Re: [考古] 88台大c的證明題
※ 引述《vivaonly (viva)》之銘言：
: 計算題乙 (2/pi)x〈 sinx 〈x , 當 0〈 x〈 (pi/2)
: 這題要怎麼證明呢??
: 請大大指教 謝謝!!!!
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 連續
x→0+ x→0+ x
π
所以 f(x) 在 [0 , ---] 上連續
2
π
Claim: x < tanx if 0 < x < ---
2
π
pf: 令 g(x) = tanx - x , 0 ≦ x ≦ ---
2
π
則 g'(x) = (secx)^2 - 1 > 0 , for all 0 < x < ---
2
π
當 0 < x < --- 時 , 由均值定理得
2
g(x) - g(0) = (g'(c))(x - 0) , 0 < c < x
因為 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
因為 f'(x) = ----------------
x^2
(cosx)(x - tanx) π
= ---------------- < 0 , for all 0 < x < ---
x^2 2
π
=> f(x) 在 [0 , ---] 上為嚴格遞減函數
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
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 61.66.173.21
討論串 (同標題文章)
本文引述了以下文章的的內容：
完整討論串 (本文為第 2 之 2 篇)：