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HMMT 二月 2005 · CALC 赛 · 第 1 题

HMMT February 2005 — CALC Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Let f ( x ) = x + ax + b , with a 6 = b , and suppose the tangent lines to the graph of f at x = a and x = b are parallel. Find f (1). ∫ ∫ ∞ ∞ cos u sin u
解析
  1. Let f ( x ) = x + ax + b , with a 6 = b , and suppose the tangent lines to the graph of f at x = a and x = b are parallel. Find f (1). Solution: 1 ′ 2 2 2 2 2 Since f ( x ) = 3 x + a , we must have 3 a + a = 3 b + a . Then a = b , and since a 6 = b , a = − b . Thus f (1) = 1 + a + b = 1. ∫ ∫ ∞ ∞ cos u sin u