7 4
113)
Find the derivative of the function. 114) y = 2 cos 9x A) dy = 18 sin 9x dx
114) B) dy = 9 sin 9x dx
C) dy = -18 sin 9x
D) dy = -2 sin 9x
dx
dx
115) y = 9 sin (7x - 5)
115)
A) dy = 7 sin (7x - 5) dx
B) dy = -7 cos (7x - 5) dx
C) dy = 63 cos (7x - 5)
D) dy = 9 cos (7x - 5)
dx
dx
116) y = 2x sin 19x
116)
A) dy = 38x cos 2x + 2 sin 19x dx
B) dy = 38x cos 19x + sin 19x dx
C) dy = -38x cos 19x + 2 sin 19x
D) dy = 38x cos 19x + 2 sin 19x
dx
117) y = sin 6x
A) dy = 6 cos 6x dx
dx
117) B) dy = cos x
C) dy = 6 cos x
dx
dx
118) y = 9 sin x5
D) dy = cos 6x dx
118)
A) dy = 45x5 cos x5 dx
B) dy = x cos x5 dx
C) dy = 45x4 cos x5
D) dy = 45x4 cos x4
dx
dx
119) y = cos x5
119)
A) dy = -5x4 sin x5
B) dy = sin x5
C) dy = 5 sin x5
D) dy = -5x5 sin x5
dx
dx
dx
dx
120) y = cos (9x2 + 2)
120)
A) dy = -18 sin 9x2 dx
B) dy = -18x sin (9x2 + 2) dx
C) dy = 18x sin (9x2 + 2)
D) dy = sin (9x2 + 2)
dx
dx
22
121) y = x3 cos 6x2
121)
A) dy = sin 6x2 + 3x2 cos 6x2
B) dy = -12x4 sin 6x2
C) dy = -12x4 sin 6x2 + 3x2 cos 6x2
D) dy = 12x4 sin 6x2 + 3x2 cos 6x2
dx
dx
dx
dx
122) y = 4 sin 9x cos x
122)
A) dy = -4 sin 9x sin x + 36 cos x cos 9x dx
B) dy = 4 sin 9x sin x + 36 cos x cos 9x dx
C) dy = -4 sin 9x sin x + cos x cos 9x
D) dy = sin 9x sin x + 36 cos x cos 9x
dx
dx
123) y = x6cos x - 5x sin x - 5 cos x
123)
A) dy = - 6x5 sin x - 5 cos x + 5 sin x dx
B) dy = - x6 sin x + 6x5 cos x - 5x cos x - 10 sin x dx
C) dy = x6 sin x - 6x5 cos x + 5x cos x dx
D) dy = - x6 sin x + 6x5 cos x - 5x cos x dx
124) y = 2 tan6 x
124)
A) dy = 12 tan5 x sec2 x
B) dy = 12 tan7 x
C) dy = 12 tan5 x
D) dy = 12 tan6 x sec x
dx
dx
dx
dx
125) y = cot (4x - 3)
125)
A) dy = 4 cot (4x - 3) csc (4x - 3) dx
B) dy = - csc2(4x - 3) dx
C) dy = -4 csc2 (4x - 3)
D) dy = -4 sec2(4x - 3)
dx
dx
126) y = 2 sec3 x
126)
A) dy = 6 tan x sec3 x dx
B) dy = 6 tan2 x sec2 x dx
C) dy = 6 sec2 x
D) dy = 6 tan2 x sec3 x
dx
dx
127) y = sec 6x
127)
A) dy = 6 sec x tan x dx
B) dy = 6 sec 6x cot 6x dx
C) dy = 6 sec 6x tan 6x
D) dy = - sec 6x tan 6x
dx
dx
23
128) y = x · csc 5x
128)
A) dy = csc 5x - 5x · csc 5x tan 5x dx
B) dy = csc 5x + 5x · csc 5x cot 5x dx
C) dy = csc 5x - 5x · csc 5x cot 5x
D) dy = csc 5x - x · csc 5x cot 5x
dx
dx
129) y = x5 tan x
129)
A) dy = x5 sec x tan x + 5x4 tan x dx
B) dy = x5 sec2 x + 5x4 tan x dx
C) dy = - x5 sec2 x + 5x4 tan x
D) dy = 5x4 sec2 x
dx
dx
130) y = x3 - csc x + 2
130)
A) dy = x2 - cot2 x + 2 dx
B) dy = 3x2 + cot2x dx
C) dy = 3x2 - csc x cot x
D) dy = 3x2 + csc x cot x
dx
dx
131) y = (csc x + cot x)(csc x - cot x)
131)
A) dy = 0 dx
B) dy = 1 dx
C) dy = -csc x cot x
D) dy = -csc2 x
dx
dx
132) y = 4 sec x + 4 cot 1 x - 7 sin(6x3 ) + e6x
132)
8
A) dy = 4 tan2 x - 1 cot 1 x csc 1 x + 126x2 cos(6x3 ) + e6x dx
2
8
8
B) dy = 4 tan2 x - 1 cot 1 x csc 1 x - 126x2 cos(6x3 ) + 6e6x dx
2
8
8
C) dy = 4 sec x tan x - 1 csc2 1 x - 126x2 cos(6x3 ) + 6e6x dx
2
8
D) dy = 4 sec x tan x - 1 csc2 1 x + 126x2 cos(6x3 ) + 6e5x dx
2
8
133) y = (cos 4x + tan x3 )5
133)
A) dy = 5(cos 4x + tan x3 )4 dx B) dy = 5(cos 4x + tan x3 )4 (-4 sin 4x + 3x2 sec2 (x3 )) dx
C) dy = 5(cos 4x + tan x3 )4 (4 sin 4x + 3x2 cot2 (x3)) dx
D) dy = 5(- sin 4x + 3x2 sec x3 tan x3)4 dx
24
134) y = cos e2x
134)
A) dy = e2x cos e2x
B) dy = -2e2xsin e2x
C) dy = 2e2x sin e2x
D) dy = -2 sin e2x
dx
dx
dx
dx
135) y = sin 4e-4x
135)
A) dy = 16e-4x cos 4e-4x dx
B) dy = e-4x cos 4e-4x dx
C) dy = cos 4e-4x
D) dy = -16e-4x cos 4e-4x
dx
dx
136) y = ecos x
136)
A) dy = sin x ecos x dx
B) dy = -sin x ecos x dx
C) dy = -cos x esin x
D) dy = -esin x
dx
dx
137) y = sin (ln 5x6 ) A) dy = dx
137)
6 cos (ln 5x6 )
B) dy = dx
5x6
6
x
6
C) dy = -6 cos (ln 5x ) dx
5 cos (ln 5x6 )
D) dy = 6 cos (ln 5x )
x
dx
138) y = ln cos 4x6
x
138)
A) dy = -tan 4x6
B) dy = tan 4x6
C) dy = -24 x5 tan 4x6
D) dy = 24 x5 tan 4x6
dx
dx
dx
139) y = ln tan2 3x
dx
139) 2
A) dy = 6 sec 3x
B) dy =
C) dy = 6 sec 3x
D) dy = sec 3x csc 3x
dx
dx
tan 3x
dx
6 tan 3x
dx
140) y = tan x
140)
3x - 6 2
A) dy = (3x - 6)csc x - 3tan x dx
B) dy = (3x - 6)sec x tan x - 3tan x dx
(3x - 6)2 2
2
C) dy = (3x - 6)sec x - 3tan x dx
(3x - 6)2
D) dy = sec x - 3tan x
(3x - 6)2
dx
25
(3x - 6)2
141) y = cos x
141)
x2
A) dy = -2 cos x - x sin x
B) dy = cos x - x sin x
C) dy = 2 cos x + x sin x
D) dy = -2 cos x - x sin x
dx
x3
dx
dx
x3
dx
x3
x2
142) y = sec (3x - 4)
142)
x
A) dy = sec (3x - 4)[3 tan(3x - 4) - x] dx
x2
B) dy = 3 sec (3x - 4) cot (3x - 4) - sec (3x - 4) dx
x2
C) dy = sec (3x - 4)[x tan(3x - 4) -1] dx
x2
D) dy = sec (3x - 4)[3x tan(3x - 4) -1] dx
x2
143) y = 3 cos x
143)
6 - sin x 2
2
2
A) dy = -18 sin x + 3 sin x - 3cos x
B) dy = -18 sin x + 3 sin x - 3 sin x cos x
C) dy = 18 sin x - 3
D) dy = -18 sin x + 3
dx
dx
(6 - sin x)2
dx
(6 - sin x)2
dx
(6 - sin x)2
(6 - sin x)2
144) y = sin 2x
144)
cos 3x
A) dy = 2 cos 2x cos 3x - 3 sin 2x sin 3x
B) dy = cos 2x cos 3x + sin 2x sin 3x
C) dy = 2 cos 2x sin 3x + 3 sin 2x cos 3x
D) dy = 2 cos 2x cos 3x + 3 sin 2x sin 3x
dx
cos2 3x
dx
dx
cos2 3x
dx
cos2 3x
cos2 3x
145) y = 9 + 1 sin x
145)
cot x
A) dy = 9 cos x - csc2 x
B) dy = 9 csc x cot x - csc2 x
C) dy = -9 csc x cot x + sec2 x
D) dy = 9 csc x cot x - sec2 x
dx
dx
dx
dx
26
146) y = sin x + 5x 5x
146)
sin x
A) dy = x cos x - sin x + 5 sin x - 5x cos x
B) dy = x cos x + sin x + 5 sin x + 5x cos x
C) dy = sin x - x cos x + 5x cos x - 5 sin x
D) dy = cos x + 5
dx
5x2
dx
sin2 x
25x2
dx
dx
sin2 x
5x2
5
sin2 x
cos x
147) y = 4 + sec x
147)
4 - sec x 2
A) dy = - 2 sec x tan x
B) dy = 8 sec x tan x
C) dy = - 8 sec x tan x
2 D) dy = 8 tan x
dx
dx
dx
(4 - sec x)2
(4 - sec x)2
dx
(4 - sec x)2
(4 - sec x)2
148) y = sec x + csc x
148)
csc x
A) dy = sec2 x + 1
B) dy = -csc x cot x
C) dy = sec x tan x
D) dy = sec2 x
dx
dx
dx
dx
Find the slope of the line tangent to the curve at the given point.
149) y = 25 sin x; x = A) 25
B) - 25
2
2
2
150)
4
B) - 17 2
A) - 17 2
151) y = 7 sin x; x =
D) 25 3
C) 1
2
150) y = 17 cos x; x =
C) 17 2
2
2
D) 17 3 2
151)
2
B) 0
A) -7
152) y = 15 tan x; x = A) 20
149)
3
C) 7
D) 7 2
152)
3
B) 15
C) 30
27
D) 60
153) y = 4 cot x; x =
153)
6
A) 16
C) 16
B) -4
3
D) -16
Solve the problem. 154) A car moves along a straight road. The distance from the starting point is given by s(t) = 2 sin t + cos t. Find the velocity at t = 0. A) 0 B) 1 C) 2 D) -1
154)
155) The electric charge q, in Coulombs, passing a given point in a circuit is given by
155)
156) The revenue received from the sale of electric heaters is seasonal, with maximum revenue in the
156)
157) The revenue received from the sale of electric heaters is seasonal, with maximum revenue in the
157)
158) The motion of a spring-mass system is described by the equation y = 13 sin
158)
q = t csc (0.5t2 + 2), where t is the time in seconds. Find the current i, to the nearest tenth of an amp, for t = 0.75 s. (i = dq/dt) A) 0.64 A B) 1.1 A C) 2.2 A D) 2.0 A
winter. Let the revenue received from the sale of heaters be approximated by R(x) = 42 cos 2 x + 600, where x is time in years, measured from January 1. Find R'(x) for March 1st. A) 0 B) -42 C) -42 3 D) -42 2
winter. Let the revenue received from the sale of heaters be approximated by R(x) = 162 cos 2 x + 600, where x is time in years, measured from January 1. Find R'(x) for August 1st. A) 162 3 B) 162 2 C) 0 D) 162 t-
4
, where y is
the distance in feet from the equilibrium position and t is time in seconds. Find the velocity of the mass at time t = 3/4 seconds. Is the mass moving toward or away from its equilibrium position or neither? A) 0 feet/second; neither
B) 13 feet/second; away from equilibrium position C) 0 feet/second; away from equilibrium position D) -13 feet/second; toward from equilibrium position 159) The motion of a spring-mass system is described by the equation y = 11 sin
t-
4
, where y is
the distance in feet from the equilibrium position and t is time in seconds. Find the acceleration of the mass at time t = 3/4 seconds. A) -11 2 feet/sec2 B) 11 2 feet/sec2
C) -11 feet/sec2
D) 0 feet/sec2
28
159)
160) Find the maximum and minimum values of y for the equation of simple harmonic motion. y = 8 sin (4x + 3) A) Maximum = 11, minimum = -8
160)
B) Maximum = 8, minimum = -8 D) Maximum = 24, minimum = -24
C) Maximum = 11, minimum = -11
161) The electric charge q, in Coulombs, passing a given point in a circuit is given by
161)
q = t csc (0.4t2 + 2), where t is the time in seconds. Find the current i as a function of t (i = dq/dt). A) dq = csc (0.4t2 + 2) - 0.8t2 csc (0.4t2 + 2) tan (0.4t2 + 2) dt
B) dq = csc (0.4t2 + 2) -t csc (0.4t2 + 2) cot (0.4t2 + 2) dt
C) dq = csc (0.4t2 + 2) - 0.8t2 csc (0.4t2 + 2) cot (0.4t2 + 2) dt
D) dq = csc (0.4t2 + 2) + 0.8t2 sec (0.4t2 + 2) cot (0.4t2 + 2) dt
162) A thief tries to enter a building by placing a ladder over a 10-foot high fence so that it rests against
162)
163) A surveyor measures the distance between two markers to be 422 m. Then, moving along a line
163)
the building which is 3 ft back from the fence. What is the length of the shortest ladder that can be used? [Hint: Let be the angle between the ladder and the ground. Express the length of the ladder in terms of and find the value of that minimizes the length of the ladder.] A) 17.07 ft B) 17.55 ft C) 17.43 ft D) 18.38 ft
1 equidistant from the markers, the distance d from the surveyor to each marker is d = 211 csc , 2 where is the angle between the lines of sight to the markers. See the figure below. By using differentials, find the change in d if changes from 94.2° to 94.35° 422 m
A) -20.07 m
B) -0.7007 m
C) -0.3504 m
29
D) 0.02034 m
164) A plant population experiences seasonal growth. At time t, the population, f(t), is modeled by
164)
f(t) = 500e1.3sin(t) . Find the maximum and minimum values of f(t) and the values of t where they occur.
A) Maximum: 1,254 when t = Minimum: 199 when t =
2
+ 2 n, where n is any integer
3 +2 n 2
B) Maximum: 500 when t =
2
Minimum: -500 when t =
+ 2 n, where n is any integer
3 +2 n 2
C) Maximum: 1,835 when t = 2 n, where n is any integer Minimum: 136 when t =
+2 n
D) Maximum: 1,835 when t = Minimum: 136 when t =
2
+ 2 n, where n is any integer
3 +2 n 2
165) The intensity of light I at time t hours after sunrise and at a depth of x feet below the ocean surface is given by
I(x,t) = Ioe-kx sin3
165)
t , D
where D is the length of daylight in hours, Io is the intensity of light on the surface of the water at midday, and k is a positive constant. Assume D = 14, Io = 50, and k = 0.006. In a study of the effect of light on ocean plants, a scientist needs to determine the minimum depth x at which a plant would never at any time be exposed to an intensity of light greater than 35. What is this minimum depth? A) 384 ft B) 108 ft C) 60 ft D) 66 ft
166) The beacon on a lighthouse 20 m from a straight shoreline rotates twice per minute. How fast is
the beam moving along the shoreline at the moment when the light beam and the shoreline are at right angles? [Hint: Find an equation relating and x where : is the angle between the beam of light and the line from the lighthouse to the shoreline x is the distance along the shoreline from the point on the shoreline closest to the lighthouse and the point where the beam hits the shoreline. Use the chain rule to find dx/dt. You need to express d /dt in radians per minute.] A) 80 m/min B) 20 3 m/min C) 80 2 m/min D) 20 m/min
30
166)
167) The beacon on a lighthouse 30 m from a straight shoreline rotates twice per minute. Find dx/dt
167)
where x is the distance along the shoreline from the point on the shoreline closest to the lighthouse and the point where the beam hits the shoreline. [Hint: Find an equation relating and x where is the angle between the beam of light and the line from the lighthouse to the shoreline. Use the chain rule and express d /dt in radians per minute.] A) dx = 30 sec2 B) dx = 120 sec tan dt dt
C) dx = 120 sec2
D) dx = -120 csc2
dt
dt
168) On March 1, high tide in Charleston was at midnight (t = 0). The water level at high tide was 9
168)
169) A population of animals varies periodically between a low of 700 on January 1 and a high of 900
169)
feet. The next high tide was exactly 12 hours later (i.e., at noon) and the water level was again 9 feet. In between, at low tide, the water level was 1 foot. Assuming that the height of the water can be described by a sine or cosine curve, find the rate of change of the water level with respect to time at 7 am. Give your answer in exact form. A) 4 ft/hr B) ft/hr C) 2 ft/hr D) ft/hr 3 3 6
on July 1. Find an equation (using a trigonometric function) to describe the size of the population at any time t, where t is measured in months from the beginning of the year, and use this equation to find the rate of change of the population on September 1. Give your answer in exact form. A) - 25 3 animals per month B) - 100 animals per month 3 6
C) - 25
3
3
D) -50 3 animals per month
animals per month
170) A wheel of radius 1 foot revolves at a rate of 10 revolutions per second. A dot is painted at a point P on the rim of the wheel (see figure). Find the rate of change in the horizontal position of the dot when = 30°. Give your answer in exact form.
A) 10
3 ft/s
B) 1 ft/s
C) 10 ft/s
2
31
D)
10
ft/s
170)
171) A piston moves back and forth in a cylindrical chamber that is filled with gas and has a radius of
171)
5 centimeters (see figure below). At time t seconds, the end of the piston is 6 + 4 sin 2t centimeters from the closed end of the chamber. What is the rate of change of the volume of the gas at time t = 4 /3? Give your answer in exact form.
A) -100 cm3/s
B) - 50 cm3/s
C) 50 cm3 /s
D) 100
3 cm3 /s
Find the integral.
172)
5 sin 3x dx
A) 5 cos 3x + C 3
173)
174)
D) - 5 cos 3x + C 3
173) B) 8 cos t - 1 sin 3t + C
C) 8 cos t + 3 sin 3t + C
D) 8 cos t + 1 sin 3t + C
3 3
-6 x sin x2 dx
174) B) 3x cos x2 + C
C) - 3 cos x2 + C
D) 6 cos x2 + C
9 csc2 6x dx 2
175) C) - 3 cot 6x + C
B) - 54 cot 6x + C
2
D) 3 cot 6x + C 2
sin2 x cos x dx 3
A) sin x + C 3
177)
C) - 15 cos 3x + C
A) -8 cos t - 3 sin 3t + C
A) - 3 tan 6x + C
176)
B) 5 cos 3x + C
(-8 sin t + cos 3t) dt
A) 3 cos x2 + C 175)
172)
176) 2
3
B) sin x + C
C) sin x + C
3
2
2
D) sin x + C 2
cos x dx sin2 x
A) csc x + C
177) B) sec x + C
C) -sec x + C
32
D) -csc x + C
178)
sin x cos7 x dx
A) 7 sin7 x + C
179)
178) B) 1 sin8 x + C
C) - 1 cos8 x + C
8
8
9x3 cos x4 dx
179)
A) 36 sin x4 + C
B) 9 sin x4 + C
C) - 9 x4 sin x4 + C
D) 9 sin x4 + C
4
4
180)
2x5 csc x 6 cot x 6 dx
180)
A) 12 csc x6 + C
B) 1 csc x6 + C
C) - 1 sec x6 + C
D) - 1 x6 csc x6 + C
3
3
181)
3
x3 tan x4 dx
181)
A) - 1 ln cos x4 + C
B) 1 sec2 (x4 ) + C
C) - 4 ln cos x4 + C
D) - 1 ln sin x4 + C
4
182)
4
4
sec2 x tan x dx
A) 1 tan2 x + C 2
183)
182) B) 1 tan x + C
C) sec 2x + C
2
cos x dx 1 + sin x
D) tan 2x + C
183) sin x +C x - cos x
A) ln 1 + sin x + C
B)
C) - sin x + C
D) - ln 1 + sin x + C
x - cos x
184)
D) -7 cos7 x + C
9 csc3 9x cot 9x dx
184)
A) - 1 csc3 9x + C
B) - 1 csc2 9x cot 9x + C
C) 1 csc3 9x + C
D) csc2 9x + C
3
2
3
33
185)
sec3 (x - 9) tan (x - 9) dx
185)
A) - 1 sec4 (x - 9) + C
B) 1 sec3 (x - 9) + C
C) 1 sec4 (9x - 9) + C 4
D) 1 sec4 (x - 9) + C
4
186)
3 4
7 csc3 x cot x dx
186)
A) - 7 cot3 x + C
B) - 7 csc3 x + C
C) - 7 csc4 x + C
D) 7 csc4 x cot x + C
3
3
3
187)
4
x x 2 tan dx 3 3
A) 3 ln sin x 2
187) 2
3
C) - 1 x2 ln cos x 6
188)
cot
2
3
2
+C
2
2
3
+C
x dx 8
188)
A) -8 csc2 x + C
B) 8 ln cos x + C
C) - 1 ln sin x + C
D) 8 ln sin x + C
8
8
8
8
e4x csc e4x cot e4x dx
189)
A) - 1 e4x csc e4x + C
B) - 1 cot e4x + C
C) - 1 csc e4x + C
D) - 1 e4x cot e4x + C
4
4
4
190)
3
D) - 3 ln cos x
+C
8
189)
2
B) 3 sec2 x
+C
4
7e-x cos e-x dx
190)
A) -7 sin e-x + C
B) 7 sin e-x + C
C) 7e-x sin e-x + C
D) - 1 sin e-x + C 7
34
191)
-5x cos 9x dx
191)
A) - 5 cos 9x - 5 x sin 5x + C
B) - 5 cos 9x - 5 sin 9x + C
C) - 5 cos 9x - 5 x sin 9x + C 81 9
D) - 5 cos 9x - 5x sin 9x + C 9
81
192)
9
192)
A) - 3 sin 4x + 3 x cos 4x + C
B) 3 sin 4x + 3 cos 4x + C
C) - 3 sin 4x + 3 x cos 4x + C
D) - 3 sin 4x + 3 cos 4x + C
4
4
16
4
193)
12x cos
B) 18 sin x - x cos x + C D) 18 sin x - 18x cos x + C
1 x dx 2
194)
A) 12 sin 1 x + 24x cos 1 x + C
B) 24x sin 1 x + 48 cos 1 x + C
C) 12x sin 1 x - 24 cos 1 x + C
D) 48 sin 1 x - 24x cos 1 x + C
2
2
8x sin
2
2
2
2
2
1 x dx 2
195)
A) 16 cos 1 x + 32x cos 1 x + C
B) 32 sin 1 x + 16x cos 1 x + C
C) 32 sin 1 x - 16x cos 1 x + C
D) 16 sin 1 x - 32x cos 1 x + C
2
2
2
196)
4
18x sin x dx
2
195)
4
16
A) 18 sin x + 18x cos x + C C) 18 sin x - 18 cos x + C
194)
9
-3x sin 4x dx
16
193)
81
2
2
2
2
2
x2 sin 3x dx
196)
A) 1 x2 cos 3x - 2 x sin 3x - 2 cos 3x + C
B) - 1 x2 cos 3x + 2 x sin 3x + 2 cos 3x + C
C) - 3 x2 cos 3x + 18 x sin 3x + 54 cos 3x + C
D) - 1 x2 cos 3x + 2 x sin 3x + C
3
9
27
3 3
35
9 9
27
Evaluate the definite integral. /2 197) 17 sin x dx 0 A) 0 3 /2
198)
197) B) 17
C) 1
D) -17
18 cos x dx
198)
/2
A) 36 /6
199)
14 tan x dx
B) 18
C) -18
D) -36
(Round to the nearest thousandth.)
199)
0
B) 2.014
A) -12.124 /3
200)
16 cot x dx
C) -2.014
D) -9.702
(Round to the nearest thousandth.)
200)
/6
A) -8.789
B) 8.789
C) -11.088
D) 13.392
4 cos x dx
201)
201)
/6
A) 8 /2
202) 0
B) 4
A) - 1 /4
D) 2
cos2 x sin x dx
3
203)
C) -2
202) C) 3
B) -3
D) 1 3
(tan x)-3 sec2 x dx
203)
/6
A) 1
C) 3
B) -1
D) 0
e-x cos 3x dx
204) 0
A) 1
10
204) C) 3
B) 1
10
36
D) Diverges
e-5x sin x dx
205) 0
A) 5
26
205) B) Diverges
C) 1
D) 1
26
Solve the problem. 206) The formula Z = R sec gives the impedance Z (in ohms) in an ac circuit with resistance R (in ohms) and phase angle (in radians). Find the average value of Z for an 89.7- resistor as ranges from /8 radians to /3 radians. A) 163.8 B) 125.2 C) 156.3 D) 81.9
206)
207) The length of a British nautical mile varies with latitude, according to the formula
207)
208) If the velocity (in ft/s) of a particle is given by v = tan 2t, find the distance traveled as time t
208)
l = 6077 - 26cos , where is the latitude in radians and l is in feet. Find the average length of a nautical mile for latitudes between /4 radians and /2 radians. A) 6,067 ft B) 9,538 ft C) 18,272 ft D) 4,769 ft
changes from 3 to 4 sec. A) 0.96 ft
B) 0.02 ft
C) -0.98 ft
D) 0.94 ft
209) The force F (in newtons) acting on an object is given by F = csc (x/3) where x is the distance (in
209)
210) The number of ducks (in thousands) counted at a certain checkpoint in their migration is given by
210)
211) The number of ducks (in thousands) counted at a certain checkpoint in their migration is given by
211)
212) The number of ducks (in thousands) counted at a certain checkpoint in their migration is given by
212)
213) The velocity of a car is 74 cos t km/hr on the time interval [0, 2] hours. Calculate the distance the
213)
meters) from the rest position. Find the work done in moving the object from x = 0.59 to x = 1.18. A) -0.6 newtons B) 11.8 newtons C) -1.8 newtons D) 2.1 newtons
O(t) = 5 + 5 cos ( t/6), where t is time in months and t = 0 is October. Find the number of ducks passing the checkpoint between October and November. A) 4775 B) 9775 C) 5796 D) 7500
O(t) = 5 + 5 cos ( t/6), where t is time in months and t = 0 is October. Find the number of ducks passing the checkpoint between October and January. A) 24,536 B) 24,549 C) 20,000 D) 17,618
O(t) = 5 + 5 cos ( t/6), where t is time in months and t = 0 is October. Find the number of ducks passing the checkpoint between October and April. A) 29,000 B) 29,965 C) 29,500 D) 30,000
car traveled in that time interval. A) 67.288 kilometers
B) 148 kilometers D) 80.712 kilometers
C) 215.288 kilometers
37
214) The velocity of a car is 81 sin t - 52 kilometers per hour on the time interval [0, 2] hours. Assuming
214)
that the odometer runs backward when the car has negative velocity, express the odometer reading at the end of the time interval in terms of its reading at the beginning, R0 .
A) 218.708 + R0
B) 162 + R0
C) 10.708 + R0
D) 13.92 + R0
215) During a one-hour race, the velocities of two cars are v1 (t) = 76(1 - cos( t)) and v2 (t) = 147t, where 0 t 1. If, at the beginning of the race, both cars were at the mile 0 mark, which car won? A) Car 2
B) It was a tie. C) Car 1 D) There is not enough information to determine the winner.
38
215)
Answer Key Testname: UNTITLED3
1) C 2) C 3) C 4) D 5) D 6) C 7) A 8) B 9) D 10) C 11) A 12) B 13) B 14) C 15) B 16) A 17) B 18) B 19) D 20) B 21) C 22) B 23) B 24) B 25) B 26) A 27) B 28) A 29) B 30) A 31) A 32) D 33) B 34) C 35) D 36) A 37) D 38) B 39) D 40) D 41) A 42) A 39
Answer Key Testname: UNTITLED3
43) A 44) C 45) C 46) A 47) C 48) D 49) A 50) D 51) C 52) D 53) D 54) A 55) D 56) A 57) A 58) D 59) A 60) B 61) B 62) B 63) B 64) D 65) B 66) D 67) A 68) A 69) B 70) A 71) A 72) A 73) A 74) C 75) B 76) B 77) B 78) D 79) A 80) B 81) D 82) B 83) A 84) A 40
Answer Key Testname: UNTITLED3
85) C 86) C 87) B 88) D 89) D 90) D 91) D 92) A 93) A 94) D 95) D 96) A 97) B 98) C 99) D 100) D 101) D 102) B 103) D 104) C 105) A 106) A 107) B 108) A 109) A 110) A 111) B 112) A 113) C 114) C 115) C 116) D 117) A 118) C 119) A 120) B 121) C 122) A 123) D 124) A 125) C 126) A 41
Answer Key Testname: UNTITLED3
127) C 128) C 129) B 130) D 131) A 132) C 133) B 134) B 135) D 136) B 137) D 138) C 139) A 140) C 141) A 142) D 143) D 144) D 145) C 146) A 147) B 148) D 149) A 150) B 151) B 152) D 153) D 154) C 155) D 156) C 157) D 158) A 159) A 160) B 161) C 162) C 163) C 164) D 165) C 166) A 167) C 168) B 42
Answer Key Testname: UNTITLED3
169) C 170) C 171) A 172) D 173) D 174) A 175) C 176) A 177) D 178) C 179) B 180) B 181) A 182) A 183) A 184) A 185) B 186) B 187) D 188) D 189) C 190) A 191) C 192) A 193) D 194) B 195) C 196) B 197) B 198) D 199) B 200) A 201) C 202) D 203) B 204) A 205) D 206) B 207) A 208) D 209) D 210) B 43
Answer Key Testname: UNTITLED3
211) B 212) D 213) A 214) C 215) C
44