Name: ______________________
Class: _________________
Date: _________
Senior Spring Final 2011 -PrecalculusMultiple Choice Identify the choice that best completes the statement or answers the question. Identify the ratio that defines the trigonometric function of the angle θ. ____
1. sin θ
a. b.
b c a c
c. d.
c a a b
Find the exact value of the function. ____
2. cos 210° a. 2 b.
−
3 2
c.
1
d.
−
c.
π 4
c.
cos θ = −
3 3
Find the reference angle θ . ____
3. θ = a.
____
5π 4 3π − 4
b.
−
π 4
7 and sin θ < 0, find cos θ . 24 7 cos θ = − 25 7 cos θ = 25
4. Given tan θ = a. b.
d.
1 /12
24 25 24 cos θ = 25
d.
5π 4
ID: A
Name: ______________________
ID: A
Identify the quadrant in which θ lies. ____
5. sin > 0 and cot > 0 a. Quadrant III b. Quadrant I
c. d.
Quadrant II Quadrant IV
Use the given measures and the Law of Cosines to solve triangle ABC. ____
6. a = 20, b = 16, c = 15 a. A = 9.7°; B = 132.3°; C = 38.0° b. A = 52.0°; B = 80.3°; C = 47.7° c. A = 132.3°; B = 38.0°; C = 9.7° d. A = 80.3°; B = 52.0°; C = 47.7°
____
7. If $5500 is invested in a long-term trust fund with an interest rate of 7% compounded continuously, what is the amount of money in the account after 25 years? a. $33,945.22 c. $31,650.31 b. $29,850.88 d. $40,639.81 Evaluate the expression.
____
____
Ê 2.3 ˆ 8. 500 ÁÁÁÁ 3 ˜˜˜˜ Ë ¯ a. 6133.301 b. 3450 c. 6256.751 d. 6083.5 9. A pole 80 feet tall is situated at the bottom of a hill that slopes up at an angle of 18.8176. A guy wire from the top of the pole to the hillside forms an angle of 34176 with the top of the pole. Find the distance from the base of the pole to the guy wire's point of attachment. a. 45.2 ft c. 36.4 ft b. 44.9 ft d. 46.4 ft Use the Law of Cosines to find the third side of the triangle.
____ 10. a. b.
14.2 11.4
c. d.
2 /12
10.3 8
Name: ______________________
ID: A
Use the information to solve the triangle. ____ 11. B = 14°, C = 31°, and a = 4 a. A = 135°, b ≈ 40.9, c ≈ 5.5 b. A = 145°, b ≈ 40.9, c ≈ 11.7 c. A = 145°, b ≈ 1.7, c ≈ 2.9 d. A = 135°, b ≈ 1.4, c ≈ 2.9 Solve the exponential equation algebraically. −0.03x
____ 12. 8e + 40 = 42.4 a. 0.343 b. 17.429 700 ____ 13. = 650 −x 1+e a. −0.657 b. 2.628
c. d.
40.132 −10.000
c. d.
−0.731 2.565
Identify the graph of the logarithmic function. ____ 14. f ( x) = log 2 ( x+ 3 )
a.
c.
b.
d.
3 /12
Name: ______________________
ID: A
Identify the logarithmic equation written in exponential form. ____ 15. log 243 81 = 81
b.
ÊÁ 4 ÁÁ ÁÁ ÁË 5
ˆ˜ 243 ˜˜ = 81 ˜˜ ˜¯
= 243
c.
ÊÁ 4 ÁÁ ÁÁ ÁË 5
ˆ˜ 81 ˜˜ = 243 ˜˜ ˜¯
d.
243
4/5
a.
4 5
4/5
= 81
Condense the expression to the logarithm of a single quantity. ____ 16. 3 log 10 x + 6 log 10 ( x + 8 ) a. b.
log 10
x
3 6
(x + 8 ) None of these
____ 17. Using a graphing utility, find lim x→0
1 − cos4x 16x
2
c.
log 10 x ( x + 8 )
d.
18 log 10 x ( x + 8 )
.
1 1 b. c. ∞ d. 2 4 ____ 18. Use the Table feature of a graphing utility to find the indicated limit. ÊÁ 3 ˆ ÁÁ e + 729 ˜˜˜ Á ˜˜ lim ÁÁ ˜˜ Á e+ 9 Á ˜ e → −9 Ë ¯ a. 243 b. 162 c. 81 d. 72 a.
−
4 /12
1 2
Name: ______________________
ID: A
Use the graph to find the limit (if it exists). ____ 19. lim f ( x) , where f ( x) ={ x→2
4 − x,
x≠2
−3,
x=2
a.
The limit does b. 4 not exist. ____ 20. Use the graph to find the limit (if it exists). lim f ( x)
c.
–3
x →−2
−6 −5 The limit does not exist. −7 f ( x + h ) − f ( x) ____ 21. Let g ( x) = lim for f ( x) = h h→0 a. b. c. d.
a.
−
b.
4
1 2
4 x+4
. Find g ( −3 ) .
c. d.
5 /12
1 4 −2
d.
2
Name: ______________________
____ 22. Find lim h→0
a. b. c. d.
ID: A
f ( 3 +h ) − f ( 3 ) 4 if f ( x) = x . h
–173 0 –135 108
____ 23. Let g ( x) = lim
h→0
f ( x + h ) − f ( x) 4 for f ( x) = Find g ( 0 ) . 5 h −5x − 2
a. −∞ b. 5 ____ 24. Find the limit (if it exists). lim f ( x) , where f ( x) ={
5 − x,
−
a. b.
6 The limit does not exist.
∞ 0
c. d.
4 2
c.
1
d.
−
x<1 2
x→3
c. d.
5x − x ,
x>1
Find the limit, if it exists. ____ 25.
lim
| − x| | − 4x|
x→3
+
a.
The limit does not exist. 1 4
b.
1 4 2
____ 26. Find the difference quotient for the function f ( x) = −2x + 6x − 6. a. b.
2
−4hx − 2h + 6h −4x − 2h + 6
c. d.
−2x − 4h + 6 2 −2hx − 4x + 6x 2
____ 27. What is the difference quotient of the function f ( x) = − 2x + 2?
a.
b.
ÈÍ ˘ ÍÍ −2 ( x + h ) 2 + 2 ˙˙˙ − ÁÊÁÁ −2x 2 + 2 ˜ˆ˜˜ ÍÍ ˙˙ Á ˜¯ Î ˚ Ë h ÊÁ ˆ Ê ˆ ÁÁ −2x 2 + 2 + h ˜˜˜ − ÁÁÁ −2x 2 + 2 ˜˜˜ ÁË ˜¯ ÁË ˜¯
c.
d.
6 /12
ÈÍ ˘ ÍÍ −2 ( x + h ) 2 + 2 ˙˙˙ − ÊÁÁÁ −2x 2 + 2 ˆ˜˜˜ ÍÍ ˙˙ Á ˜¯ Î ˚ Ë ÊÁ ˆ Ê ˆ ÁÁ −2x 2 + 2 + h ˜˜˜ − ÁÁÁ −2x 2 + 2 ˜˜˜ ÁË ˜¯ ÁË ˜¯ h
Name: ______________________
ID: A
____ 28. What is the expression that can be used to find the equation for the slope of the graph of the 2 function f ( x) = − 2x + 2 at a given x-value?
a.
b.
ÈÍ ˘ ÍÍ −2 ( x + h ) 2 + 2 ˙˙˙ − ÊÁÁÁ −2x 2 + 2 ˆ˜˜˜ ÍÍ ˙˙ Á ˜¯ Î ˚ Ë lim h h→0 ÈÍ ˘ ÍÍ −2 ( x + h ) 2 + 2 ˙˙˙ − ÊÁÁÁ −2x 2 + 2 ˆ˜˜˜ ÍÍ ˙˙ Á ˜¯ Î ˚ Ë h
c.
d.
ÊÁ ˆ Ê ˆ ÁÁ −2x 2 + 2 + h ˜˜˜ − ÁÁÁ −2x 2 + 2 ˜˜˜ ÁË ˜¯ ÁË ˜¯ h ÊÁ ˆ Ê ˆ ÁÁ −2x 2 + 2 + h ˜˜˜ − ÁÁÁ −2x 2 + 2 ˜˜˜ ÁË ˜¯ ÁË ˜¯ lim h h→0
Use the limit definition to find the derivative of the function. ____ 29. f ( x) =
1 −2x − 2 2
a.
f ( x) =
b.
f ( x) = − 2 ( −2x − 2 )
( −2x − 2 )
2 2
c.
f ( x) =
d.
f ( x) =
4 −2x − 2 4 ( −2x − 2 )
2
Use a graphing utility to find the line that appears to be tangent to the graph at ÊÁË 0, − 2 ˆ˜¯ . 3
2
____ 30. f ( x) = 2x + x − x − 2
a.
y = x−4
b.
y = x−2
c.
7 /12
y = −x − 4
d.
y = −x − 2
Name: ______________________
ID: A
Use a graphing utility to find the line that appears to be tangent to the graph at ÁÊË 2, 4 ˜ˆ¯ . 3
____ 31. f ( x) = −x + 3x
a.
2
y=2
b.
y = 2x + 2
c.
y=4
c. d.
–4 –8
c.
8
d.
0
c.
The limit does not exist.
d.
8
d.
y = 2x + 4
Find the limit, if it exists. 2
____ 32. lim x→2
a. b. ____ 33. lim x→2
a. b. ____ 34. lim
x + 4x − 12 x−2 The limit does not exist. 8 x−2 3
2
x − 2x + 4x − 8 1 8 The limit does not exist. x−2 2
x→2
x − 8x + 12
a.
−
b.
1 6 1 − 4
8 /12
Name: ______________________
ID: A
____ 35. Find the slope of the line tangent to the graph of f(x) at the indicated point. .fx = −8x + 4, at (4, −28)
a.
−
1 7
b.
4
c.
-8
d.
−
1 2
e.
None of these
Find the limit, if it exists. x+4 − 2 x
____ 36. lim x→0
a. b.
1 4 1 2
c.
The limit does not exist.
d.
0
Use a calculator to evaluate the expression. ____ 37. cot 5.6 a. −1.2286 b. −2.4572 c. 2.2764 d. −0.8139 ____ 38. A point on the rim of a wheel has a linear speed of 50 cm/s. If the radius of the wheel is 45 cm, what is the angular speed of the wheel in radians per second? a. 0.6 rad/s b. 2.2 rad/s c. 3.5 rad/s d. 1.1 rad/s Express the angle in radian measure in terms of π. Do not use a calculator. ____ 39. 150° a.
5π 6
b.
5π 3
c.
3π 5
In which quadrant is the terminal side of the angle θ? ____ 40. θ = − 225° a. Quadrant b. Quadrant c. Quadrant d. Quadrant
IV III I II
9 /12
d.
6π 5
Name: ______________________
ID: A
Let θ be an acute angle. Use the given function value and trigonometric identities to find the indicated trigonometric function. ____ 41. If cot θ =
3 , find tan θ. 4
4 3 c. 5 5 5 4 b. d. 3 3 ____ 42. The cable supporting a ski lift rises 2 feet for each 5 feet of horizontal length. The top of the cable is fastened 880 feet above the cable's lowest point. Find the lengths b and c, and find the measure of angle θ. a.
b = 2369 ft a.
b.
c = 2200 ft
b = 2200 ft c.
c = 2369 ft
θ = 23.6°
θ = 21.8°
b = 352 ft
b = 948 ft
c = 948 ft
d.
θ = 68.2°
c = 352 ft
θ = 0.4°
Short Answer Sketch a right triangle corresponding to the trigonometric function of the acute angle θ. Use the Pythagorean Theorem to determine the third side and then find the indicated trigonometric function of θ. 43. If cos θ =
3 , find tan θ . 2
Find the exact value of the function. 44. cot
7π 6
10 /12
Name: ______________________
ID: A
Find the indicated trigonometric value in the specified quadrant. 45. θ is in Quadrant II and cos θ = −
2 . Find sin θ. 3
Use a calculator to evaluate the function. Round your answer to four decimal places. (Be sure the calculator is in the correct angle mode.) 46. csc 10.47° Let θ be an acute angle. Use the given function value and trigonometric identities to find the indicated trigonometric function. 47. If cos θ =
1 , find sin θ. 7
48. Given a triangle with a = 14, A = 25°, and B = 17°, what is the length of c? Round your answer to two decimal places. Write the exponential equation in logarithmic form. 2
49. 3 = 9 Find the value of x. Round to three decimal places. 50. ln
3x − 4 = 3.9
51. A 60-foot long irrigation sprinkler line rotates around one end as shown. The sprinkler moves through an arc of 140176 in 1.05 hours. Find the speed of the moving end of the sprinkler in feet per minute. Round your answer to the nearest tenth.
Indicate whether the following statement is true or false. If it is false, correct the statement. 52. log 10
8 = log 10 8 ÷ log 10 7 7
11 /12
Name: ______________________
ID: A
Evaluate the logarithm using the change-of-base formula. Find the value to three decimal places. 53. log 2/5 34 Find the value of x. 54. 9 ln ( 4x) = 11 55. A flat roof rises at a 25176 angle from the front wall of a storage shed to the back wall. The front wall is 9.5 feet tall and the back wall is 15.6 feet tall. Find the length of the roof line and the depth of the shed from front to back. Round your answers to the nearest tenth of a foot. Sketch the angle in standard position. 56.
2π 3
Rewrite the indicated trigonometric function in terms of the angle's reference angle. Use the same function. 57. sin ( −173°) 58. A photographer points a camera at a window in a nearby building forming an angle of 41° with the camera platform. If the camera is 59 meters from the building, how high above the platform is the window? Round to two decimal places.
12 /12
ID: A
Senior Spring Final 2011 -PrecalculusAnswer Section MULTIPLE CHOICE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
B B C C B D C C D C D C D D D B D A D A D D D A B B A A A D C B A B C A A D A D 1 /2
ID: A 41. D 42. C SHORT ANSWER 43. Answers may vary. Sample answer58 tan θ =
44.
3 . 3
3
45. sin θ =
5 3
46. 5.5030 3 7 48. 22.17 49. log 3 9 = 2 47.
4
50. 814.867 51. 2.3 ft/min 52. False. Answers may vary. Sample answer: log 10
8 = log 10 8 − log 10 7. 7
53. –3.849 54. 0.849 55. roof line= 14.4 ft long, shed depth = 13.1 ft
56. 57. − sin 7° 58. 51.29 m
2 /2