HMMT 二月 2009 · CALC 赛 · 第 1 题

HMMT February 2009 — CALC Round — Problem 1

[ 3 ] Let f be a differentiable real-valued function defined on the positive real numbers. The tangent lines to the graph of f always meet the y -axis 1 unit lower than where they meet the function. If f (1) = 0, what is f (2)?

解析

[ 3 ] Let f be a differentiable real-valued function defined on the positive real numbers. The tangent lines to the graph of f always meet the y -axis 1 unit lower than where they meet the function. If f (1) = 0, what is f (2)? Answer: ln 2 Solution: The tangent line to f at x meets the y -axis at f ( x ) − 1 for any x , so the slope of the ′ 1 tangent line is f ( x ) = , and so f ( x ) = ln( x ) + C for some a . Since f (1) = 0, we have C = 0, and so x f ( x ) = ln( x ). Thus f (2) = ln(2).