MIT18_014F10_final_pr_ex

MIT18_014F10_final_pr_ex - PRACTICE PROBLEMS FOR THE FINAL...

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Unformatted text preview: PRACTICE PROBLEMS FOR THE FINAL EXAM (1) Determine each limit, if it exists: (a) lim x →∞ x sin(1 /x ) cos( π/ 2+1 /x ) (b) lim x e x − e − x → sin(3 x ) x sin( x 2 ) (c) lim x 2 log( x 3 +1) . (Hint: Use Taylor approximations rather than L’Hopital → here. You’ll save 20 minutes.) (2) Evaluate each integral or find an antiderivative: (a) x sin x cos xdx x +1 (b) 1 ( x 2 +2 x +2) 3 dx (c) − 1 x − 1 / 5 dx (d) sin 3 xdx dx (e) ∞ √ x (3) Determine whether the following series converge absolutely, converge con- ditionally, or diverge: (a) (log 1 n ) 5 (b) ( 2 n n !) 2 2 3 n (c) n n (d) ( − 1) n log n 2 (e) ( − 1) n 5 n n 3 +10 (4) Determine the radius of convergence for each of the following series: (a) x n n n ! n (b) 2 n 2 n x n n (c) ( n !) 2 x (2 n )! (5) Using power series already familiar to you from class, determine the power series each of the following functions. Also determine the radius of conver- gence....
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This note was uploaded on 01/18/2012 for the course MATH 18.014 taught by Professor Christinebreiner during the Fall '10 term at MIT.

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MIT18_014F10_final_pr_ex - PRACTICE PROBLEMS FOR THE FINAL...

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