※ 引述《victor7935 (victor)》之銘言：
原 PO 真的有想過問題 ?
這些題目是作業吧?
問題咀嚼過後有問題再發問吧
: 1. Show that
: x - (x^3)/6 < sinx for all x > 0
: 微1次後
: f(x) = sinx -x + (1/6)x^3
: f'(x)= cosx +1 + (1/2)x^2
: 然後.....
Note that f is continuous and f(0) = 0 .
f'(x) > 0 for every x .
: 2.
: Suppose that f is continuous on [a,b] and f(a)=f(b).
: Show that f has at least one critical point in (a,b).
f is continuous on [a,b] => f attains its maximum M & minimum m on [a,b]
If f has no critical point in (a,b) , then M = m ( ∵ f(a) = f(b) ) ,
i.e. f is a const. function on [a,b] →←
: 3.
: (a)
: Give an example of a nonconstant function that takes on both its
: absolute max and absolute min on every interval.
取 Dirichlet function D(x) 即可
: (b)
: Give an example of a nonconstant function that has an iinfinite
: number of distinct local max and an infinite number of distinct
: local min.
x , x 屬於 Z and x is even
f(x) = {
-x , x 屬於 Z and x is odd
0 , x 屬於 R\Z
: 4.Use
: 1 - (1/2)x^2 < cosx < 1- (1/2)x^2 + (1/24)x^4 for all x>0.
: to estimate cos 6度(the x above is in radians.)
π
6 ° = ------ rad ≒ 0.105 rad
30
=> cos 6 ° ≒ 0.989
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