1020考研总结

几道积分不等式习题的解题思路

pingf posted @ Sat, 21 Aug 2010 03:59:22 -1100 in 考研 with tags 微积分 考研 , 5890 readers

例题

例1. f(x)在(a,b)上可导, \underset{[a,b]}{max}f^{\,\prime}(x)=M 且有f(a)=0证明\int_{a}^{b}f(x)dx<=\frac{M}{2}(b-a)^{2}

例2. f(x)在(a,b)上导函数连续,\underset{[a,b]}{max}f^{\,\prime}(x)=M 且有f(a)=f(b)=0证明 \int_{a}^{b}f(x)dx<=\frac{M}{4}(b-a)^{2}

例3. f(x)在(a,b)上二阶导函数连续,\underset{[a,b]}{max}f^{\,\prime\,\prime}(x)=M 且有f(a)=f(b)=0证明\vert\int_{a}^{b}f(x)dx\vert<=\frac{M}{12}(b-a)^{3}
解题思路

• 拉格朗日中值定理

f(x)-f(a)=(x-a)f^{\,\prime}(\xi)

• 分部积分法

\int_{a}^{b}udv=uv\vert_{a}^{b}-\int_{a}^{b}vdu 对于上面这类题目还要进行一些配凑,如\int_{a}^{b}f(x)d(x-a)或者是\int_{a}^{b}f(x)d(x-b)

• 泰勒公式

若某f(x)在(a,b)上有n+1阶导数,且有x_{0}\epsilon(a,b)时的\forall x\epsilon(a,b)f(x)=\underset{i=0}{\overset{n}{\sum}}\frac{f^{\,\prime}(x_{0})}{i\,!}(x-x_{0})^{i}+\frac{f^{\,(n+1)}(\xi)}{(n+1)!}(x-x_{0})^{n+1}其中\xi在x到x_{0}之间

然而对于上面的几题,泰勒公式要向下面这样用,

\forall x\epsilon(a,b)在x点展开,对于某个t\epsilon(a,b),则f(t)可表示为f(t)=\underset{i=0}{\overset{n}{\sum}}\frac{f^{\,\prime}(x)}{i\,!}(t-x)^{i}+\frac{f^{\,(n+1)}(\xi)}{(n+1)!}(t-x)^{n+1}其中\xi在x到t之间

而在上述几题中,将已知的f(x)=0点带入(如令t=a或t=a,t=b联立求解),求得f(x)与其高阶导数的关系,联立方程组,比较不等式后可得出结果

另外,上面三道题都可用泰勒公式求解

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