HMMT 二月 2012 · 代数 · 第 1 题
HMMT February 2012 — Algebra — Problem 1
题目详情
英文原题
Let fbe the function such that f(x) =/braceleftbigg2x ifx≤1 2 2−2xifx >1 2 What is the total length of the graph of f(f(...f/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 2012 f′s(x)...)) from x= 0 to x= 1?
解析
英文解析
- Let f be the function such that
{
2 x if x ≤1
f ( x ) =2
2 − 2 x if x >1
What is the total length of the graph of f ( f ( ...f ( x ) ... )) from x = 0 to x = 1?2
︸ ︷︷ ︸
′
2012 f s
√
2012 n
Answer: 4 + 1 When there are n copies of f , the graph consists of 2 segments, each of whichngoes 1 / 2 units to the right, and alternately 1 unit up or down. So, the length is
√
√
n 1
2 1 + = 4 + 1 n
2 n
Taking n = 2012, the answer is 2
√
2012
4 + 1