21C-WQ02 - You must state the test used a n = 1-1 n 2 n n 3...

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Final 21C Winter 2002 1. (9) Compute f y where f (x,y) = xy 2 e x 3 sin( π y) . 2. a. (8) Suppose you know at some point that f x = 2, f y = -3, x s = -5, x t = 5, y s = 4, y t = -2. What is the value of f t ? b. (8) Suppose in addition that you know s u = 2, t u = 3, what is the value of f u ? 3. (13) Change the order of integration for the following integral and evaluate. 2 4 1 1 + y 2 dy dx. 0 2x 4. (12) Write the following integral in spherical coordinates. DO NOT EVALUATE 2 p 1 r r 2 dz dr d q . 0 0 r 2 5. (12) Find the critical points of f(x,y) = 6xy 2 - 2x 3 - 3y 4 + 1 6. (13) Given that (0,0) and (0,4) are critical points for z = 4y 3 + 12yx 2 - 24x 2 - 24 y 2 , determine whether each is a local min, max, or saddle.
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2 7. (25) Determine whether the following series converge absolutely, conditionally, or diverge.
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Unformatted text preview: You must state the test used. a. n = 1 (-1) n 2 n n 3 b. n = 1 n e-n 2 c. n = 1 sin ( n ) 8. (13) Find the Maclaurin series for f(x) = (5 + 2x)-1 using the Maclaurin formula for the terms a n 9. (12) For the series n = 1 (-1) n x n n 3 n a. What is the radius of convergence? b. Where does the series converge absolutely? c. Where does the series converge conditionally? 10. (10) Use Maclaurin series to compute lim x -> 0 1 - cos(x 2 ) x 4 11. (15) a. Use the principles developed in this course to estimate 1/2 1 1 + x 3 dx. You don't have to complete any computations that would normally be done on a calculator. b. Estimate the size of the error that results from doing your estimate in part a....
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