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+重点是找到两点$f(a)=f(b)$。 + +\subsubsection{判断不等式} + +\paragraph{寻找原函数} \leavevmode \medskip 通过乘积求导公式$(uv)'=u'v+uv'$的逆运算来构造辅助函数。 @@ -174,6 +178,24 @@ $\therefore a=1,b=e$。 即证明什么就构造他的原函数为函数式子。 +\paragraph{介值定理} \leavevmode \medskip + +\textbf{例题:}设$f(x)$在$[0,3]$上连续,$(0,3)$上可导,且$f(0)+f(1)+f(2)=3$,$f(3)=1$,证明存在$\varepsilon\in(0,3)$,使得$f'(\varepsilon)=0$。 + +解:对于一个导数$f'(\varepsilon)=0$,只会想到罗尔定理和拉格朗日中值定理两种方式。由于这里没有差式,所以很可能是罗尔定理。所以必须找到$f(a)=f(b)$。 + +由于题目只给出特定条件而没有给出$f(x)$的表达形式,所以无法使用求原函数的方式证明。$f(3)=1$是唯一已知固定常数值的,所以想在$(0,3)$上找到一点$c$,使得$f(c)=f(3)=1$。 + +所以此时能用的条件就只有$f(0)+f(1)+f(2)=3$,如何转换为存在$f(c)=1$呢? + +已知$f(x)$在$[0,3]$上连续,$(0,3)$上可导,则一定存在最大值$M$,最小值$m$,使得$m\leqslant f(0)\leqslant M$,$m\leqslant f(1)\leqslant M$,$m\leqslant f(2)\leqslant M$。\medskip + +所以相加$m\leqslant\dfrac{f(0)+f(1)+f(2)}{3}=1\leqslant M$。 + +由介值定理,可得一定存在某点$c\in[0,2]$,使得$f(c)=\dfrac{f(0)+f(1)+f(2)}{3}=1$。 + +所以$f(c)=f(3)=1$,根据罗尔定理,得证。 + \subsubsection{零点情况} \paragraph{直接式子} \leavevmode \medskip @@ -354,7 +376,7 @@ $(1,-10)$为拐点,代入:$y''\vert_{x=1}=6a+2b=0$,$y\vert_{x=1}=a+b+c+d=- \subsubsection{零点定理} -若$f(x)$在$[a,b]$上连续,且$f(a)f(b)<0$,则$f(x)=0$在$(a,b)$内至少有一个根。其中$ab$是具体数也可以是无穷大。 +若$f(x)$在$[a,b]$上连续,且$f(a)f(b)<0$,则$f(x)=0$在$(a,b)$内至少有一个根。其中$ab$是具体数也可以是无穷大。如果是无穷大时需要求$f(x)$到正负无穷的极限值。 用于证明存在某一个零点。 @@ -364,11 +386,17 @@ $(1,-10)$为拐点,代入:$y''\vert_{x=1}=6a+2b=0$,$y\vert_{x=1}=a+b+c+d=- 用于证明只有一个零点。 +当用于证明有且仅有一个实根时需要将零点定理和单调性一同使用。 + +\textbf{例题:} + \subsubsection{罗尔原话} 若$f^{(n)}(x)=0$至多有$k$个根,则$f(x)=0$至多有$k+n$个根。是罗尔定理的推论。 -即若$f(x)=0$至少有两个根,则$f'(x)$至少有一个根。 +即若$f(x)=0$至少有两个根,则$f'(x)$至少有一个根。一般证明$f(x)$至多有$k$个实根,就要求$f(x)$的$k$次导。 + +由于罗尔原话只能推断出至多多少个实根,所以我们往往要带值进入$f(x)$计算出至少多少个实根。 \textbf{例题:}证明方程$2^x-x^2=1$有且仅有3个实根。 @@ -382,7 +410,9 @@ $f(4)=-1$,$f(5)=6$,所以$(4,5)$内存在一个实根,从而一共与三 \subsubsection{实系数奇次方程} -实系数奇次方程至少与一个实根。即$x^{2n+1}+a_1x^{2n}+\cdots+a_{2n}x+a_{2n+1}=0$至少与一个实根。 +实系数奇次方程至少有一个实根。即$x^{2n+1}+a_1x^{2n}+\cdots+a_{2n}x+a_{2n+1}=0$至少与一个实根。 + +这个判断法则往往要和罗尔原话一起使用确定是否有且仅有常数个实根。 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a/advanced-math/exercise/3-differentiation-of-functions-of-single-variable/differentiation-of-functions-of-single-variable.tex +++ b/advanced-math/exercise/3-differentiation-of-functions-of-single-variable/differentiation-of-functions-of-single-variable.tex @@ -403,7 +403,7 @@ $\varphi''(y)=\dfrac{\textrm{d}}{\textrm{d}x}\left[\dfrac{1}{f'(x)}\right]\cdot\ 若$y=1$,则$x=\dfrac{1}{2}$,则$\varphi''(1)=-\dfrac{f''\left(\dfrac{1}{2}\right)}{\left[f'\left(\dfrac{1}{2}\right)\right]^3}=-\dfrac{8e}{8e^3}=-\dfrac{1}{e^2}$。 -\section{微分} +\section{微分不等式} 微分若是出单独的计算很可能是物理应用问题,如计算速度增量、面积增量等,但是对于数一而言单独考的概率不大。 diff --git a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.pdf b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.pdf index 542c69323e2888359d613532afcffe5a0a20ad71..7c743bcbe61de6b52a4cef3794cea05b7d9b382d 100644 GIT binary patch delta 134094 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中值定理一般是在微分中使用,积分中也可能考到,但是重点是将定限积分化为两个常数的差的形式,所以基本上使用拉格朗日中值定理。 +或者使用积分中值定理:若$f(x)$在$[a,b]$上连续,则存在$\varepsilon\in[a,b]$,使得$\int_a^bf(x)\,\textrm{d}x=f(\varepsilon)(b-a)$。 + \textbf{例题:}设函数$f(x)$在$[0,3]$上连续,在$(0,3)$内有二阶导数,且$2f(0)=\int_0^2f(x)\,\textrm{d}x=f(2)+f(3)$。 证明:(1)存在$\eta\in(0,2)$,使得$f(\eta)=f(0)$;(2)存在$\varepsilon\in(0,3)$,使得$f''(\varepsilon)=0$。