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Update differential-mean-value-theorem-and-applications-of-derivatives.tex

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Didnelpsun
2021-02-12 00:14:02 +08:00
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@@ -520,7 +520,7 @@ $\forall x\in U(x_0,\delta)$恒有$f(x)\leqslant f(x_0)$,则$f(x)$在$x_0$取
\item 若参数方程下:$x=\phi(t),y=\psi(t)$$\rm{d}s=\sqrt{\left(\dfrac{\rm{d}x}{\rm{d}t}\right)^2+\left(\dfrac{\rm{d}y}{\rm{d}t}\right)^2}\rm{d}t$\medskip\\$=\sqrt{\psi'^2(t)+\phi'^2(t)}\rm{d}t$,即$\rm{d}s=\sqrt{\psi'^2(t)+\phi'^2(t)}\rm{d}t$
\end{itemize}
\subsection{曲率与曲率半径}
\subsection{曲率}
曲率\textcolor{violet}{\textbf{定义:}}表明曲线在某一点的弯曲程度的数值,针对曲线上某个点的切线方向角对弧长的转动率,通过微分来定义,表明曲线偏离直线的程度。曲率越大,表示曲线的弯曲程度越大。
@@ -590,6 +590,7 @@ $\therefore\dfrac{\rm{d}\alpha}{\rm{d}x}=\dfrac{y''}{1+y'^2}\Rightarrow\rm{d}\al
$\therefore k=\left\lvert\dfrac{\rm{d}\alpha}{\rm{d}s}\right\rvert=\dfrac{\vert y''\vert}{(1+y'^2)^{\frac{3}{2}}}$
\subsection{曲率半径}
\begin{minipage}{0.5\linewidth}
$\bigcirc O$为函数$L$在点$X$处的曲率圆,该圆与$L$$X$处相切,切线为$T$