HMMT 二月 2008 · CALC 赛 · 第 5 题

HMMT February 2008 — CALC Round — Problem 5

[ 4 ] Let f ( x ) = sin + cos for all real numbers x . Determine f (0) (i.e., f differentiated 4 4 2008 times and then evaluated at x = 0). ( ) n − 1 ∑ n

解析

[ 4 ] Let f ( x ) = sin + cos for all real numbers x . Determine f (0) (i.e., f differentiated 4 4 2008 times and then evaluated at x = 0). 3 Answer: We have 8 6 6 2 2 3 2 2 2 2 sin x + cos x = (sin x + cos x ) − 3 sin x cos x (sin x + cos x ) ( ) 3 3 1 − cos 4 x 2 2 2 = 1 − 3 sin x cos x = 1 − sin 2 x = 1 − 4 4 2 5 3 = + cos 4 x. 8 8 5 3 (2008) 3 3 It follows that f ( x ) = + cos x . Thus f ( x ) = cos x . Evaluating at x = 0 gives . 8 8 8 8 ( ) n − 1 ∑ n