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HMMT 二月 2010 · CALC 赛 · 第 2 题

HMMT February 2010 — CALC Round — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 3 ] Let f be a function such that f (0) = 1, f (0) = 2, and ′′ ′ f ( t ) = 4 f ( t ) − 3 f ( t ) + 1 for all t . Compute the 4th derivative of f , evaluated at 0. ′
解析
  1. [ 3 ] Let f be a function such that f (0) = 1, f (0) = 2, and ′′ ′ f ( t ) = 4 f ( t ) − 3 f ( t ) + 1 for all t . Compute the 4th derivative of f , evaluated at 0. ′′ (3) ′′ Answer: 54 Putting t = 0 gives f (0) = 6. By differentiating both sides, we get f ( t ) = 4 f ( t ) − ′ (3) (4) (3) ′′ (4) 3 f ( t ) and f (0) = 4 · 6 − 3 · 2 = 18. Similarly, f ( t ) = 4 f ( t ) − 3 f ( t ) and f (0) = 4 · 18 − 3 · 6 = 54. ′