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[再寄小读者之数学篇](2014-06-03 微分、积分中值定理的应用)

时间:2014-06-06 20:56:57      阅读:358      评论:0      收藏:0      [点我收藏+]

fbubuko.com,布布扣bubuko.com,布布扣 [0,1]bubuko.com,布布扣bubuko.com,布布扣 上连续, 在 (0,1)bubuko.com,布布扣bubuko.com,布布扣 内二阶可导, 且

limbubuko.com,布布扣x0bubuko.com,布布扣bubuko.com,布布扣f(x)bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣xbubuko.com,布布扣2bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣 存在,bubuko.com,布布扣1bubuko.com,布布扣0bubuko.com,布布扣f(x)dx=f(1).bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣
证明: 存在 ξ(0,1)bubuko.com,布布扣bubuko.com,布布扣 , 使得 fbubuko.com,布布扣′′bubuko.com,布布扣(ξ)+2ξfbubuko.com,布布扣bubuko.com,布布扣(ξ)=0bubuko.com,布布扣bubuko.com,布布扣 .   

 

证明: 由 limbubuko.com,布布扣x0bubuko.com,布布扣bubuko.com,布布扣f(x)bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣xbubuko.com,布布扣2bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣 存在知 f(0)=0bubuko.com,布布扣bubuko.com,布布扣 , 而

fbubuko.com,布布扣bubuko.com,布布扣(0)=limbubuko.com,布布扣x0bubuko.com,布布扣bubuko.com,布布扣f(x)bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣xbubuko.com,布布扣2bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣?x=0.bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣
又由积分中值定理 (与书上的不同, 要变形, 证明利用微分中值定理),
? η(0,1),s.t. f(η)=bubuko.com,布布扣1bubuko.com,布布扣0bubuko.com,布布扣f(x)dx=f(1).bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣
再据 Rolle 定理,
? ζ(η,1),s.t. fbubuko.com,布布扣bubuko.com,布布扣(ζ)=0.bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣
F(x)=ebubuko.com,布布扣2xbubuko.com,布布扣fbubuko.com,布布扣bubuko.com,布布扣(x)bubuko.com,布布扣bubuko.com,布布扣 , 则
F(0)=F(ζ)=0.bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣
由 Rolle 定理,
? ξ(0,ζ),s.t. Fbubuko.com,布布扣bubuko.com,布布扣(ξ)=0.bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣

[再寄小读者之数学篇](2014-06-03 微分、积分中值定理的应用),布布扣,bubuko.com

[再寄小读者之数学篇](2014-06-03 微分、积分中值定理的应用)

原文:http://www.cnblogs.com/zhangzujin/p/3765824.html

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