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HMMT 二月 2008 · CALC 赛 · 第 5 题

HMMT February 2008 — CALC Round — Problem 5

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 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).
    ( ) − 1 n
    ∑ n
解析

英文解析

  1. [ 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).
    Answer: We have 3
    6 6 2 2 3 2 2 2 28
    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
    ( ) − 1 n
    ∑ n