作者LuisSantos ( )
看板trans_math
标题Re: [考古] 88台大c的证明题
时间Wed Jul 9 09:18:34 2008
※ 引述《vivaonly (viva)》之铭言：
: 计算题乙 (2/pi)x〈 sinx 〈x , 当 0〈 x〈 (pi/2)
: 这题要怎麽证明呢??
: 请大大指教 谢谢!!!!
sinx π
------ , 0 < x ≦ ---
x 2
令 f(x) =
1 , x = 0
sinx
lim f(x) = lim ------ = 1 = f(0) => f(x) 在 x = 0 连续
x→0+ x→0+ x
π
所以 f(x) 在 [0 , ---] 上连续
2
π
Claim: x < tanx if 0 < x < ---
2
π
pf: 令 g(x) = tanx - x , 0 ≦ x ≦ ---
2
π
则 g'(x) = (secx)^2 - 1 > 0 , for all 0 < x < ---
2
π
当 0 < x < --- 时 , 由均值定理得
2
g(x) - g(0) = (g'(c))(x - 0) , 0 < c < x
因为 g(0) = 0 , 且 g'(c) > 0 , x > 0
π
所以 g(x) = (g'(c))(x) > 0 , for all 0 < x < ---
2
π
tanx - x > 0 , for all 0 < x < ---
2
π
x < tanx , for all 0 < x < ---
2
(x)(cosx) - sinx
因为 f'(x) = ----------------
x^2
(cosx)(x - tanx) π
= ---------------- < 0 , for all 0 < x < ---
x^2 2
π
=> f(x) 在 [0 , ---] 上为严格递减函数
2
2 π π
=> --- = f(---) < f(x) < f(0) = 1 , for all 0 < x < ---
π 2 2
2 sinx π
=> --- < ---- < 1 , for all 0 < x < ---
π x 2
2 π
=> (---)(x) < sinx < x , for all 0 < x < ---
π 2
--
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