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b/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex @@ -337,10 +337,60 @@ $\dfrac{\textrm{d}^2y}{\textrm{d}x^2}=\dfrac{\textrm{d}\left(\dfrac{\textrm{d}y} \section{导数应用} -\subsection{单调性与凹凸性} +\subsection{单调性} + +\textbf{例题:}求$y=x+\vert\sin 2x\vert$的单调区间。 + +因为函数的定义域为$R$。 + +又$y=\left\{\begin{array}{lcl} + x+\sin 2x, & & n\pi\leqslant x\leqslant n\pi+\dfrac{\pi}{2} \\ + x-\sin 2x, & &n\pi+\dfrac{\pi}{2}\leqslant x\leqslant (n+1)\pi +\end{array}\right.$($n=0,\pm 1,\pm2,\cdots$)。 + +$\therefore y'=\left\{\begin{array}{lcl} + 1+2\cos 2x, & & n\pi\leqslant x\leqslant n\pi+\dfrac{\pi}{2} \\ + 1-2\cos 2x, & &n\pi+\dfrac{\pi}{2}\leqslant x\leqslant (n+1)\pi +\end{array}\right.$($n=0,\pm 1,\pm2,\cdots$)。 + +令$y'=0$,所以得到驻点为$x=n\pi+\dfrac{\pi}{3}$和$x=n\pi+\dfrac{5\pi}{6}$。 + +分割区间:$\left[n\pi,n\pi+\dfrac{\pi}{3}\right]$,$\left[n\pi+\dfrac{\pi}{3},n\pi+\dfrac{\pi}{2}\right]$,$\left[n\pi+\dfrac{\pi}{2},x=n\pi+\dfrac{5\pi}{6}\right]$, + +$\left[x=n\pi+\dfrac{5\pi}{6},(n+1)\pi\right]$($n=0,\pm 1,\pm2,\cdots$)。 + +当$x\in\left[n\pi,n\pi+\dfrac{\pi}{3}\right]$,$y'>0$,所以函数在区间上单调递增。 + +当$x\in\left[n\pi+\dfrac{\pi}{3},n\pi+\dfrac{\pi}{2}\right]$,$y'<0$,所以函数在区间上单调递减。 + +当$x\in\left[n\pi+\dfrac{\pi}{2},x=n\pi+\dfrac{5\pi}{6}\right]$,$y'>0$,所以函数在区间上单调递增。 + +当$x\in\left[x=n\pi+\dfrac{5\pi}{6},(n+1)\pi\right]$,$y'<0$,所以函数在区间上单调递减。 + +从而函数在$\left[\dfrac{k\pi}{2},\dfrac{k\pi}{2}+\dfrac{\pi}{3}\right]$时单调增加,在$\left[\dfrac{k\pi}{2}+\dfrac{\pi}{3},\dfrac{k\pi}{2}+\dfrac{\pi}{2}\right]$上单调减少($k=0,\pm 1,\pm2,\cdots$)。 + +\subsection{凹凸性} + +二阶导数为0处就是拐点。 + +\textbf{例题:}决定曲线$y=ax^3+bx^2+cx+d$中参数,使得$x=-2$处曲线有水平切线,$(1,-10)$为拐点,且点$(-2,44)$在曲线上。 + +$y'=3ax^2+2bx+c$,$y''=6ax+2b$。 + +因为$x=-2$处曲线有水平切线,即$y'\vert_{x=-2}=12a-4b+c=0$。 + +$(1,-10)$为拐点,代入:$y''\vert_{x=1}=6a+2b=0$,$y\vert_{x=1}=a+b+c+d=-10$。 + +又点$(-2,44)$在曲线上,所以$y\vert_{x=-2}=-8a+4b-2c+d=44$。 + +解得四个方程:$a=1$,$b=-3$,$c=-24$,$d=16$。 \subsection{极值与最值} +求极值需要考虑$y'$与点两边正负号,如果$y''$存在则可以考虑,$y''<0$则取极大值,$y''>0$则取极小值。 + +对于最值需要考虑极值和闭区间端点两个部分。 + \subsection{函数图像} \subsection{曲率}