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Rec. Time CALCULUS II - FINAL EXAM May 9, 2007
Show all work for full credit. No books, notes or calculators are permitted. The point value of each problem is given in the left-hand margin. You have 1 hour and 50 minutes. Z sec x dx = ln | sec x + tan x| + C
Z
√
dx x = arcsin( ) + C 2 a −x
a2
Z
a2
dx 1 x = arctan( ) + C 2 +x a a
Z
1 |x| dx = arcsec( )+C a a x x 2 − a2
Z
√
Z
√
√
a2
−
u2
1 √ 2 u du = u a − u2 + a2 arcsin +C, 2 a
u2 ± a2 du =
√ 1 √ 2 u u ± a2 ± a2 ln |u + u2 ± a2 | + C 2
CentroidR for the region trapped between y = f (x) , y = g(x) , a ≤ x ≤ b , (with ρ = 1 ) Rb 1 b 2 2 Mx = 2 a f (x) − g(x) dx , My = a x(f (x) − g(x)) dx Maclaurin Series: ex = ln(1 + x) =
P∞
n=1
(−1)n+1 xn , n
sin x =
P∞
(−1)n x2n+1 , (2n+1)!
cos x =
P∞
(−1)n x2n , (2n)!
n=0
tan−1 x =
n=0
P∞
n=0
xn n=0 n! ,
P∞
(−∞, ∞)
(−1, 1]
(−∞, ∞) (−∞, ∞)
(−1)n x2n+1 , 2n+1
(−1, 1]
page 2 of 10 1. Evaluate the following integrals. (12) a)
(12) b)
Z
Z
(x + 1) ln(x) dx
x3 + 2x − 1 dx x2 − 1
page 3 of 10 (12) c)
Z
√
dx 4 + x2
2. Let R be the region bounded below the curves y = x and y = 1/x over the interval [0,2], as shown below. (6) a) Find the area of the region R. (Start by labelling the graph.)
(8) b) Find x , the x coordinate of the centroid of R. (Do not calculate y . ) Hint: See cover sheet.
page 4 of 10 3. Evaluate the following limits or indicate that they diverge. Show all work. 1 − cos(3x) x→0 x2
(8) a) lim
(8) b) lim+ x · ln(x) x→0
(12) 4. A pot of boiling water (100 ◦ C) is placed in a room with constant temperature 10 ◦ C. After an hour it cools to 20 ◦ C. Find the temperature T at any time t ≥ 0 . Assume Newton’s cooling law dT /dt = −k(T − 10) . ( t = hours)
page 5 of 10 (6) 5. a) Sketch the graph of the curve given by the polar equation r = cos(θ)−sin(θ) , 0 ≤ θ ≤ π . Start by identifying all angles θ on the interval [0, π ] where r = 0 .
(6) b) Convert the equation in part (a) to a Cartesian equation in x and y and identify the curve.
(10) 6. Find the equation of the tangent line to the cycloid x = t − sin(t) , y = 1 − cos t , at the point where t = π2 .
page 6 of 10 (8) 7. a) Set up an integral for the arc length of the curve y = sin x , 0 ≤ x ≤ π . DO NOT EVALUATE.
(8) b) Set up an integral for the surface area of the surface obtained by rotating the curve y = sin(x) , 0 ≤ x ≤ π , around the y -axis. DO NOT EVALUATE.
(12) 8. Find the interval of convergence of the power series of any end points.)
∞ X
(x − 3)n . (Make clear the status n 3n n=1
page 7 of 10 9. Let S =
P∞
n=1
(−1)n+1 2n−1
.
(3) a) Explain why the sum converges.
(4) b) Mark the approximate positions of the partial sums S1 , S2 , S3 and the sum S on the number line below.
−1
0
1
(4) c) How many terms are required to estimate the sum S with an error less than .01 , using the n -th partial sum Sn ?
(2) d) Evaluate the sum S by making use of a series on the cover page.
10. Determine whether the following series converge or diverge. State clearly which test you are using and implement the test as clearly as you can. The answer for each problem is worth 2 points and the work you show 5 points. (7) a)
∞ X n=2
e1/n
page 8 of 10 (7) b)
∞ X n=3
(7) c)
∞ X
√ n2
n −n
1 n=2 n ln(n)
page 9 of 10 11. Let T2 (x) be the second degree Taylor polynomial for the function f (x) = ln(x) centered at a = 2 and R2 (x) = f (x) − P2 (x) . (8) a) Calculate T2 (x) .
(4) b) Find an upper bound on the remainder R2 (x) = f (x) − T2 (x) valid for any x ≥ 1 , using Taylor’s inequality.
(3) c) T2 (x) is the unique quadratic polynomial satisfying what three properties in terms of the graph of f (x) . (These are the defining properties of a Taylor polynomial.)
(4) 12. a) Use the geometric series formula to find a power series expansion for
(4) b) Use the expansion in part (a) to evaluate numbers.)
R 1 dx 0 1+x3
1 1+x3
.
. (Your answer will be an infinite sum of
page 10 of 10 (8) 13. Use series given on the cover sheet to find the first two nonzero terms of the Maclaurin series for xex 1 + x − x2
(7) 14. Use√a binomial expansion to find the first three nonzero terms of the Maclaurin series for f (x) = 1 + x2 .