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 , 5906 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 $\forall x\epsilon(a,b)$ 有 $f(x)=\underset{i=0}{\overset{n}{\sum}}\frac{f^{\,\prime}(x_{0})}{i\,!}(x-x_{0})^{i}+\frac{f^{\,(n+1)}(\xi)}{(n+1)!}(x-x_{0})^{n+1}$ 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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