HMMT 二月 2010 · CALC 赛 · 第 2 题
HMMT February 2010 — CALC Round — Problem 2
题目详情
英文原题
- [ 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 4 th derivative of f , evaluated at 0.
′
解析
英文解析
- [ 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 4 th 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.
′