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Final Examination of Calculus, Jan. 9, 2004
Each question is 10 points.
1. Evaluate the following limits.
(a) lim
x→0+
1
x2 − 1 xtan x
. (b) lim
x→0
cos x cos 3x
1/x2
.
2. Find Z
x sec−1x2
dx.
3. Find
Z dx
ex2 + ex3 + ex6.
4. Find the volume of a solid torus which is obtained by rotating the circle (x − R)2+ y2 = r2, R > r > 0, about the y-axis.
5. If the infinite curve y = e−x, x ≥ 0, is rotated about the x-axis, find the area of the resulting surface.
6. Determine whether the series
∞
X
n=1
(−1)[√n] (n!)k
(kn)! is absolutely convergent, condi- tionally convergent, or divergent, where [√
n] is the largest integer no greater than √n, and k is a positive integer.
7. Determine whether the series
∞
X
n=1
(−1)n−1 √n
n − 1 is absolutely convergent, con- ditionally convergent, or divergent.
8. Compute f(13)(0) when f (x) = ex5ex.
9. Find the Taylor series of ln (2 + 2x − x2) centered at 1, and determine the interval of convergence of this Taylor series.
10. Use power series to approximate Z 1
1 2
cos x2 dxcorrect up to three decimal places.
Carefully justify that your answer is indeed correct up to three decimal places.(~
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