HMMT 二月 2012 · 代数 · 第 1 题
HMMT February 2012 — Algebra — Problem 1
题目详情
- Let f be the function such that { 1 2 x if x ≤ 2 f ( x ) = 1 2 − 2 x if x > 2 What is the total length of the graph of f ( f ( ...f ( x ) ... )) from x = 0 to x = 1? ︸ ︷︷ ︸ ′ 2012 f s
解析
- Let f be the function such that { 1 2 x if x ≤ 2 f ( x ) = 1 2 − 2 x if x > 2 What is the total length of the graph of f ( f ( ...f ( x ) ... )) from x = 0 to x = 1? ︸ ︷︷ ︸ ′ 2012 f s √ n 2012 Answer: 4 + 1 When there are n copies of f , the graph consists of 2 segments, each of which n goes 1 / 2 units to the right, and alternately 1 unit up or down. So, the length is √ √ 1 n n 2 1 + = 4 + 1 2 n 2 Taking n = 2012, the answer is √ 2012 4 + 1