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2c9a7bd..4aeff2d 100644 --- a/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex +++ b/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex @@ -241,7 +241,22 @@ $=e^{2\lim\limits_{x\to 0}\frac{f(x)-f(0)}{x-0}}=e^{2f'(0)}=e^6$。 \section{高阶导数} -\subsection{导数存在性} +求高阶导数基本上使用归纳法或莱布尼茨公式。 + +高阶导数基本公式: + +\begin{enumerate} + \item $(e^x)^{(n)}=e^x$。 + \item $(a^x)^{(n)}=a^x(\ln a)^n$。 + \item $(\ln x)^{(n)}=(-1)^{n-1}(n-1)!x^{-n}$。 + \item $\left(\dfrac{1}{1+x}\right)^{(n)}=(-1)^n\dfrac{n!}{(1+x)^n}$。 + \item $\left(\dfrac{1}{1-x}\right)^{(n)}=\dfrac{n!}{(1-x)^n}$。 + \item $(\sin x)^{(n)}=\sin\left(x+\dfrac{n\pi}{2}\right)$。 + \item $(\cos x)^{(n)}=\cos\left(x+\dfrac{n\pi}{2}\right)$。 + \item $\{f(ax+b)\}^{(n)}=a^nf^{(n)}(ax+b)$。 +\end{enumerate} + +\subsection{高阶导数存在性} \subsection{携带未知数的多项式求高阶导} @@ -303,6 +318,52 @@ $\dfrac{\textrm{d}^3x}{\textrm{d}y^3}=\dfrac{\textrm{d}\dfrac{\textrm{d}^2x}{\te \section{微分} +微分若是出单独的计算很可能是物理应用问题,如计算速度增量、面积增量等,但是对于数一而言单独考的概率不大。 + +微分一般都是微分不等式的形式进行出题,即含有微分的不等式证明。 + +\subsection{函数性态} + +包括单调性、凹凸性与最值。 + +\textbf{例题:}证明当$x>0$时$\ln\left(1+\dfrac{1}{x}\right)<\dfrac{1}{\sqrt{x(x+1)}}$。 + +证明:令$F(x)=\ln\left(1+\dfrac{1}{x}\right)-\dfrac{1}{\sqrt{x(x+1)}}$。 + +即证明$\sqrt{x(x+1)}\ln\left(1+\dfrac{1}{x}\right)<1$。 + +令$t=\dfrac{1}{x}$,即证$\ln(1+t)\sqrt{\dfrac{1}{t}\left(\dfrac{1}{t}+1\right)}<1$,$\ln(1+t)\sqrt{1+t}0$。\medskip + +则$F'(t)=1-\dfrac{\sqrt{1+t}}{1+t}-\dfrac{\ln(1+t)}{2\sqrt{1+t}}=1-\dfrac{2+\ln(1+t)}{2\sqrt{1+t}}>0$。 + +$F(t)$递增,所以$F(t)>F(0)=0$。 + +\subsection{常数变量} + +如果不等式中都是常数,可以将其中的一个或几个常数变量化再利用导数去证明。 + +\textbf{例题:}设$02\dfrac{b-a}{a+b}$。 + +因为左边含有$\dfrac{b}{a}$不好处理,所以右边分子分母同时除以$a$全部变成统一变量:$\ln\dfrac{b}{a}>2\dfrac{\dfrac{b}{a}-1}{1+\dfrac{b}{a}}$。然后令$x=\dfrac{b}{a}$,所以即需要证明$\ln x>2\dfrac{x-1}{1+x}$,$x>1$。 + +\subsection{中值定理} + +一般使用拉格朗日中值定理或泰勒公式。 + +\textbf{例题:}设$f(x)$在闭区间$[0,c]$上连续,其导数$f'(x)$在开区间$(0,c)$内存在且单调减少,又$f(0)=0$,证明$f(a+b)\leqslant f(a)+f(b)$,$0\leqslant a\leqslant b\leqslant a+b\leqslant c$。 + +因为所要证明的式子中含有$a$、$b$、$a+b$,$f(0)=0$,所以对这几个区间进行拉格朗日中值定理。 + +$f(a)-f(0)=f'(\xi_1)(a-0)$,$f(a+b)-f(b)=f'(\xi_2)(a+b-b)$。 + +从而$f(a)=f'(\xi_1)a$,$f(a+b)-f(b)=f'(\xi_2)a$。 + +又$f'(x)$单调减少,所以$f'(\xi_1)>f'(\xi_2)$。 + +$f(a)\geqslant f(a+b)-f(b)$,所以$f(a+b)\leqslant f(a)+f(b)$。 + \section{隐函数与参数方程} 隐函数与参数方程求导基本上只用记住:\medskip @@ -419,6 +480,84 @@ $\therefore k=\dfrac{\dfrac{\sqrt{2}}{2}}{\dfrac{3}{2}\cdot\sqrt{\dfrac{3}{2}}}= 当$2x-4=0$时即在$(2,7)$时曲率最大为2。 +\subsection{零点问题} +\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。 + +用于证明存在某一个零点。 + +\subsubsection{单调性} + +若$f(x)$在$(a,b)$内单调($f'(x)$存在且不恒等于0),则$f(x)=0$在$(a,b)$内至多有一个根。 + +用于证明只有一个零点。 + +\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。 + +\textbf{例题:}证明方程$2^x-x^2=1$有且仅有3个实根。 + +解:令$f(x)=2^x-x^2-1$,则$f'(x)=\ln22^x-2x$,$f''(x)=(\ln2)^22^x-2$,$f'''(x)=(\ln2)^32^x\neq 0$。 + +所以$f'''(x)=0$至多0个根。所以根据罗尔原话$f(x)=0$至多三个根。 + +又观察法$f(0)=0$,$f(1)=0$得到两个实根。 + +$f(4)=-1$,$f(5)=6$,所以$(4,5)$内存在一个实根,从而一共与三个根。 + +\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。 + +\textbf{例题:}若$3a^2-5b<0$,则方程$x^5+2ax^3+3bx+4c=0$()。 + +$A.\text{无实根}$\qquad$B.\text{有唯一实根}$\qquad$C.\text{有三个不同实根}$\qquad$D.\text{与五个不同实根}$ + +解:令$f(x)=x^5+2ax^3+3bx+4c$,该实系数奇次方程至少有一个根。 + +$f'(x)=5x^4+6ax^2+3b$,令$t=x^2$,$5t^2+6at+3b=0$。 + +$\Delta=36a^2-4\cdot5\cdot3b=36a^2-60b=12(3a^2-5b)<0$。 + +$\therefore f'(x)$无实根,所以$t=x^2$解不出来,所以$f'(x)\neq0$。 + +$f'(x)=0$至多0个根。所以根据罗尔原话$f(x)=0$至多一个根,又由上面至少一个根,所以只有一个根,选择$B$。 + +\subsubsection{函数含参导数不含参} + +参数是一个加在式子上的常数,函数求导后参数就被消掉了,所以可以在计算过程中不考虑参数,等到了最后的结果再讨论参数。 + +\textbf{例题:}设常数$k>0$,函数$f(x)=\ln x-\dfrac{x}{e}+k$在$(0,+\infty)$内的零点个数为()。 + +$A.3$\qquad$B.2$\qquad$C.1$\qquad$D.0$ + +解:$f'(x)=\dfrac{1}{x}-\dfrac{1}{e}$,令其为0,则$x=e$。 + +$x\in(0,e)$,$f'(x)>0$,$f(x)\nearrow$,$x\in(e,+\infty)$,$f'(x)<0$,$f(x)\searrow$。 + +又$f(e)=k>0$,$\lim\limits_{x\to0^+}f(x)=\lim\limits_{x\to0^+}(\ln x-\dfrac{x}{e}+k)=-\infty$,所以左边有一个根,$\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}(\ln x-\dfrac{x}{e}+k)=-\infty$,所以一共有两个根。 + +\subsubsection{函数导数含参} + +参数与自变量进行运算,从而求导后参数仍在式子中,计算时需要携带参数来思考。 + +\textbf{例题:}求方程$k\arctan x-x=0$的不同实根的个数,其中$k$为参数。 + +解:令$f(x)=k\arctan x-x$$,\because f(-x)=-f(x)$,所以$f(x)$是一个奇函数,所以可以只要考虑一边的情况。$x=0$是函数的一个根。 + +$f'(x)=\dfrac{k}{1+x^2}-1=\dfrac{k-1-x^2}{1+x^2}$。 + +若$k-1\leqslant0$即$k<1$则$f'(x)\leqslant0$,所以$f(x)$单调减少,从而只有一个根。 + +若$k-1>0$即$k>1$,令$f'(x)=0$,即$k-1-x^2=0$,$x=\sqrt{k-1}$。 + +$x\in(0,\sqrt{k-1})$,$f'(x)>0$,$f(x)\nearrow$。$x\in(\sqrt{k-1},+\infty)$,$f'(x)<0$,$f(x)\searrow$。 + +$\lim\limits_{x\to+\infty}(k\arctan x-x)=-\infty$,所以在0的右侧一定存在一个零点,同理左边也因为奇函数对称存在一个零点,所以一共有三个根。 \end{document} diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf 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