P317练习3
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# P317练习3
设 $\displaystyle x\neq 0$, $\displaystyle f(x)=\int\_0^x \sin\frac\{1\}\{t\}\mathrm\{ d\} t$, $\displaystyle f(0)=0$, 求证:$\displaystyle f'(0)=0$. (北京大学习题)
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\begin\{aligned\} \mbox\{原式\}=&\lim\_\{x\to 0\}\frac\{f(x)-f(0)\}\{x-0\} =\lim\_\{x\to 0\} \int\_0^x t^2\cdot \frac\{1\}\{t^2\}\sin\frac\{1\}\{t\}\mathrm\{ d\} t =\lim\_\{x\to 0\}\frac\{1\}\{x\}\int\_0^x t^2\mathrm\{ d\} \cos\frac\{1\}\{t\}\\\\ &=\lim\_\{x\to 0\}\frac\{1\}\{x\} \left\[\left.t^2\cos\frac\{1\}\{t\}\right|\_0^x -\int\_0^x 2t\cos\frac\{1\}\{t\} \mathrm\{ d\} t\right\] =\lim\_\{x\to 0\}\frac\{1\}\{x\}\left\\\\\ &\xlongequal[\tiny\mbox\{分中值\}]\{\tiny\mbox\{第一积\}\}\lim\_\{x\to 0\}\left\ =\lim\_\{x\to 0\}\left\ \xlongequal[\tiny\mbox\{无穷小\}]\{\tiny\mbox\{有界 $\displaystyle \cdot$\}\} 0. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
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