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Unformatted text preview: 1. Find the areas between these curves x = y 2 + 2y - 4 and x = y + 2. 2. Find the volume of the solid formed by rotating the region between y = x2 and y = 2x about y = -2. 3. (a) Find the area enclosed by the curve r = 1 - cos(), 0 2. (b) Find its arclength. 4. Evaluate these integrals ex dx (a) e2x - 1 (b) (c)
0 2x sin-1 (x)dx
1 ln(x)dx 5. Determine whether these series converge or diverge. (a)
n=1 n sin 1 n (b) (c) (d) 1 n+ n n=1 n2 2n n=1 n=1 n+1 n -n2 x 6. Find the Maclaurin series of the function f (x) = integral. You need not prove this fact. 0 et - 1 dt. Note this is a proper t 7. Find the sum 5n . (Hint: express this as a value of a power series.) (n2 - n)6n n=2 8. (a) Find the Maclaurin series expansion of f (x) = arctan(x). Show how you arrived at your answer. "I memorized it," is NOT sufficient justification. (Hint: Differentiation formula 3 in the table.) (b) Use (a) to evaluate the derivatives (arctan(x))(n) (0) for all positive integers n. 9. (a) Find the radius of convergence of (n!)2 n x . (2n)! n=0 (b) Find the radius of convergence of the series of part (a). 2x t2 1 + u4 du, find F (x) and F (x). 10. If F (x) = f (t)dt where f (t) = u 1 1 ...
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This note was uploaded on 05/26/2008 for the course MCDB 120 taught by Professor Johncarlsoncarolbascom-slackfrankslack during the Fall '07 term at Yale.
- Fall '07