diff --git a/Math.lyx b/Math.lyx index 05534c7..bfb10ed 100644 --- a/Math.lyx +++ b/Math.lyx @@ -45723,15 +45723,69 @@ y=x+1. 曲率 \end_layout +\begin_layout Standard +弧微分（arc differential）通常指的是弧长的微小变化， +数学上一般用 +\end_layout + +\begin_layout Standard +\begin_inset Formula $ds$ +\end_inset + + 表示。 +在二维和三维空间中， +弧微分可以通过微积分的方法来求得。 + +\end_layout + \begin_layout Subsection 弧微分 \end_layout +\begin_layout Subsubsection +二维平面上的弧微分 +\end_layout + +\begin_layout Standard +在二维平面上， +给定曲线 +\begin_inset Formula $y=f(x)$ +\end_inset + +， +其弧长微分 +\begin_inset Formula $ds$ +\end_inset + + 由以下公式给出： + +\end_layout + \begin_layout Standard -弧微分公式: \begin_inset Formula \[ -ds=\sqrt{1+y'^{2}}dx +ds=\sqrt{dx^{2}+dy^{2}} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +由于 +\begin_inset Formula $dy=f'(x)dx$ +\end_inset + +， +所以可以改写为： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +ds=\sqrt{1+(f'(x))^{2}}\,dx \] \end_inset @@ -45997,10 +46051,6 @@ wide false sideways false status open -\begin_layout Plain Layout - -\end_layout - \begin_layout Plain Layout \begin_inset Graphics filename image/弧微分.png @@ -46021,10 +46071,118 @@ status open \end_layout -\begin_layout Plain Layout +\end_inset + + +\end_layout + +\begin_layout Paragraph +参数方程形式 +\end_layout + +\begin_layout Standard +如果曲线由参数方程 +\begin_inset Formula $x=x(t),y=y(t)$ +\end_inset + + 给出， +则弧微分为： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +ds=\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\,dt +\] + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +三维空间中的弧微分 +\end_layout + +\begin_layout Standard +对于三维曲线， +设其参数方程为： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +x=x(t),\quad y=y(t),\quad z=z(t) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +则弧微分为： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +ds=\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}+\left(\frac{dz}{dt}\right)^{2}}\,dt +\] + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +极坐标系中的弧微分 +\end_layout + +\begin_layout Standard +在极坐标系 +\begin_inset Formula $\ensuremath{(r,\theta)}$ +\end_inset + +中， +若曲线由 +\begin_inset Formula $\ensuremath{r=r(\theta)}$ +\end_inset + +表示， +则弧微分为： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +ds=\sqrt{\left(\frac{dr}{d\theta}\right)^{2}+r^{2}}\,d\theta +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +或者若直接给出 +\begin_inset Formula $\ensuremath{(r,\theta)}$ +\end_inset + +作为参数方程： \end_layout +\begin_layout Standard +\begin_inset Formula +\[ +ds=\sqrt{r^{2}+\left(\frac{dr}{d\theta}\right)^{2}}\,d\theta +\] + \end_inset @@ -46035,21 +46193,54 @@ status open \end_layout \begin_layout Standard -在 -\begin_inset Formula $xOy$ +曲率描述的是曲线在某一点的弯曲程度， +通常用 +\begin_inset Formula $\kappa$ +\end_inset + +（kappa）表示。 +曲率的倒数称为 曲率半径 +\begin_inset Formula $R$ +\end_inset + +， +即： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +R=\frac{1}{\kappa} +\] + \end_inset -坐标系中, + +\end_layout + +\begin_layout Standard +曲率可以帮助分析曲线的几何性质， +例如曲线的“弯曲程度”、拐点、轨迹的变化趋势等。 + +\end_layout + +\begin_layout Subsubsection +直角坐标系下的曲率 +\end_layout + +\begin_layout Standard +在直角坐标系中, 设曲线 \begin_inset Formula $C$ \end_inset 是光滑的(连续可导), - 在曲线 + 在曲线 \begin_inset Formula $C$ \end_inset - 上选定一点 + 上选定一点 \begin_inset Formula $M_{0}$ \end_inset @@ -46166,7 +46357,7 @@ K=\underset{\Delta s\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid \end_inset -.曲率是曲线弯曲程度的定量描述, +曲率是曲线弯曲程度的定量描述, 曲率大, 曲线的弯曲程度大； 去率小曲线的, @@ -46179,230 +46370,186 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \end_inset - + \end_layout -\begin_layout Subsection -曲率圆与曲率半径 +\begin_layout Standard +其中： + \end_layout \begin_layout Standard -设曲线 -\begin_inset Formula $y=f(x)$ +- +\begin_inset Formula $f'(x)$ \end_inset -在点 -\begin_inset Formula $M(x,y)$ -\end_inset + 是曲线的一阶导数， +表示切线的斜率。 -处的曲率为 -\begin_inset Formula $K(K\neq0)$ +\end_layout + +\begin_layout Standard +- +\begin_inset Formula $f''(x)$ \end_inset -. - 在点 -\begin_inset Formula $M$ + 是曲线的二阶导数， +表示曲率的变化趋势。 + +\end_layout + +\begin_layout Subparagraph +参数方程形式 +\end_layout + +\begin_layout Standard +如果曲线用参数方程表示： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +x=x(t),\quad y=y(t) +\] + \end_inset - 处的曲线的法线上, - 在凹的一侧取一点 -\begin_inset Formula $D$ + +\end_layout + +\begin_layout Standard +则曲率为： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\kappa=\frac{|x'y''-y'x''|}{(x'^{2}+y'^{2})^{\frac{3}{2}}} +\] + \end_inset -, - 使 -\begin_inset Formula $|DM|=\frac{1}{K}=\rho$ + +\end_layout + +\begin_layout Standard +其中： + +\end_layout + +\begin_layout Standard +- +\begin_inset Formula $x',y'$ \end_inset -. - 以 -\begin_inset Formula $D$ + 是对参数 +\begin_inset Formula $\ensuremath{t}$ \end_inset -为圆心, - -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none +的一阶导数（速度分量）。 -\begin_inset Formula $\rho$ +\end_layout + +\begin_layout Standard +- +\begin_inset Formula $x'',y''$ \end_inset + 是对参数 +\begin_inset Formula $\ensuremath{t}$ +\end_inset -\family default -\series default -\shape default -\size default -\emph default -\bar default -\strikeout default -\xout default -\uuline default -\uwave default -\noun default -\color inherit -为半径作圆, - 这个圆叫做曲线在点 -\begin_inset Formula $M$ -\end_inset +的二阶导数（加速度分量）。 -处的圆, - 曲率圆的圆心 -\begin_inset Formula $D$ -\end_inset +\end_layout -叫做曲线在点M处 -\series bold -曲率中心 -\series default -, - 曲率圆的半径 -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none +\begin_layout Subsubsection +三维曲线的曲率 +\end_layout -\begin_inset Formula $\rho$ -\end_inset +\begin_layout Standard +对于三维曲线， +设其参数方程为： +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\mathbf{r}(t)=(x(t),y(t),z(t)) +\] -\family default -\series default -\shape default -\size default -\emph default -\bar default -\strikeout default -\xout default -\uuline default -\uwave default -\noun default -\color inherit -叫做曲线在点 -\begin_inset Formula $M$ \end_inset -处的 -\series bold -曲率半径 -\series default -. + \end_layout \begin_layout Standard +则曲率的计算公式为： -\series bold -曲线上一点处的曲率半径与曲线在 该点处的曲率互为倒数. -\end_layout - -\begin_layout Subsection -渐屈线与渐伸线 \end_layout \begin_layout Standard -设已知曲线的方程是 -\begin_inset Formula $y=f(x)$ -\end_inset +\begin_inset Formula +\[ +\kappa=\frac{|\mathbf{r}'(t)\times\mathbf{r}''(t)|}{|\mathbf{r}'(t)|^{3}} +\] -, - 且其二阶导数 -\begin_inset Formula $y"$ \end_inset -在点 -\begin_inset Formula $x$ -\end_inset -不为零， -则曲线在对应点 -\begin_inset Formula $M(x,y)$ -\end_inset +\end_layout -的曲率中心的坐标为 -\begin_inset Formula -\[ -\left\{ \begin{array}{cc} -\alpha & =x-\frac{1+(y')^{2}}{y''}\\ -\beta & =y+\frac{1+(y')^{2}}{y''} -\end{array}\right. -\] +\begin_layout Standard +其中： + +\end_layout +\begin_layout Standard +\begin_inset Formula $\mathbf{r}'(t)$ \end_inset + 是速度向量。 \end_layout \begin_layout Standard -当点 -\begin_inset Formula $(x,f(x))$ +\begin_inset Formula $\mathbf{r}''(t)$ \end_inset - 沿曲线 -\begin_inset Formula $C$ -\end_inset + 是加速度向量。 - 移动时， -相应的曲率中心 -\begin_inset Formula $D$ -\end_inset +\end_layout -的轨迹曲线 -\begin_inset Formula $G$ +\begin_layout Standard +\begin_inset Formula $\times$ \end_inset -称为曲线 C的 -\series bold -渐屈线 -\series default -， -而曲线 -\begin_inset Formula $C$ -\end_inset + 表示向量叉积， +其结果的模长表示法向加速度的大小。 -称为曲线 -\begin_inset Formula $G$ -\end_inset +\end_layout -的 -\series bold -渐伸线 -\series default -. - +\begin_layout Paragraph +极坐标系下的曲率 \end_layout \begin_layout Standard -所以曲线 -\begin_inset Formula $y=f(x)$ +如果曲线在极坐标系中表示为 +\begin_inset Formula $r=r(\theta)$ \end_inset -的渐屈线的参数方程为 +， +则曲率为： + \end_layout \begin_layout Standard \begin_inset Formula \[ -\left\{ \begin{array}{cc} -\alpha & =x-\frac{1+y^{3}}{y''}\\ -\beta & =y+\frac{1+y^{3}}{y''} -\end{array}\right. +\kappa=\frac{|r^{2}+2(r')^{2}-rr''|}{(r^{2}+(r')^{2})^{\frac{3}{2}}} \] \end_inset @@ -46410,242 +46557,211 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \end_layout -\begin_layout Chapter -微分 +\begin_layout Standard +其中： + \end_layout -\begin_layout Definition -设函数 -\begin_inset Formula ${\displaystyle y=f(x)}$ +\begin_layout Standard +\begin_inset Formula $r'=\frac{dr}{d\theta}$ \end_inset -在某区间 -\begin_inset Formula $I$ -\end_inset + 是径向变化率。 -内有定义. - 对于 -\begin_inset Formula $I$ -\end_inset +\end_layout -内一点 -\begin_inset Formula $x_{0}$ +\begin_layout Standard +\begin_inset Formula $\ensuremath{r''=\frac{d^{2}r}{d\theta^{2}}}$ \end_inset -, - 当 -\begin_inset Formula $x_{0}$ -\end_inset +是径向加速度。 -变动到附近的 -\begin_inset Formula $x_{0}+\Delta x$ -\end_inset +\end_layout -(也在此区间内)时, - 如果函数的 -\series bold -增量 -\series default - -\begin_inset Formula $\Delta y=f(x_{0}+\Delta x)-f(x_{0})$ -\end_inset +\begin_layout Standard +\begin_inset Tabular + + + + + + +\begin_inset Text - 可表示为 -\begin_inset Formula $\Delta y=A\Delta x+o(\Delta x)$ -\end_inset +\begin_layout Plain Layout +曲线表示方式 +\end_layout -(其中 -\begin_inset Formula $A$ \end_inset + + +\begin_inset Text -是不依赖于 -\begin_inset Formula $\Delta x$ -\end_inset +\begin_layout Plain Layout +曲率公式 +\end_layout -的常数, - -\begin_inset Formula $o(\Delta x)$ \end_inset + + + + +\begin_inset Text -是比 -\begin_inset Formula $\Delta x$ +\begin_layout Plain Layout +\begin_inset Formula $\ensuremath{y=f(x)}$ \end_inset -\series bold -高阶的无穷小 -\series default -) 那么称函数 -\begin_inset Formula $f(x)$ -\end_inset +\end_layout -在点 -\begin_inset Formula $x_{0}$ \end_inset + + +\begin_inset Text -是可微的, - 且 -\begin_inset Formula $A\Delta x$ +\begin_layout Plain Layout +\begin_inset Formula $\kappa=\frac{|f''(x)|}{(1+(f'(x))^{2})^{\frac{3}{2}}}$ \end_inset -称作函数在点 -\begin_inset Formula $x_{0}$ -\end_inset -相应于自变量增量 -\begin_inset Formula $\Delta x$ +\end_layout + \end_inset + + + + +\begin_inset Text -的微分, - 记作 -\begin_inset Formula $\textrm{d}y$ +\begin_layout Plain Layout +参数方程 +\begin_inset Formula $\ensuremath{x(t),y(t)}$ \end_inset -, - 即 -\begin_inset Formula -\[ -{\displaystyle \textrm{d}y=A\Delta x} -\] + +\end_layout \end_inset + + +\begin_inset Text -, - -\begin_inset Formula $\textrm{d}y$ +\begin_layout Plain Layout +\begin_inset Formula $\kappa=\frac{|x'y''-y'x''|}{(x'^{2}+y'^{2})^{\frac{3}{2}}}$ \end_inset -是 -\begin_inset Formula $\Delta y$ -\end_inset + +\end_layout -的线性主部, -\begin_inset Formula $\Delta y$ \end_inset + + + + +\begin_inset Text -是变化的精确值， - -\begin_inset Formula $\textrm{d}y$ +\begin_layout Plain Layout +三维曲线 +\begin_inset Formula $\mathbf{r}(t)=(x(t),y(t),z(t))$ \end_inset -是变化的近似值 -\begin_inset Formula $\Delta y\approx\textrm{d}y$ -\end_inset -. \end_layout -\begin_layout Standard -通常把自变量 -\begin_inset Formula $x$ -\end_inset - -的增量 -\begin_inset Formula $\Delta x(\Delta x\rightarrow0)$ -\end_inset - -称为自变量的微分,记作 -\begin_inset Formula $\textrm{d}x$ \end_inset + + +\begin_inset Text -,即 -\begin_inset Formula ${\displaystyle \textrm{d}x=\Delta x}$ +\begin_layout Plain Layout +\begin_inset Formula $\kappa=\frac{|\mathbf{r}'(t)\times\mathbf{r}''(t)|}{|\mathbf{r}'(t)|^{3}}$ \end_inset -。 \end_layout -\begin_layout Standard -可微的函数, - 其微分等于导数乘以自变量的微分 -\begin_inset Formula $\textrm{d}x$ \end_inset + + + + +\begin_inset Text -, - 函数的微分与自变量的微分之商等于该函数的导数(因此,导数也叫做微商). - 即 -\begin_inset Formula $\textrm{d}y=f'(x)Δx=f'(x)\textrm{d}x=A\Delta x\approx\Delta y$ +\begin_layout Plain Layout +极坐标 +\begin_inset Formula $\ensuremath{r=r(\theta)}$ \end_inset - . -\begin_inset Formula $A$ -\end_inset -就是 -\begin_inset Formula $f(x)$ +\end_layout + \end_inset + + +\begin_inset Text -在 -\begin_inset Formula $x_{0}$ +\begin_layout Plain Layout +\begin_inset Formula $\ensuremath{\kappa=\frac{|r^{2}+2(r')^{2}-rr''|}{(r^{2}+(r')^{2})^{\frac{3}{2}}}}$ \end_inset -处的导数。 \end_layout -\begin_layout Standard -\begin_inset Formula $\frac{\textrm{d}y}{\textrm{d}x}=f'(x)$ \end_inset + + + + +\begin_inset Text -， -导数也叫做微商。 - -\end_layout +\begin_layout Plain Layout +圆（半径 +\begin_inset Formula $R$ +\end_inset -\begin_layout Standard -对于一元函数来说,可微与可导是完全等价的概念. +） \end_layout -\begin_layout Standard -对于一元函数: -\begin_inset Formula -\[ -\Delta y=A\Delta x+o(\Delta x) -\] - \end_inset + + +\begin_inset Text - -\begin_inset Formula -\[ -\frac{\Delta y}{\Delta x}=\frac{A\Delta x+o(\Delta x)}{\Delta x} -\] - +\begin_layout Plain Layout +\begin_inset Formula $\kappa=\frac{1}{R}$ \end_inset -\begin_inset Formula -\[ -\frac{\Delta y}{\Delta x}=\frac{A\Delta x+o(\Delta x)}{\Delta x} -\] +\end_layout \end_inset + + + + +\begin_inset Text - +\begin_layout Plain Layout +直线 \end_layout -\begin_layout Standard -\begin_inset Formula -\[ -\frac{\Delta y}{\Delta x}=A+\frac{o(\Delta x)}{\Delta x} -\] +\end_inset + + +\begin_inset Text +\begin_layout Plain Layout +\begin_inset Formula $\kappa=0$ \end_inset \end_layout -\begin_layout Standard -当 -\begin_inset Formula $\Delta x\rightarrow0$ \end_inset - -时 -\begin_inset Formula -\[ -\underset{\Delta x\rightarrow0}{\lim}\frac{\Delta y}{\Delta x}=A=f'(x_{0}) -\] + + + \end_inset @@ -46653,187 +46769,126 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \end_layout \begin_layout Example -有一块正方形的金属片， -它的边长原来是3， -受热后增加了 -\begin_inset Formula $\Delta x$ +求椭圆 +\begin_inset Formula $4x^{2}+y^{2}=4$ \end_inset - ， -问这块金属片的面积增加了多少? +在点 +\begin_inset Formula $(0,2)$ +\end_inset + +处的曲率. \end_layout -\begin_layout Standard -设加热前正方形金属片的面积为 ， -即 -\begin_inset Formula $A$ +\begin_layout Solution +由隐函数求导法,得 +\begin_inset Formula $8x+2yy'=0$ \end_inset -， -加热后正方形金属片的面积为 -\begin_inset Formula $A+\Delta A$ +, +\begin_inset Formula $y'=-\frac{4x}{y}$ \end_inset -， -即 -\begin_inset Formula $A+\Delta A=(x+\Delta x)^{2}$ +, +\begin_inset Formula $y'(0,2)=0$ \end_inset \end_layout -\begin_layout Standard -因此正方形金属片在加热后面积增加了 ： - -\begin_inset Formula $(x+\Delta x)^{2}-x^{2}=2x\Delta x+(\Delta x)^{2}\approx2x\Delta x$ +\begin_layout Solution +又 +\begin_inset Formula $y''=-\frac{4y-4xy'}{y^{2}}=-\frac{16}{y^{3}}$ \end_inset -. - -\end_layout - -\begin_layout Standard -在实际问题的计算中， -如果遇到几个不同阶的无穷小量之和， -为了简化计算, - 常常把高阶无穷小忽略不计, -\begin_inset Formula $(\Delta x)^{2}$ +, +\begin_inset Formula $y''(0,2)=-2$ \end_inset -是 -\begin_inset Formula $2x\Delta x\text{的高阶无穷小, 所以忽略(\Delta x)^{2}}$ -\end_inset -. - 即 -\begin_inset Formula -\[ -\Delta A=2x\Delta x -\] +\end_layout +\begin_layout Solution +\begin_inset Formula $\kappa=|\frac{-2}{(1+0^{2})^{\frac{3}{2}}}|=2$ \end_inset \end_layout \begin_layout Example -\begin_inset Formula $f(x)=x^{2}$ -\end_inset - -,求 -\begin_inset Formula $x=2$ +\begin_inset Formula $x^{2}+y^{2}=R^{2}$ \end_inset -处微分， - -\begin_inset Formula $\Delta x=0.01$ +的圆的曲率为 +\begin_inset Formula $\frac{1}{R}$ \end_inset -,改变量和微分值。 \end_layout \begin_layout Example -微分： - -\begin_inset Formula $dy=2x\Delta x$ +由隐函数求导法,得 +\begin_inset Formula $y'=-\frac{x}{y}$ \end_inset , -\begin_inset Formula $dy|_{x=2}4\Delta x$ +\begin_inset Formula $y''=-\frac{R^{3}}{y^{3}}$ \end_inset \end_layout \begin_layout Example -改变量： - -\begin_inset Formula $\Delta y=0.0401$ +\begin_inset Formula $\kappa=|\frac{y''}{(1+y'^{2})^{\frac{3}{2}}}|=\frac{1}{R}$ \end_inset \end_layout -\begin_layout Example -微分： - -\begin_inset Formula $dy|_{x=2,\Delta x=0.01}=0.04$ -\end_inset - - +\begin_layout Subsubsection +曲率圆与曲率半径 \end_layout \begin_layout Standard -\begin_inset Separator plain +设曲线 +\begin_inset Formula $y=f(x)$ \end_inset - -\end_layout - -\begin_layout Example -计算 -\begin_inset Formula $\sqrt{2}$ +在点 +\begin_inset Formula $M(x,y)$ \end_inset -的近似值 -\end_layout - -\begin_layout Example -因为 -\begin_inset Formula $\sqrt{2}=\sqrt{1.96+0.04}$ +处的曲率为 +\begin_inset Formula $K(K\neq0)$ \end_inset -， -取函数 -\begin_inset Formula $f(x)=\sqrt{x}$ +. + 在点 +\begin_inset Formula $M$ \end_inset -,设 -\begin_inset Formula $x_{0}=1.96$ + 处的曲线的法线上, + 在凹的一侧取一点 +\begin_inset Formula $D$ \end_inset -， -由于 -\begin_inset Formula $f'(x)=\frac{1}{2\sqrt{x}}$ +, + 使 +\begin_inset Formula $|DM|=\frac{1}{K}=\rho$ \end_inset -,所以 -\begin_inset Formula $f'(1.96)=\frac{1}{2.8}$ +. + 以 +\begin_inset Formula $D$ \end_inset - -\end_layout - -\begin_layout Example - -\family roman -\series medium -\shape up -\size normal -\emph off -\nospellcheck off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none -\begin_inset Formula $\textrm{d}y=f'(x)Δx=f'(x)\textrm{d}x=A\Delta x\approx\Delta y$ -\end_inset - - -\end_layout - -\begin_layout Example - +为圆心, + \family roman \series medium \shape up \size normal \emph off -\nospellcheck off \bar no \strikeout off \xout off @@ -46841,34 +46896,44 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \uwave off \noun off \color none -\begin_inset Formula $\Delta y=f(x_{0}+\Delta x)-f(x_{0})$ -\end_inset - - -\end_layout -\begin_layout Example -\begin_inset Formula $f(x)=f(x_{0}+\Delta x)=f(x_{0})+dy\approx f(x_{0})+\Delta y=f(x_{0})+f'(x_{0})Δx$ +\begin_inset Formula $\rho$ \end_inset -\end_layout - -\begin_layout Example -\begin_inset Formula $f(2)\approx f(1.96)+f'(1.96)\cdot(2-1.96)=1.4143$ +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +为半径作圆, + 这个圆叫做曲线在点 +\begin_inset Formula $M$ \end_inset +处的圆, + 曲率圆的圆心 +\begin_inset Formula $D$ +\end_inset -\end_layout - -\begin_layout Example - +叫做曲线在点M处 +\series bold +曲率中心 +\series default +, + 曲率圆的半径 \family roman \series medium \shape up \size normal \emph off -\nospellcheck off \bar no \strikeout off \xout off @@ -46876,582 +46941,706 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \uwave off \noun off \color none -\begin_inset Formula $\varDelta x$ + +\begin_inset Formula $\rho$ \end_inset -取值要小， -\begin_inset Formula $\varDelta x$ +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +叫做曲线在点 +\begin_inset Formula $M$ \end_inset -越大误差越大。 - +处的 +\series bold +曲率半径 +\series default +. \end_layout \begin_layout Standard -\begin_inset Separator plain -\end_inset - +\series bold +曲线上一点处的曲率半径与曲线在 该点处的曲率互为倒数. \end_layout -\begin_layout Example -\begin_inset Formula $\ln1.03=\ln(1+0.03)=\ln1+(\ln1)Δx\approx0.03$ -\end_inset +\begin_layout Subsection +渐屈线与渐伸线 +\end_layout +\begin_layout Standard +渐屈线和渐伸线是微分几何中的一对互相关联的曲线， +主要用于描述曲线的局部几何特性。 \end_layout -\begin_layout Section -微分的几何意义 -\end_layout +\begin_layout Standard +在曲线上选一定点 +\begin_inset Formula $S$ +\end_inset -\begin_layout Enumerate -函数 -\begin_inset Formula $y=f(x)$ +。 +有一动点 +\begin_inset Formula $P$ \end_inset -在点 -\begin_inset Formula $x_{0}$ +由 +\begin_inset Formula $S$ \end_inset -处的微分 -\begin_inset Formula $dy$ +出发沿曲线移动， +选在 +\begin_inset Formula $P$ \end_inset -在几何意义上表示曲线 -\begin_inset Formula $y=f(x)$ +的切线上的 +\begin_inset Formula $Q$ \end_inset - 在点 -\begin_inset Formula $M_{0}$ +， +使得曲线长 +\begin_inset Formula $SP$ \end_inset -处 +和直线段长 +\begin_inset Formula $PQ$ +\end_inset + +相同。 + \series bold -切线 +渐伸线 \series default -的 +就是Q的轨迹。 + +\end_layout + +\begin_layout Standard +曲线的 \series bold -纵坐标 +渐屈线 \series default -的改变量. +是该曲线每点的 +\series bold +曲率中心 +\series default +的集。 + \end_layout -\begin_layout Enumerate -当 -\begin_inset Formula $\mid\Delta x\mid$ +\begin_layout Standard +设已知曲线的方程是 +\begin_inset Formula $y=f(x)$ \end_inset -很小时, - 也就是说, - 在点 -\begin_inset Formula $M_{0}$ +, + 且其二阶导数 +\begin_inset Formula $y"$ \end_inset -的附近, - 可以用切线的纵坐标的增量 -\begin_inset Formula $dy$ +在点 +\begin_inset Formula $x$ \end_inset -来近似替代曲线纵坐标的改变量 -\begin_inset Formula $\Delta y$ +不为零， +则曲线在对应点 +\begin_inset Formula $M(x,y)$ \end_inset -,即 -\begin_inset Formula $\Delta y≈dy$ +的曲率中心的坐标为 +\begin_inset Formula +\[ +\left\{ \begin{array}{cc} +\alpha & =x-\frac{1+(y')^{2}}{y''}\\ +\beta & =y+\frac{1+(y')^{2}}{y''} +\end{array}\right. +\] + \end_inset -. + \end_layout \begin_layout Standard -微分是对函数的局部变化的一种线性描述。 -微分可以近似地描述当函数自变量的变化量取值作足够小时, - 函数的值是怎样改变的. +所以曲线 +\begin_inset Formula $y=f(x)$ +\end_inset + +的渐屈线的参数方程为 \end_layout -\begin_layout Section -运算法则 +\begin_layout Standard +\begin_inset Formula +\[ +\left\{ \begin{array}{cc} +\alpha & =x-\frac{1+y^{3}}{y''}\\ +\beta & =y+\frac{1+y^{3}}{y''} +\end{array}\right. +\] + +\end_inset + + \end_layout -\begin_layout Subsection -基本初等函数微分公式 +\begin_layout Subsubsection +圆的渐伸线 \end_layout \begin_layout Standard -\begin_inset Tabular - - - - - - -\begin_inset Text +圆的渐伸线会形成一个类似阿基米德螺线的图形。 -\begin_layout Plain Layout -导数公式 \end_layout -\end_inset - - -\begin_inset Text +\begin_layout Standard +在笛卡儿坐标系中， +一个圆的渐开线的参数方程可以写成： -\begin_layout Plain Layout -微分公式 -\begin_inset Formula ${\displaystyle \textrm{d}x=\Delta x}$ +\end_layout + +\begin_layout Standard +\begin_inset Formula ${\displaystyle \,x=a\left(\cos\ t+t\sin\ t\right)}$ \end_inset \end_layout -\end_inset - - - - -\begin_inset Text - -\begin_layout Plain Layout -\begin_inset Formula $(x^{a})'=ax^{a-1}(a为非零常数,x>0)$ +\begin_layout Standard +\begin_inset Formula ${\displaystyle \,y=a\left(\sin\ t-t\cos\ t\right)}$ \end_inset \end_layout +\begin_layout Standard +其中 +\begin_inset Formula ${\displaystyle \,a}$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(x^{a})=ax^{a-1}\cdot dx(a\text{为非零常数},x>0)$ -\end_inset +是圆的半径， +\begin_inset Formula ${\displaystyle \,t}$ +\end_inset +为参数 \end_layout -\end_inset - - - - -\begin_inset Text +\begin_layout Standard +在极坐标系中， -\begin_layout Plain Layout -\begin_inset Formula $(\sin x)'=\cos x$ +\begin_inset Formula ${\displaystyle \,r,\theta}$ \end_inset + 一个圆的渐开线的参数方程可以写成： \end_layout -\end_inset - - -\begin_inset Text - -\begin_layout Plain Layout - -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none -\begin_inset Formula $\mathrm{d}(\sin x)=\cos(x\cdot dx)$ +\begin_layout Standard +\begin_inset Formula ${\displaystyle \,r=a\sec\alpha}$ \end_inset \end_layout -\end_inset - - - - -\begin_inset Text - -\begin_layout Plain Layout -\begin_inset Formula $(\cos x)'=-\sin x$ +\begin_layout Standard +\begin_inset Formula ${\displaystyle \,\theta=\tan\alpha-\alpha}$ \end_inset \end_layout +\begin_layout Standard +其中 +\begin_inset Formula ${\displaystyle \,a}$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\cos x)=-\sin(x\cdot dx)$ +是圆的半径 +\begin_inset Formula ${\displaystyle \,\alpha}$ \end_inset - +为参数 \end_layout -\end_inset - - - - -\begin_inset Text +\begin_layout Standard +通常， +一个圆的渐开线常被写成写成： -\begin_layout Plain Layout -\begin_inset Formula $(\tan x)'=\sec^{2}x$ +\end_layout + +\begin_layout Standard +\begin_inset Formula ${\displaystyle \,r=a\sqrt{1+t^{2}}}$ \end_inset \end_layout +\begin_layout Standard +\begin_inset Formula ${\displaystyle \,\theta=\arctan\frac{\cos t+t\sin t}{\sin t-t\cos t}}$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\tan x)=\sec^{2}(x\cdot dx)$ -\end_inset +. +\end_layout +\begin_layout Standard +欧拉建议使用圆的渐开线作为齿轮的形状, + 这个设计普遍存在于目前使用， +称为渐开线齿轮。 \end_layout -\end_inset - - - - -\begin_inset Text +\begin_layout Chapter +微分 +\end_layout -\begin_layout Plain Layout -\begin_inset Formula $(\cot x)'=-\csc^{2}x$ +\begin_layout Definition +设函数 +\begin_inset Formula ${\displaystyle y=f(x)}$ \end_inset - -\end_layout - +在某区间 +\begin_inset Formula $I$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\cot x)=-\csc^{2}(x\cdot dx)$ +内有定义. + 对于 +\begin_inset Formula $I$ \end_inset +内一点 +\begin_inset Formula $x_{0}$ +\end_inset -\end_layout +, + 当 +\begin_inset Formula $x_{0}$ +\end_inset +变动到附近的 +\begin_inset Formula $x_{0}+\Delta x$ \end_inset - - - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $(\sec x)'=-\sec x\tan x$ +(也在此区间内)时, + 如果函数的 +\series bold +增量 +\series default + +\begin_inset Formula $\Delta y=f(x_{0}+\Delta x)-f(x_{0})$ \end_inset + 可表示为 +\begin_inset Formula $\Delta y=A\Delta x+o(\Delta x)$ +\end_inset -\end_layout +(其中 +\begin_inset Formula $A$ +\end_inset +是不依赖于 +\begin_inset Formula $\Delta x$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\sec x)=-\sec x\tan(x\cdot dx)$ +的常数, + +\begin_inset Formula $o(\Delta x)$ \end_inset +是比 +\begin_inset Formula $\Delta x$ +\end_inset -\end_layout +\series bold +高阶的无穷小 +\series default +) 那么称函数 +\begin_inset Formula $f(x)$ \end_inset - - - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $(\csc x)'=-\csc x\cot x$ +在点 +\begin_inset Formula $x_{0}$ \end_inset +是可微的, + 且 +\begin_inset Formula $A\Delta x$ +\end_inset -\end_layout +称作函数在点 +\begin_inset Formula $x_{0}$ +\end_inset +相应于自变量增量 +\begin_inset Formula $\Delta x$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\csc x)=-\csc x\cot(x\cdot dx)$ +的微分, + 记作 +\begin_inset Formula $\textrm{d}y$ \end_inset +, + 即 +\begin_inset Formula +\[ +{\displaystyle \textrm{d}y=A\Delta x} +\] -\end_layout +\end_inset +, + +\begin_inset Formula $\textrm{d}y$ \end_inset - - - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $(a^{x})'=a^{x}\ln a(a>0,a\neq1)$ +是 +\begin_inset Formula $\Delta y$ \end_inset +的线性主部, +\begin_inset Formula $\Delta y$ +\end_inset -\end_layout +是变化的精确值， +\begin_inset Formula $\textrm{d}y$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(a^{x})=a^{x}\ln(a\cdot dx)(a>0,a\neq1)$ +是变化的近似值 +\begin_inset Formula $\Delta y\approx\textrm{d}y$ \end_inset - +. \end_layout +\begin_layout Standard +通常把自变量 +\begin_inset Formula $x$ \end_inset - - - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $(e^{x})'=e^{x}$ +的增量 +\begin_inset Formula $\Delta x(\Delta x\rightarrow0)$ \end_inset - -\end_layout - +称为自变量的微分,记作 +\begin_inset Formula $\textrm{d}x$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(e^{x})=e^{x}\cdot dx$ +,即 +\begin_inset Formula ${\displaystyle \textrm{d}x=\Delta x}$ \end_inset +。 \end_layout +\begin_layout Standard +可微的函数, + 其微分等于导数乘以自变量的微分 +\begin_inset Formula $\textrm{d}x$ \end_inset - - - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $(log_{a}x)'=\frac{1}{x\ln a}(a>0,a\neq1)$ +, + 函数的微分与自变量的微分之商等于该函数的导数(因此,导数也叫做微商). + 即 +\begin_inset Formula $\textrm{d}y=f'(x)Δx=f'(x)\textrm{d}x=A\Delta x\approx\Delta y$ \end_inset + . +\begin_inset Formula $A$ +\end_inset -\end_layout - +就是 +\begin_inset Formula $f(x)$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(log_{a}x)=\frac{1}{x\ln a}\cdot dx(a>0,a\neq1)$ +在 +\begin_inset Formula $x_{0}$ \end_inset +处的导数。 \end_layout +\begin_layout Standard +\begin_inset Formula $\frac{\textrm{d}y}{\textrm{d}x}=f'(x)$ \end_inset - - - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $(\ln x)'=\frac{1}{x}$ -\end_inset +， +导数也叫做微商。 +\end_layout +\begin_layout Standard +对于一元函数来说,可微与可导是完全等价的概念. \end_layout +\begin_layout Standard +对于一元函数: +\begin_inset Formula +\[ +\Delta y=A\Delta x+o(\Delta x) +\] + \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\ln x)=\frac{1}{x}\cdot dx$ + +\begin_inset Formula +\[ +\frac{\Delta y}{\Delta x}=\frac{A\Delta x+o(\Delta x)}{\Delta x} +\] + \end_inset -\end_layout +\begin_inset Formula +\[ +\frac{\Delta y}{\Delta x}=\frac{A\Delta x+o(\Delta x)}{\Delta x} +\] \end_inset - - - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $(\arcsin x)'=-\frac{1}{\sqrt{1-x^{2}}}$ + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\frac{\Delta y}{\Delta x}=A+\frac{o(\Delta x)}{\Delta x} +\] + \end_inset \end_layout +\begin_layout Standard +当 +\begin_inset Formula $\Delta x\rightarrow0$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\arcsin x)=-\frac{1}{\sqrt{1-x^{2}}}\cdot dx$ +时 +\begin_inset Formula +\[ +\underset{\Delta x\rightarrow0}{\lim}\frac{\Delta y}{\Delta x}=A=f'(x_{0}) +\] + \end_inset \end_layout -\end_inset - - - - -\begin_inset Text +\begin_layout Standard +\begin_inset Float figure +placement document +alignment document +wide false +sideways false +status open \begin_layout Plain Layout -\begin_inset Formula $(\arccos x)'=-\frac{1}{\sqrt{1-x^{2}}}$ + +\end_layout + +\begin_layout Plain Layout +\begin_inset Graphics + filename image/Dydx_zh.svg + \end_inset +\begin_inset Caption Standard + +\begin_layout Plain Layout + \end_layout \end_inset - - -\begin_inset Text + + +\end_layout \begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\arccos x)=-\frac{1}{\sqrt{1-x^{2}}}\cdot dx$ + +\end_layout + \end_inset \end_layout +\begin_layout Example +有一块正方形的金属片， +它的边长原来是3， +受热后增加了 +\begin_inset Formula $\Delta x$ \end_inset - - - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $(\arctan x)'=\frac{1}{1+x^{2}}$ + ， +问这块金属片的面积增加了多少? +\end_layout + +\begin_layout Standard +设加热前正方形金属片的面积为 ， +即 +\begin_inset Formula $A$ +\end_inset + +， +加热后正方形金属片的面积为 +\begin_inset Formula $A+\Delta A$ +\end_inset + +， +即 +\begin_inset Formula $A+\Delta A=(x+\Delta x)^{2}$ \end_inset \end_layout +\begin_layout Standard +因此正方形金属片在加热后面积增加了 ： + +\begin_inset Formula $(x+\Delta x)^{2}-x^{2}=2x\Delta x+(\Delta x)^{2}\approx2x\Delta x$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\arctan x)=\frac{1}{1+x^{2}}\cdot dx$ +. + +\end_layout + +\begin_layout Standard +在实际问题的计算中， +如果遇到几个不同阶的无穷小量之和， +为了简化计算, + 常常把高阶无穷小忽略不计, +\begin_inset Formula $(\Delta x)^{2}$ +\end_inset + +是 +\begin_inset Formula $2x\Delta x\text{的高阶无穷小, 所以忽略(\Delta x)^{2}}$ +\end_inset + +. + 即 +\begin_inset Formula +\[ +\Delta A=2x\Delta x +\] + \end_inset \end_layout +\begin_layout Example +\begin_inset Formula $f(x)=x^{2}$ \end_inset - - - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $(\mathrm{arccot}x)'=-\frac{1}{1+x^{2}}$ +,求 +\begin_inset Formula $x=2$ +\end_inset + +处微分， + +\begin_inset Formula $\Delta x=0.01$ \end_inset +,改变量和微分值。 \end_layout +\begin_layout Example +微分： + +\begin_inset Formula $dy=2x\Delta x$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout -\begin_inset Formula $\mathrm{d}(\mathrm{arccot}x)=-\frac{1}{1+x^{2}}\cdot dx$ +, +\begin_inset Formula $dy|_{x=2}4\Delta x$ \end_inset \end_layout -\end_inset - - - +\begin_layout Example +改变量： +\begin_inset Formula $\Delta y=0.0401$ \end_inset \end_layout -\begin_layout Subsection -四则运算法则 +\begin_layout Example +微分： + +\begin_inset Formula $dy|_{x=2,\Delta x=0.01}=0.04$ +\end_inset + + \end_layout \begin_layout Standard -如果设函数 -\begin_inset Formula $u$ +\begin_inset Separator plain \end_inset -、 -\begin_inset Formula $v$ + +\end_layout + +\begin_layout Example +计算 +\begin_inset Formula $\sqrt{2}$ \end_inset -可微,可导,那么: +的近似值 \end_layout -\begin_layout Standard -\begin_inset Tabular - - - - - - -\begin_inset Text +\begin_layout Example +因为 +\begin_inset Formula $\sqrt{2}=\sqrt{1.96+0.04}$ +\end_inset -\begin_layout Plain Layout +， +取函数 +\begin_inset Formula $f(x)=\sqrt{x}$ +\end_inset + +,设 +\begin_inset Formula $x_{0}=1.96$ +\end_inset + +， +由于 +\begin_inset Formula $f'(x)=\frac{1}{2\sqrt{x}}$ +\end_inset + +,所以 +\begin_inset Formula $f'(1.96)=\frac{1}{2.8}$ +\end_inset + + +\end_layout + +\begin_layout Example \family roman \series medium \shape up \size normal \emph off +\nospellcheck off \bar no \strikeout off \xout off @@ -47459,21 +47648,55 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \uwave off \noun off \color none -导数公式 +\begin_inset Formula $\textrm{d}y=f'(x)Δx=f'(x)\textrm{d}x=A\Delta x\approx\Delta y$ +\end_inset + + +\end_layout + +\begin_layout Example + +\family roman +\series medium +\shape up +\size normal +\emph off +\nospellcheck off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $\Delta y=f(x_{0}+\Delta x)-f(x_{0})$ +\end_inset + + \end_layout +\begin_layout Example +\begin_inset Formula $f(x)=f(x_{0}+\Delta x)=f(x_{0})+dy\approx f(x_{0})+\Delta y=f(x_{0})+f'(x_{0})Δx$ \end_inset - - -\begin_inset Text -\begin_layout Plain Layout + +\end_layout + +\begin_layout Example +\begin_inset Formula $f(2)\approx f(1.96)+f'(1.96)\cdot(2-1.96)=1.4143$ +\end_inset + + +\end_layout + +\begin_layout Example \family roman \series medium \shape up \size normal \emph off +\nospellcheck off \bar no \strikeout off \xout off @@ -47481,6 +47704,129 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \uwave off \noun off \color none +\begin_inset Formula $\varDelta x$ +\end_inset + +取值要小， + +\begin_inset Formula $\varDelta x$ +\end_inset + +越大误差越大。 + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Example +\begin_inset Formula $\ln1.03=\ln(1+0.03)=\ln1+(\ln1)Δx\approx0.03$ +\end_inset + + +\end_layout + +\begin_layout Section +微分的几何意义 +\end_layout + +\begin_layout Enumerate +函数 +\begin_inset Formula $y=f(x)$ +\end_inset + +在点 +\begin_inset Formula $x_{0}$ +\end_inset + +处的微分 +\begin_inset Formula $dy$ +\end_inset + +在几何意义上表示曲线 +\begin_inset Formula $y=f(x)$ +\end_inset + + 在点 +\begin_inset Formula $M_{0}$ +\end_inset + +处 +\series bold +切线 +\series default +的 +\series bold +纵坐标 +\series default +的改变量. +\end_layout + +\begin_layout Enumerate +当 +\begin_inset Formula $\mid\Delta x\mid$ +\end_inset + +很小时, + 也就是说, + 在点 +\begin_inset Formula $M_{0}$ +\end_inset + +的附近, + 可以用切线的纵坐标的增量 +\begin_inset Formula $dy$ +\end_inset + +来近似替代曲线纵坐标的改变量 +\begin_inset Formula $\Delta y$ +\end_inset + +,即 +\begin_inset Formula $\Delta y≈dy$ +\end_inset + +. +\end_layout + +\begin_layout Standard +微分是对函数的局部变化的一种线性描述。 +微分可以近似地描述当函数自变量的变化量取值作足够小时, + 函数的值是怎样改变的. +\end_layout + +\begin_layout Section +运算法则 +\end_layout + +\begin_layout Subsection +基本初等函数微分公式 +\end_layout + +\begin_layout Standard +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +导数公式 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout 微分公式 \begin_inset Formula ${\displaystyle \textrm{d}x=\Delta x}$ \end_inset @@ -47496,7 +47842,33 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \begin_inset Text \begin_layout Plain Layout -\begin_inset Formula $(u\pm v)'=u'\pm v'$ +\begin_inset Formula $(x^{a})'=ax^{a-1}(a为非零常数,x>0)$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(x^{a})=ax^{a-1}\cdot dx(a\text{为非零常数},x>0)$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $(\sin x)'=\cos x$ \end_inset @@ -47521,7 +47893,7 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \uwave off \noun off \color none -\begin_inset Formula $d(u+v)=du+dv$ +\begin_inset Formula $\mathrm{d}(\sin x)=\cos(x\cdot dx)$ \end_inset @@ -47535,7 +47907,7 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \begin_inset Text \begin_layout Plain Layout -\begin_inset Formula $(uv)'=u'v+uv'$ +\begin_inset Formula $(\cos x)'=-\sin x$ \end_inset @@ -47547,7 +47919,7 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \begin_inset Text \begin_layout Plain Layout -\begin_inset Formula $d(uv)=v(du)+u(dv)$ +\begin_inset Formula $\mathrm{d}(\cos x)=-\sin(x\cdot dx)$ \end_inset @@ -47561,7 +47933,7 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \begin_inset Text \begin_layout Plain Layout -\begin_inset Formula $(Cu)'=Cu'$ +\begin_inset Formula $(\tan x)'=\sec^{2}x$ \end_inset @@ -47569,11 +47941,11 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \end_inset - + \begin_inset Text \begin_layout Plain Layout -\begin_inset Formula $d(Cu)=C(du)$ +\begin_inset Formula $\mathrm{d}(\tan x)=\sec^{2}(x\cdot dx)$ \end_inset @@ -47587,6 +47959,9 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \begin_inset Text \begin_layout Plain Layout +\begin_inset Formula $(\cot x)'=-\csc^{2}x$ +\end_inset + \end_layout @@ -47596,7 +47971,7 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \begin_inset Text \begin_layout Plain Layout -\begin_inset Formula ${\displaystyle d(au+bv)=d(au)+d(bv)=a(du)+b(dv)}$ +\begin_inset Formula $\mathrm{d}(\cot x)=-\csc^{2}(x\cdot dx)$ \end_inset @@ -47606,11 +47981,11 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le - + \begin_inset Text \begin_layout Plain Layout -\begin_inset Formula $(\frac{u}{v})'=\frac{u'v-uv'}{v^{2}}(v\neq0)$ +\begin_inset Formula $(\sec x)'=-\sec x\tan x$ \end_inset @@ -47618,11 +47993,11 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \end_inset - + \begin_inset Text \begin_layout Plain Layout -\begin_inset Formula $d\left(\frac{u}{v}\right)=\frac{v(du)-u(dv)}{v^{2}}(v\neq0)$ +\begin_inset Formula $\mathrm{d}(\sec x)=-\sec x\tan(x\cdot dx)$ \end_inset @@ -47631,442 +48006,516 @@ K=\underset{\Delta x\rightarrow0}{\lim}\mid\frac{\Delta\alpha}{\Delta s}\mid=\le \end_inset - + + +\begin_inset Text +\begin_layout Plain Layout +\begin_inset Formula $(\csc x)'=-\csc x\cot x$ \end_inset \end_layout -\begin_layout Example -求微分 -\begin_inset Formula $y=x^{2}+\ln x-3^{x}$ +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(\csc x)=-\csc x\cot(x\cdot dx)$ \end_inset \end_layout -\begin_layout Example -\begin_inset Formula $dy=dx^{2}+d\ln x-d3^{x}=2xdx+\frac{1}{x}dx-3^{x}\ln3dx=(2x+\frac{1}{x}-3^{x}\ln3)dx$ +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $(a^{x})'=a^{x}\ln a(a>0,a\neq1)$ \end_inset \end_layout -\begin_layout Example -\begin_inset Formula $dy=y'dx=(2x+\frac{1}{x}-3^{x}\ln3)dx$ +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(a^{x})=a^{x}\ln(a\cdot dx)(a>0,a\neq1)$ \end_inset \end_layout -\begin_layout Standard -\begin_inset Separator plain \end_inset + + + + +\begin_inset Text +\begin_layout Plain Layout +\begin_inset Formula $(e^{x})'=e^{x}$ +\end_inset -\end_layout -\begin_layout Subsection -复合函数微分法则 \end_layout -\begin_layout Standard -设 -\begin_inset Formula $y=f(u)$ \end_inset + + +\begin_inset Text -及 -\begin_inset Formula $u=g(x)$ +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(e^{x})=e^{x}\cdot dx$ \end_inset -都可导,则复合函数 -\begin_inset Formula $y=f[g(x)]$ -\end_inset -的微分为 -\begin_inset Formula -\[ -\mathrm{dy}=y',\mathrm{d}x=f'(u)g'(x)\mathrm{d}x -\] +\end_layout \end_inset + + + + +\begin_inset Text - -\begin_inset Formula -\[ -\mathrm{d}y=f'(u)du -\] - +\begin_layout Plain Layout +\begin_inset Formula $(log_{a}x)'=\frac{1}{x\ln a}(a>0,a\neq1)$ \end_inset \end_layout -\begin_layout Standard -一阶微分形式不变形 -\series bold +\end_inset + + +\begin_inset Text -\begin_inset Formula $\rightarrow$ +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(log_{a}x)=\frac{1}{x\ln a}\cdot dx(a>0,a\neq1)$ \end_inset -全微分形式不变性： \end_layout -\begin_layout Standard -\begin_inset Formula $u$ \end_inset + + + + +\begin_inset Text -是自变量： - -\begin_inset Formula $dy=f'(u)du$ +\begin_layout Plain Layout +\begin_inset Formula $(\ln x)'=\frac{1}{x}$ \end_inset \end_layout -\begin_layout Standard -\begin_inset Formula $u$ -\end_inset - -是函数， - -\begin_inset Formula $u=\varphi(x)$ \end_inset + + +\begin_inset Text -： - -\begin_inset Formula $dy=y_{x}^{'}dx=f'(u)\varphi'(x)dx=f'(u)du$ +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(\ln x)=\frac{1}{x}\cdot dx$ \end_inset \end_layout -\begin_layout Section -微分与导数的区别 -\end_layout +\end_inset + + + + +\begin_inset Text -\begin_layout Standard -导数是函数在一点处的变化率, - 代表曲线上相应点处切线的斜率, - 而微分是函数在一点处由自变量的增量引起的函数值变化量的主要部分. -\end_layout +\begin_layout Plain Layout +\begin_inset Formula $(\arcsin x)'=-\frac{1}{\sqrt{1-x^{2}}}$ +\end_inset -\begin_layout Section -全微分 -\end_layout -\begin_layout Subsection -二元函数全微分的概念 \end_layout -\begin_layout Standard -一般地, - 如果二元函数 -\begin_inset Formula $z=f(x,y)$ \end_inset + + +\begin_inset Text -在点 -\begin_inset Formula $(x,y)$ +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(\arcsin x)=-\frac{1}{\sqrt{1-x^{2}}}\cdot dx$ \end_inset -处的增量为 -\begin_inset Formula $\Delta x$ -\end_inset -与 -\begin_inset Formula $\Delta y$ -\end_inset +\end_layout -, - 当 -\begin_inset Formula $\Delta x\rightarrow0$ \end_inset + + + + +\begin_inset Text -与 -\begin_inset Formula $\Delta y\rightarrow0$ +\begin_layout Plain Layout +\begin_inset Formula $(\arccos x)'=-\frac{1}{\sqrt{1-x^{2}}}$ \end_inset -时, - 称 -\begin_inset Formula $A\Delta x+B\Delta y$ -\end_inset - (其中 -\begin_inset Formula $A$ -\end_inset +\end_layout -, - -\begin_inset Formula $B$ \end_inset + + +\begin_inset Text -与 -\begin_inset Formula $\Delta x$ +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(\arccos x)=-\frac{1}{\sqrt{1-x^{2}}}\cdot dx$ \end_inset -及 -\begin_inset Formula $\Delta y$ -\end_inset -无关)是函数 -\begin_inset Formula $z=f(x,y)$ -\end_inset +\end_layout -在点 -\begin_inset Formula $(x,y)$ \end_inset + + + + +\begin_inset Text -处的全微分, - 记作 -\begin_inset Formula $dz$ +\begin_layout Plain Layout +\begin_inset Formula $(\arctan x)'=\frac{1}{1+x^{2}}$ \end_inset - 或 -\begin_inset Formula $df(x,y)$ -\end_inset -, - 即 -\begin_inset Formula -\[ -dz=A\Delta x+B\Delta y -\] +\end_layout + +\end_inset + + +\begin_inset Text +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(\arctan x)=\frac{1}{1+x^{2}}\cdot dx$ \end_inset \end_layout -\begin_layout Standard -与一元函数类似,全微分 -\begin_inset Formula $dz$ \end_inset + + + + +\begin_inset Text -是全增量 -\begin_inset Formula $\Delta x$ +\begin_layout Plain Layout +\begin_inset Formula $(\mathrm{arccot}x)'=-\frac{1}{1+x^{2}}$ \end_inset - 的近似值, - 即 -\begin_inset Formula $\Delta z=dz$ -\end_inset -. - \end_layout -\begin_layout Standard -对于二元函数 -\begin_inset Formula $z=f(x,y)$ \end_inset + + +\begin_inset Text -的全微分,我们有以下结论: -\end_layout - -\begin_layout Standard -如果函数 -\begin_inset Formula $z=f(x,y)$ +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{d}(\mathrm{arccot}x)=-\frac{1}{1+x^{2}}\cdot dx$ \end_inset -在点 -\begin_inset Formula $(x,y)$ -\end_inset -处具有连续的偏导数, - 令 -\begin_inset Formula $A=\frac{\partial z}{\partial x}$ -\end_inset +\end_layout -, -\begin_inset Formula $B=\frac{\partial z}{\partial y}$ \end_inset + + + - , - 则二元函数 -\begin_inset Formula $z=f(x,y)$ \end_inset -在点 -\begin_inset Formula $(x,y)$ -\end_inset -处的全微分可以表示为 -\begin_inset Formula $dz=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y$ -\end_inset +\end_layout -. +\begin_layout Subsection +四则运算法则 \end_layout \begin_layout Standard -类似于一元函数,记 -\begin_inset Formula $\Delta x=dx$ +如果设函数 +\begin_inset Formula $u$ \end_inset -, -\begin_inset Formula $\Delta y=dy$ +、 +\begin_inset Formula $v$ \end_inset -,从而有 -\begin_inset Formula -\[ -dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy -\] +可微,可导,那么: +\end_layout -\end_inset +\begin_layout Standard +\begin_inset Tabular + + + + + + +\begin_inset Text +\begin_layout Plain Layout +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +导数公式 \end_layout -\begin_layout Standard -通常把二元函数的全微分等于它的两个偏微分之和称为二元函数的微分符合叠加原理. +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +微分公式 +\begin_inset Formula ${\displaystyle \textrm{d}x=\Delta x}$ +\end_inset + + \end_layout -\begin_layout Standard -二元函数全微分的定义及上述有关结论都可以推广到三元及三元以上的函数. +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $(u\pm v)'=u'\pm v'$ +\end_inset + + \end_layout -\begin_layout Standard +\end_inset + + +\begin_inset Text -\series bold -全微分形式不变性 -\series default -: - 设函数 -\begin_inset Formula $z=f(u,v)$ +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $d(u+v)=du+dv$ \end_inset -具有连续偏导数， -则有全微分: - + \end_layout -\begin_layout Standard -\begin_inset Formula -\[ -\mathrm{d}z=\frac{\partial z}{\partial x}\mathrm{d}u+\frac{\partial z}{\partial y}\mathrm{d}v -\] +\end_inset + + + + +\begin_inset Text +\begin_layout Plain Layout +\begin_inset Formula $(uv)'=u'v+uv'$ \end_inset \end_layout -\begin_layout Standard -不论 -\begin_inset Formula $u$ \end_inset + + +\begin_inset Text -、 -\begin_inset Formula $v$ +\begin_layout Plain Layout +\begin_inset Formula $d(uv)=v(du)+u(dv)$ \end_inset -是中间变量还是自变量， -全微分都保持上述形式， -这称为全微分形式不变性. + \end_layout -\begin_layout Example -求函数 -\begin_inset Formula $z=\ln(\frac{y}{x}+xy)$ +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $(Cu)'=Cu'$ \end_inset -的全微分。 \end_layout -\begin_layout Solution -\begin_inset Formula $\frac{\partial z}{\partial x}=\frac{-\frac{y}{x^{2}}+y}{\frac{y}{x}+xy}=\frac{x^{2}-1}{x+x^{3}}$ \end_inset + + +\begin_inset Text -, -\begin_inset Formula $\frac{\partial z}{\partial y}=\frac{-\frac{1}{x}+x}{\frac{y}{x}+xy}=\frac{x^{2}+1}{y+x^{2}y}$ +\begin_layout Plain Layout +\begin_inset Formula $d(Cu)=C(du)$ \end_inset \end_layout -\begin_layout Solution -\begin_inset Formula $dz=\frac{x^{2}-1}{x+x^{3}}dx+\frac{x^{2}+1}{y+x^{2}y}dy$ \end_inset + + + + +\begin_inset Text +\begin_layout Plain Layout \end_layout -\begin_layout Section -微分中值定理简介 +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula ${\displaystyle d(au+bv)=d(au)+d(bv)=a(du)+b(dv)}$ +\end_inset + + \end_layout -\begin_layout Subsection -费马引理 +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $(\frac{u}{v})'=\frac{u'v-uv'}{v^{2}}(v\neq0)$ +\end_inset + + \end_layout -\begin_layout Standard -设函数 -\begin_inset Formula $f(x)$ \end_inset + + +\begin_inset Text -在点 -\begin_inset Formula $x_{0}$ +\begin_layout Plain Layout +\begin_inset Formula $d\left(\frac{u}{v}\right)=\frac{v(du)-u(dv)}{v^{2}}(v\neq0)$ \end_inset -的某邻域 -\begin_inset Formula $U(x_{0})$ + +\end_layout + \end_inset + + + -内有定义 ( 邻域 -\begin_inset Formula $U(x_{0})$ \end_inset -的意思为以点 -\begin_inset Formula $x_{0}$ + +\end_layout + +\begin_layout Example +求微分 +\begin_inset Formula $y=x^{2}+\ln x-3^{x}$ \end_inset -为中心的开区间) , - 并且在x处可导,如果对任意的 -\begin_inset Formula $x\in U(x)$ + +\end_layout + +\begin_layout Example +\begin_inset Formula $dy=dx^{2}+d\ln x-d3^{x}=2xdx+\frac{1}{x}dx-3^{x}\ln3dx=(2x+\frac{1}{x}-3^{x}\ln3)dx$ \end_inset -,有 -\begin_inset Formula -\[ -f(x)\leqslant f(x_{0})\text{或}f(x)\geqslant f(x_{0}) -\] +\end_layout + +\begin_layout Example +\begin_inset Formula $dy=y'dx=(2x+\frac{1}{x}-3^{x}\ln3)dx$ \end_inset \end_layout \begin_layout Standard -那么 -\begin_inset Formula $f'(x_{0})=0$ +\begin_inset Separator plain \end_inset -. + \end_layout -\begin_layout Proof +\begin_layout Subsection +复合函数微分法则 +\end_layout + +\begin_layout Standard 设 -\begin_inset Formula $x\in U(x_{0})$ +\begin_inset Formula $y=f(u)$ \end_inset -, - -\begin_inset Formula $f(x)\leqslant f(x_{0})$ +及 +\begin_inset Formula $u=g(x)$ \end_inset -. - 对于 -\begin_inset Formula $x_{0}+\Delta x\in U(x_{0})$ +都可导,则复合函数 +\begin_inset Formula $y=f[g(x)]$ \end_inset -, - 如果有 +的微分为 \begin_inset Formula \[ -f(x_{0}+\Delta x)\leqslant f(x_{0}) +\mathrm{dy}=y',\mathrm{d}x=f'(u)g'(x)\mathrm{d}x \] \end_inset @@ -48074,7 +48523,7 @@ f(x_{0}+\Delta x)\leqslant f(x_{0}) \begin_inset Formula \[ -f(x_{0}+\Delta x)-f(x_{0})\leqslant0 +\mathrm{d}y=f'(u)du \] \end_inset @@ -48082,336 +48531,303 @@ f(x_{0}+\Delta x)-f(x_{0})\leqslant0 \end_layout -\begin_layout Proof -当 -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none - -\begin_inset Formula $\Delta x>0$ -\end_inset - -时 -\begin_inset Formula -\[ -\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}\leqslant0 -\] +\begin_layout Standard +一阶微分形式不变形 +\series bold +\begin_inset Formula $\rightarrow$ \end_inset +全微分形式不变性： \end_layout -\begin_layout Proof -这个比值的极限 -\begin_inset Formula $\underset{\Delta x\rightarrow0^{+}}{\lim}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$ +\begin_layout Standard +\begin_inset Formula $u$ \end_inset -存在且为 -\begin_inset Formula ${\displaystyle f'(x_{0})}$ +是自变量： + +\begin_inset Formula $dy=f'(u)du$ \end_inset -(根据导数的概念) -\end_layout -\begin_layout Proof -当 -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none +\end_layout -\begin_inset Formula $\Delta x<0$ +\begin_layout Standard +\begin_inset Formula $u$ \end_inset -时 -\begin_inset Formula -\[ -\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}\geqslant0 -\] +是函数， +\begin_inset Formula $u=\varphi(x)$ \end_inset +： -\end_layout - -\begin_layout Proof -这个比值的极限存在且为 -\begin_inset Formula ${\displaystyle f'(x_{0})}$ +\begin_inset Formula $dy=y_{x}^{'}dx=f'(u)\varphi'(x)dx=f'(u)du$ \end_inset -. - 所以 -\begin_inset Formula ${\displaystyle f'(x_{0})=0}$ -\end_inset -. \end_layout -\begin_layout Standard -通常称导数等于零的点称为函数的驻点(或稳定点,临界点). +\begin_layout Section +微分与导数的区别 \end_layout \begin_layout Standard -费马引理仅仅给出了函数在某个点为极值的必要条件。 -也就是说, - 些驻点不是极值, - 它们是拐点. +导数是函数在一点处的变化率, + 代表曲线上相应点处切线的斜率, + 而微分是函数在一点处由自变量的增量引起的函数值变化量的主要部分. \end_layout -\begin_layout Standard -\begin_inset Float figure -placement document -alignment document -wide false -sideways false -status open - -\begin_layout Plain Layout +\begin_layout Section +全微分 +\end_layout +\begin_layout Subsection +二元函数全微分的概念 \end_layout -\begin_layout Plain Layout -\begin_inset Graphics - filename image/费马引理.png - lyxscale 50 - scale 50 +\begin_layout Standard +一般地, + 如果二元函数 +\begin_inset Formula $z=f(x,y)$ +\end_inset +在点 +\begin_inset Formula $(x,y)$ \end_inset +处的增量为 +\begin_inset Formula $\Delta x$ +\end_inset -\begin_inset Caption Standard +与 +\begin_inset Formula $\Delta y$ +\end_inset -\begin_layout Plain Layout +, + 当 +\begin_inset Formula $\Delta x\rightarrow0$ +\end_inset -\end_layout +与 +\begin_inset Formula $\Delta y\rightarrow0$ +\end_inset +时, + 称 +\begin_inset Formula $A\Delta x+B\Delta y$ \end_inset + (其中 +\begin_inset Formula $A$ +\end_inset -\end_layout +, + +\begin_inset Formula $B$ +\end_inset -\begin_layout Plain Layout +与 +\begin_inset Formula $\Delta x$ +\end_inset -\end_layout +及 +\begin_inset Formula $\Delta y$ +\end_inset +无关)是函数 +\begin_inset Formula $z=f(x,y)$ \end_inset +在点 +\begin_inset Formula $(x,y)$ +\end_inset -\end_layout +处的全微分, + 记作 +\begin_inset Formula $dz$ +\end_inset -\begin_layout Subsection -罗尔定理 -\end_layout + 或 +\begin_inset Formula $df(x,y)$ +\end_inset + +, + 即 +\begin_inset Formula +\[ +dz=A\Delta x+B\Delta y +\] -\begin_layout Standard -设函数 -\begin_inset Formula $f(x)$ \end_inset \end_layout \begin_layout Standard -在闭区间 -\begin_inset Formula $[a,b]$ +与一元函数类似,全微分 +\begin_inset Formula $dz$ \end_inset -上连续, -\end_layout +是全增量 +\begin_inset Formula $\Delta x$ +\end_inset -\begin_layout Standard -在开区间 -\begin_inset Formula $(a,b)$ + 的近似值, + 即 +\begin_inset Formula $\Delta z=dz$ \end_inset -内可导, +. \end_layout \begin_layout Standard -且 -\begin_inset Formula $f(a)=f(b)$ +对于二元函数 +\begin_inset Formula $z=f(x,y)$ \end_inset -, - +的全微分,我们有以下结论: \end_layout \begin_layout Standard - -\series bold -则至少存在一点 -\begin_inset Formula $\xi\in(a,b)$ +如果函数 +\begin_inset Formula $z=f(x,y)$ \end_inset -,使得 -\begin_inset Formula $f'(\xi)=0$ +在点 +\begin_inset Formula $(x,y)$ \end_inset -. - -\end_layout - -\begin_layout Standard -罗尔定理的几何意义是非常明显的： -光滑连续曲线 -\begin_inset Formula $y=f(x),x\in[a,b]$ +处具有连续的偏导数, + 令 +\begin_inset Formula $A=\frac{\partial z}{\partial x}$ \end_inset -,其两端点的纵坐标相等, - 则曲线 -\begin_inset Formula $y=f(x)$ +, +\begin_inset Formula $B=\frac{\partial z}{\partial y}$ \end_inset -在区间 -\begin_inset Formula $(a,b)$ + , + 则二元函数 +\begin_inset Formula $z=f(x,y)$ \end_inset -内至少有一点 -\begin_inset Formula $\xi$ +在点 +\begin_inset Formula $(x,y)$ \end_inset -,使曲线在 -\begin_inset Formula $(\xi,f(\xi))$ +处的全微分可以表示为 +\begin_inset Formula $dz=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y$ \end_inset -处的切线平行于x轴. +. \end_layout \begin_layout Standard -罗尔定理并没有指明 -\begin_inset Formula $\xi$ -\end_inset - -的具体值, - 只说明了 -\begin_inset Formula $\xi$ +类似于一元函数,记 +\begin_inset Formula $\Delta x=dx$ \end_inset -的存在性,位于开区间 -\begin_inset Formula $(a,b)$ +, +\begin_inset Formula $\Delta y=dy$ \end_inset -之中, -\begin_inset Formula $\xi$ +,从而有 +\begin_inset Formula +\[ +dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy +\] + \end_inset -可以不唯一. + \end_layout -\begin_layout Proof -由于 -\begin_inset Formula $f(x)$ -\end_inset +\begin_layout Standard +通常把二元函数的全微分等于它的两个偏微分之和称为二元函数的微分符合叠加原理. +\end_layout -在闭区间 -\begin_inset Formula $[a,b]$ +\begin_layout Standard +二元函数全微分的定义及上述有关结论都可以推广到三元及三元以上的函数. +\end_layout + +\begin_layout Standard + +\series bold +全微分形式不变性 +\series default +: + 设函数 +\begin_inset Formula $z=f(u,v)$ \end_inset -上连续, - 根据闭区间上连续函数的最大值最小值定理, +具有连续偏导数， +则有全微分: -\begin_inset Formula $f(x)$ -\end_inset +\end_layout -在闭区间 -\begin_inset Formula $[a,b]$ -\end_inset +\begin_layout Standard +\begin_inset Formula +\[ +\mathrm{d}z=\frac{\partial z}{\partial x}\mathrm{d}u+\frac{\partial z}{\partial y}\mathrm{d}v +\] -上必定取得它的最大值 -\begin_inset Formula $M$ \end_inset -和最小值 -\begin_inset Formula $m$ -\end_inset -.这样,只有两种可能情形: \end_layout -\begin_layout Proof -1. - -\begin_inset Formula $M=m$ +\begin_layout Standard +不论 +\begin_inset Formula $u$ \end_inset -. - 这时 -\begin_inset Formula $f(x)$ +、 +\begin_inset Formula $v$ \end_inset -区间 -\begin_inset Formula $[a,b]$ -\end_inset +是中间变量还是自变量， +全微分都保持上述形式， +这称为全微分形式不变性. +\end_layout -上必然取相同的数值: - -\begin_inset Formula $f(x)=M$ +\begin_layout Example +求函数 +\begin_inset Formula $z=\ln(\frac{y}{x}+xy)$ \end_inset - . - 由此, -\begin_inset Formula $f'(x)$ -\end_inset +的全微分。 - = 0. \end_layout -\begin_layout Proof -2. - -\begin_inset Formula $M>m$ +\begin_layout Solution +\begin_inset Formula $\frac{\partial z}{\partial x}=\frac{-\frac{y}{x^{2}}+y}{\frac{y}{x}+xy}=\frac{x^{2}-1}{x+x^{3}}$ \end_inset , - 因为 -\begin_inset Formula $f(a)=f(b)$ +\begin_inset Formula $\frac{\partial z}{\partial y}=\frac{-\frac{1}{x}+x}{\frac{y}{x}+xy}=\frac{x^{2}+1}{y+x^{2}y}$ \end_inset \end_layout -\begin_layout Standard -罗尔定理中的条件 -\begin_inset Formula $f(a)=f(b)$ +\begin_layout Solution +\begin_inset Formula $dz=\frac{x^{2}-1}{x+x^{3}}dx+\frac{x^{2}+1}{y+x^{2}y}dy$ \end_inset -不易被满足, - 去掉这个条件, - 可得到应用范围更广的拉格朗日中值定理 -\end_layout - -\begin_layout Standard -\begin_inset Graphics - filename image/罗尔定理.png - lyxscale 50 - scale 50 - -\end_inset +\end_layout +\begin_layout Section +微分中值定理简介 \end_layout \begin_layout Subsection -拉格朗日中值定理(微分中值定理) +费马引理 \end_layout \begin_layout Standard @@ -48419,114 +48835,124 @@ status open \begin_inset Formula $f(x)$ \end_inset +在点 +\begin_inset Formula $x_{0}$ +\end_inset -\end_layout - -\begin_layout Standard -在闭区间 -\begin_inset Formula $[a,b]$ +的某邻域 +\begin_inset Formula $U(x_{0})$ \end_inset -上连续, -\end_layout +内有定义 ( 邻域 +\begin_inset Formula $U(x_{0})$ +\end_inset -\begin_layout Standard -在开区间 -\begin_inset Formula $(a,b)$ +的意思为以点 +\begin_inset Formula $x_{0}$ \end_inset -内可导, -\end_layout +为中心的开区间) , + 并且在x处可导,如果对任意的 +\begin_inset Formula $x\in U(x)$ +\end_inset -\begin_layout Standard +,有 +\begin_inset Formula +\[ +f(x)\leqslant f(x_{0})\text{或}f(x)\geqslant f(x_{0}) +\] -\series bold -则至少存在一点 -\begin_inset Formula $\xi\in(a,b)$ \end_inset -,使得 -\begin_inset Formula $f'(\xi)=\frac{f(b)-f(a)}{b-a}$ -\end_inset - \end_layout \begin_layout Standard -即如果曲线 -\begin_inset Formula $y=f(x)$ +那么 +\begin_inset Formula $f'(x_{0})=0$ \end_inset -满足拉格朗日中值定理, - 则曲线上必定存在一点的切线, - 与通过点 -\begin_inset Formula $(a,f(a))$ -\end_inset +. +\end_layout -和点 -\begin_inset Formula $(b,f(b))$ +\begin_layout Proof +设 +\begin_inset Formula $x\in U(x_{0})$ \end_inset -的直线的斜率相等. +, -\end_layout +\begin_inset Formula $f(x)\leqslant f(x_{0})$ +\end_inset -\begin_layout Standard -\begin_inset Float figure -placement document -alignment document -wide false -sideways false -status open +. + 对于 +\begin_inset Formula $x_{0}+\Delta x\in U(x_{0})$ +\end_inset -\begin_layout Plain Layout +, + 如果有 +\begin_inset Formula +\[ +f(x_{0}+\Delta x)\leqslant f(x_{0}) +\] -\end_layout +\end_inset -\begin_layout Plain Layout -\begin_inset Graphics - filename image/拉格朗日中值定理.jpg - lyxscale 50 - scale 50 -\end_inset +\begin_inset Formula +\[ +f(x_{0}+\Delta x)-f(x_{0})\leqslant0 +\] +\end_inset -\begin_inset Caption Standard -\begin_layout Plain Layout -拉格朗日中值定理 \end_layout -\end_inset +\begin_layout Proof +当 +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $\Delta x>0$ +\end_inset -\end_layout +时 +\begin_inset Formula +\[ +\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}\leqslant0 +\] \end_inset \end_layout -\begin_layout Standard -设 -\begin_inset Formula $x$ -\end_inset - -为区间 -\begin_inset Formula $[a,b]$ +\begin_layout Proof +这个比值的极限 +\begin_inset Formula $\underset{\Delta x\rightarrow0^{+}}{\lim}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$ \end_inset -内一点, - -\begin_inset Formula $x+\Delta x$ +存在且为 +\begin_inset Formula ${\displaystyle f'(x_{0})}$ \end_inset -为这区间内的另一点( -\begin_inset Formula $\Delta x>0$ -\end_inset +(根据导数的概念) +\end_layout -或 +\begin_layout Proof +当 \family roman \series medium \shape up @@ -48543,48 +48969,10 @@ status open \begin_inset Formula $\Delta x<0$ \end_inset - -\family default -\series default -\shape default -\size default -\emph default -\bar default -\strikeout default -\xout default -\uuline default -\uwave default -\noun default -\color inherit -), - 则公 式 -\series bold - -\begin_inset Formula $f'(\xi)=\frac{f(b)-f(a)}{b-a}$ -\end_inset - - -\series default -在区间 -\begin_inset Formula $[x,x+\Delta x]$ -\end_inset - -(当 -\begin_inset Formula $\Delta x>0$ -\end_inset - -时)或在区间 -\begin_inset Formula $[x+\Delta x,x]$ -\end_inset - -(当 -\begin_inset Formula $\Delta x<0$ -\end_inset - -时)上就成为 +时 \begin_inset Formula \[ -f(x+\Delta x)-f(x)=f'(x+\theta\Delta x)\cdot\Delta x\text{ }(0<\theta<1) +\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}\geqslant0 \] \end_inset @@ -48592,178 +48980,73 @@ f(x+\Delta x)-f(x)=f'(x+\theta\Delta x)\cdot\Delta x\text{ }(0<\theta<1) \end_layout -\begin_layout Standard -这个式子叫做限增量公式, - 给出了自变量取得有限增量 -\begin_inset Formula $\Delta x$ -\end_inset - -( -\begin_inset Formula $|\Delta x|$ +\begin_layout Proof +这个比值的极限存在且为 +\begin_inset Formula ${\displaystyle f'(x_{0})}$ \end_inset -不一定很小)时, - 函数增量 -\begin_inset Formula $\Delta y$ +. + 所以 +\begin_inset Formula ${\displaystyle f'(x_{0})=0}$ \end_inset -的准确表达因此, - 这个定理也叫做有限增量定理. +. \end_layout -\begin_layout Corollary -如果函数 -\begin_inset Formula $f(x)$ -\end_inset - -在区间 -\begin_inset Formula $I$ -\end_inset - -内的导数恒为0,那么函数 -\begin_inset Formula $f(x)$ -\end_inset +\begin_layout Standard +通常称导数等于零的点称为函数的驻点(或稳定点,临界点). +\end_layout -在该区间内恒为常数 +\begin_layout Standard +费马引理仅仅给出了函数在某个点为极值的必要条件。 +也就是说, + 些驻点不是极值, + 它们是拐点. \end_layout \begin_layout Standard -\begin_inset Separator plain -\end_inset +\begin_inset Float figure +placement document +alignment document +wide false +sideways false +status open +\begin_layout Plain Layout \end_layout -\begin_layout Corollary -如果函数 -\begin_inset Formula $f(x)$ -\end_inset +\begin_layout Plain Layout +\begin_inset Graphics + filename image/费马引理.png + lyxscale 50 + scale 50 -和 -\begin_inset Formula $g(x)$ \end_inset -在区间I内每一点的导数都相等,那么这两个函数在区间 -\begin_inset Formula $I$ -\end_inset -上至多相差一个常数 -\begin_inset Formula $C$ -\end_inset +\begin_inset Caption Standard -, - 即 -\begin_inset Formula $f(x)=g(x)+C,x\in1$ -\end_inset +\begin_layout Plain Layout - . \end_layout -\begin_layout Example -证 -\begin_inset Formula $\frac{x}{1+x}<\ln(1+x)0)$ \end_inset \end_layout -\begin_layout Example -设肺 -\begin_inset Formula $f(t)=\ln t,t\in[1,1+x]$ -\end_inset - -,则 -\begin_inset Formula $f(t)$ -\end_inset - -在闭区间 -\begin_inset Formula $[1,1+x](x>0)$ -\end_inset - -上满足拉格朗日中值定理的条件， -因此至少存在一点 -\family roman -\series medium -\shape up -\size normal -\emph off -\nospellcheck off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none - -\begin_inset Formula $\xi\in(1,1+x)$ -\end_inset - - -\family default -\series bold -\shape default -\size default -\emph default -\nospellcheck default -\bar default -\strikeout default -\xout default -\uuline default -\uwave default -\noun default -\color inherit - , -\series default -使得 -\series bold - -\begin_inset Formula $f'(\xi)=\frac{f(1+x)-f(x)}{1+x-1}$ -\end_inset - - , -\series default -即 -\family roman -\series medium -\shape up -\size normal -\emph off -\nospellcheck off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none - -\begin_inset Formula $\frac{1}{\xi}=\frac{\ln(1+x)}{x}$ -\end_inset - -. - 由于 -\begin_inset Formula $1<\xi<1+x$ -\end_inset +\begin_layout Plain Layout -,则 -\begin_inset Formula $\frac{1}{1+x}<\frac{1}{\xi}<1$ -\end_inset +\end_layout -,于是 -\begin_inset Formula $\frac{1}{1+x}<\frac{1}{\xi}=\frac{\ln(1+x)}{x}<1$ \end_inset -. - 因此 -\begin_inset Formula $\frac{x}{1+x}<\frac{\ln(1+x)}{x}m$ +\end_inset + +, + 因为 +\begin_inset Formula $f(a)=f(b)$ \end_inset \end_layout \begin_layout Standard -该定理说明对于 -\begin_inset Formula $\frac{0}{0}$ -\end_inset - -型(或 -\begin_inset Formula $\frac{\infty}{\infty}$ +罗尔定理中的条件 +\begin_inset Formula $f(a)=f(b)$ \end_inset -型)的未定型极限, - 在符合定理的条件下, - 可以通过对分子及分母分别求导数, - 然后再求极限的方法来确定. - 这种方法称为洛必达(L' Hospital)法则 +不易被满足, + 去掉这个条件, + 可得到应用范围更广的拉格朗日中值定理 \end_layout -\begin_layout Proof -\begin_inset Formula $\frac{0}{0}$ -\end_inset - -证明， -因为求 -\begin_inset Formula $\frac{f(x)}{g(x)}$ -\end_inset +\begin_layout Standard +\begin_inset Graphics + filename image/罗尔定理.png + lyxscale 50 + scale 50 -当 -\begin_inset Formula $x\rightarrow a$ \end_inset -时的极限与 -\begin_inset Formula $f(a)$ -\end_inset -及 -\begin_inset Formula $g(a)$ -\end_inset +\end_layout -无关, - 所以可以假定 -\begin_inset Formula $f(a)=g(a)=0$ -\end_inset +\begin_layout Subsection +拉格朗日中值定理(微分中值定理) +\end_layout -. - 由条件1,2 可知, - +\begin_layout Standard +设函数 \begin_inset Formula $f(x)$ \end_inset -和 -\begin_inset Formula $g(x)$ -\end_inset -点 -\begin_inset Formula $a$ -\end_inset +\end_layout -的某一邻域内是连续的.设 -\begin_inset Formula $x$ +\begin_layout Standard +在闭区间 +\begin_inset Formula $[a,b]$ \end_inset -是这邻域内的一点,那么在以 -\begin_inset Formula $x$ -\end_inset +上连续, +\end_layout -及 -\begin_inset Formula $a$ +\begin_layout Standard +在开区间 +\begin_inset Formula $(a,b)$ \end_inset -为端点的区间上,柯西中值定理的条件均满足,因此有 -\begin_inset Formula -\[ -\frac{f(x)}{g(x)}=\frac{f(x)-f(a)}{g(x)-g(a)}=\frac{f'(\xi)}{g'(\xi)}\text{ }(\xi\text{在}x\text{与}a\text{之间}) -\] +内可导, +\end_layout + +\begin_layout Standard +\series bold +则至少存在一点 +\begin_inset Formula $\xi\in(a,b)$ \end_inset -令 -\begin_inset Formula $x\rightarrow a$ +,使得 +\begin_inset Formula $f'(\xi)=\frac{f(b)-f(a)}{b-a}$ \end_inset -,并对上式两端求极限. + \end_layout -\begin_layout Proof -如果计算过程中 -\begin_inset Formula $\lim\frac{f'(x)}{g'(x)}$ -\end_inset - -仍然是 -\begin_inset Formula $\frac{0}{0}$ +\begin_layout Standard +即如果曲线 +\begin_inset Formula $y=f(x)$ \end_inset -型或者 -\begin_inset Formula $\frac{\infty}{\infty}$ +满足拉格朗日中值定理, + 则曲线上必定存在一点的切线, + 与通过点 +\begin_inset Formula $(a,f(a))$ \end_inset -,且 -\begin_inset Formula $f'(x)$ +和点 +\begin_inset Formula $(b,f(b))$ \end_inset -， +的直线的斜率相等. + +\end_layout -\begin_inset Formula $g'(x)$ -\end_inset +\begin_layout Standard +\begin_inset Float figure +placement document +alignment document +wide false +sideways false +status open -仍然满足洛必达法则中的条件， -可以继续使用洛必达法则中的条件。 +\begin_layout Plain Layout \end_layout -\begin_layout Proof -\begin_inset Formula $lim\frac{f(x)}{g(x)}=lim\frac{f'(x)}{g'(x)}=lim\frac{f''(x)}{g''(x)}$ +\begin_layout Plain Layout +\begin_inset Graphics + filename image/拉格朗日中值定理.jpg + lyxscale 50 + scale 50 + \end_inset -. + +\begin_inset Caption Standard + +\begin_layout Plain Layout +拉格朗日中值定理 \end_layout -\begin_layout Example -\begin_inset Formula $\underset{x\rightarrow1}{\lim}\frac{x^{5}-1}{2x^{5}-x-1}$ \end_inset \end_layout -\begin_layout Example -该极限为 -\begin_inset Formula $\frac{0}{0}$ -\end_inset - -型， -因此 -\begin_inset Formula $\underset{x\rightarrow1}{\lim}\frac{x^{5}-1}{2x^{5}-x-1}=\underset{x\rightarrow1}{\lim}\frac{(x^{5}-1)'}{(2x^{5}-x-1)'}=\frac{5}{9}$ \end_inset \end_layout \begin_layout Standard -\begin_inset Separator plain +设 +\begin_inset Formula $x$ \end_inset - -\end_layout - -\begin_layout Example -\begin_inset Formula $\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{\tan x-1}{\sec x+2}$ +为区间 +\begin_inset Formula $[a,b]$ \end_inset - -\end_layout - -\begin_layout Example -该极限为 -\begin_inset Formula $\frac{\infty}{\infty}$ +内一点, + +\begin_inset Formula $x+\Delta x$ \end_inset -型， -因此 -\begin_inset Formula $\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{\sec^{2}x}{\sec x\cdot\tan x}=\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{1}{\sin x}=1$ +为这区间内的另一点( +\begin_inset Formula $\Delta x>0$ \end_inset +或 +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none -\end_layout - -\begin_layout Standard -\begin_inset Separator plain +\begin_inset Formula $\Delta x<0$ \end_inset -\end_layout +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +), + 则公 式 +\series bold -\begin_layout Example -\begin_inset Formula $\underset{x\rightarrow\infty}{\lim}\frac{x^{n}}{e^{\lambda x}}$ +\begin_inset Formula $f'(\xi)=\frac{f(b)-f(a)}{b-a}$ \end_inset -, -\begin_inset Formula $n\in R^{+}$ + +\series default +在区间 +\begin_inset Formula $[x,x+\Delta x]$ \end_inset - -\end_layout - -\begin_layout Example -该极限为 -\begin_inset Formula $\frac{\infty}{\infty}$ +(当 +\begin_inset Formula $\Delta x>0$ \end_inset -型， -因此 -\begin_inset Formula $\underset{x\rightarrow\infty}{\lim}\frac{x^{n}}{e^{\lambda x}}=\underset{x\rightarrow\infty}{\lim}\frac{nx^{n-1}}{\lambda e^{\lambda x}}$ +时)或在区间 +\begin_inset Formula $[x+\Delta x,x]$ \end_inset +(当 +\begin_inset Formula $\Delta x<0$ +\end_inset -\end_layout +时)上就成为 +\begin_inset Formula +\[ +f(x+\Delta x)-f(x)=f'(x+\theta\Delta x)\cdot\Delta x\text{ }(0<\theta<1) +\] -\begin_layout Example -\begin_inset Formula $=\underset{x\rightarrow\infty}{\lim}\frac{n(n-2)x^{n-2}}{\lambda^{2}e^{\lambda x}}=\cdots=\underset{x\rightarrow\infty}{\lim}\frac{n!}{\lambda^{n}e^{\lambda x}}=\frac{n!}{\infty}=0$ \end_inset \end_layout -\begin_layout Example -\begin_inset Formula $\underset{x\rightarrow\infty}{\lim}\ln x<\underset{x\rightarrow\infty}{\lim}\sqrt{x}$ +\begin_layout Standard +这个式子叫做限增量公式, + 给出了自变量取得有限增量 +\begin_inset Formula $\Delta x$ \end_inset -, - -\begin_inset Formula $\underset{x\rightarrow\infty}{\lim}x^{n}<\underset{x\rightarrow\infty}{\lim}e^{\lambda x}$ +( +\begin_inset Formula $|\Delta x|$ \end_inset +不一定很小)时, + 函数增量 +\begin_inset Formula $\Delta y$ +\end_inset +的准确表达因此, + 这个定理也叫做有限增量定理. \end_layout -\begin_layout Example -对数 -\begin_inset Formula $<$ -\end_inset - -幂 -\begin_inset Formula $<$ +\begin_layout Corollary +如果函数 +\begin_inset Formula $f(x)$ \end_inset -指数 -\end_layout - -\begin_layout Enumerate -使用洛必达法则前必须检验极限类型, - 只有 -\begin_inset Formula $\frac{0}{0}$ +在区间 +\begin_inset Formula $I$ \end_inset -型或者 -\begin_inset Formula $\frac{\infty}{\infty}$ +内的导数恒为0,那么函数 +\begin_inset Formula $f(x)$ \end_inset -才可以使用洛必达法则. - +在该区间内恒为常数 \end_layout -\begin_layout Example -\begin_inset Formula $\underset{x\rightarrow0}{\lim}\frac{e^{x}-e^{-x}}{\sin x}$ +\begin_layout Standard +\begin_inset Separator plain \end_inset \end_layout -\begin_layout Example -\begin_inset Formula $\underset{x\rightarrow0}{\lim}\frac{e^{x}-e^{-x}}{\sin x}=\underset{x\rightarrow0}{\lim}\frac{e^{x}-e^{-x}}{x}=2$ +\begin_layout Corollary +如果函数 +\begin_inset Formula $f(x)$ \end_inset +和 +\begin_inset Formula $g(x)$ +\end_inset -\end_layout +在区间I内每一点的导数都相等,那么这两个函数在区间 +\begin_inset Formula $I$ +\end_inset -\begin_layout Standard -\begin_inset Separator plain +上至多相差一个常数 +\begin_inset Formula $C$ \end_inset +, + 即 +\begin_inset Formula $f(x)=g(x)+C,x\in1$ +\end_inset + . \end_layout \begin_layout Example -\begin_inset Formula $\underset{x\rightarrow0}{\lim}\frac{\tan x-x}{x-\sin x}$ +证 +\begin_inset Formula $\frac{x}{1+x}<\ln(1+x)0)$ \end_inset \end_layout \begin_layout Example -\begin_inset Formula $\underset{x\rightarrow0}{\lim}\frac{\tan x-x}{x-\sin x}=\underset{x\rightarrow0}{\lim}\frac{\sec^{2}x-1}{1-\cos x}=\underset{x\rightarrow0}{\lim}\frac{\tan^{2}x}{1-\cos x}=\underset{x\rightarrow0}{\lim}\frac{x^{2}}{\frac{1}{2}x^{2}}=2$ +设肺 +\begin_inset Formula $f(t)=\ln t,t\in[1,1+x]$ \end_inset - -\end_layout - -\begin_layout Enumerate -在极限计算过程中, - 无穷小量等价代换和洛必达法则应结合使用, - 能用等价代换则先用等价代换, - 这样可以简化计算过程. - -\end_layout - -\begin_layout Example -\begin_inset Formula $\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{\tan3x}{\tan x}$ +,则 +\begin_inset Formula $f(t)$ \end_inset +在闭区间 +\begin_inset Formula $[1,1+x](x>0)$ +\end_inset -\end_layout +上满足拉格朗日中值定理的条件， +因此至少存在一点 +\family roman +\series medium +\shape up +\size normal +\emph off +\nospellcheck off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none -\begin_layout Example -\begin_inset Formula $\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{\tan3x}{\tan x}=\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{3\sec^{2}3x}{sec^{2}x}=3\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{\cos^{2}x}{\cos^{2}3x}$ +\begin_inset Formula $\xi\in(1,1+x)$ \end_inset -\end_layout +\family default +\series bold +\shape default +\size default +\emph default +\nospellcheck default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit + , +\series default +使得 +\series bold -\begin_layout Example -\begin_inset Formula $\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{-2\cos x\cdot\sin x}{-2\cos3x\cdot\sin3x}=\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{\sin2x}{\sin6x}=\underset{x\rightarrow\frac{\pi}{2}}{\lim}\frac{2\cos2x}{6\cos6x}=\frac{1}{3}$ +\begin_inset Formula $f'(\xi)=\frac{f(1+x)-f(x)}{1+x-1}$ \end_inset + , +\series default +即 +\family roman +\series medium +\shape up +\size normal +\emph off +\nospellcheck off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none -\end_layout +\begin_inset Formula $\frac{1}{\xi}=\frac{\ln(1+x)}{x}$ +\end_inset -\begin_layout Itemize -如果计算过程中 -\begin_inset Formula $\lim\frac{f'(x)}{g'(x)}$ +. + 由于 +\begin_inset Formula $1<\xi<1+x$ \end_inset -仍然是 -\begin_inset Formula $\frac{0}{0}$ +,则 +\begin_inset Formula $\frac{1}{1+x}<\frac{1}{\xi}<1$ \end_inset -型或者 -\begin_inset Formula $\frac{\infty}{\infty}$ -\end_inset - -,且 -\begin_inset Formula $f'(x)$ +,于是 +\begin_inset Formula $\frac{1}{1+x}<\frac{1}{\xi}=\frac{\ln(1+x)}{x}<1$ \end_inset -， - -\begin_inset Formula $g'(x)$ +. + 因此 +\begin_inset Formula $\frac{x}{1+x}<\frac{\ln(1+x)}{x}0,a\neq1) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +三角函数积分 +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\sin x\,dx=-\cos x+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\cos x\,dx=\sin x+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\tan x\,dx=\ln|\sec x|+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\cot x\,dx=\ln|\sin x|+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\sec x\,dx=\ln|\sec x+\tan x|+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\csc x\,dx=\ln|\csc x-\cot x|+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +反三角函数积分 +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\frac{1}{\sqrt{1-x^{2}}}\,dx=\arcsin x+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\frac{1}{1+x^{2}}\,dx=\arctan x+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\frac{1}{|x|\sqrt{x^{2}-1}}\,dx=\text{arcsec}x+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +双曲函数积分 +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\sinh x\,dx=\cosh x+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\cosh x\,dx=\sinh x+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\tanh x\,dx=\ln|\cosh x|+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\coth x\,dx=\ln|\sinh x|+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +对数函数积分 +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int\ln x\,dx=x\ln x-x+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int x^{m}\ln x\,dx=\frac{x^{m+1}\ln x}{m+1}-\frac{x^{m+1}}{(m+1)^{2}}+C,\quad(m\neq-1) +\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +换元积分法 +\end_layout + +\begin_layout Standard +根据复合函数的微分法则推导. +\end_layout + +\begin_layout Subsubsection +第一换元法(凑微分法) +\end_layout + +\begin_layout Standard +设 +\begin_inset Formula $f(u)$ +\end_inset + +具有原函数 +\begin_inset Formula $F(u)$ +\end_inset + +, + 即 +\begin_inset Formula +\[ +F'(u)=f(u),\int f(u)du=F(u)+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +如果 +\begin_inset Formula $u$ +\end_inset + +是中间变量且 +\begin_inset Formula $u=\varphi(x)$ +\end_inset + +,且设 +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $\varphi(x)$ +\end_inset + +可微, + 那么, + 根据 +\family default +\series bold +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +复合函数微分法 +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +, + 有 +\begin_inset Formula +\[ +dF[\varphi(x)]=f[\varphi(x)]\varphi'(x)dx +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $dy=y'dx$ +\end_inset + + +\end_layout + +\begin_layout Standard +从而根据不定积分的定义就得 +\begin_inset Formula +\[ +\int f[\varphi(x)]\varphi'(x)dx=F[\varphi(x)]+C=[\int f(u)du]_{u=\varphi(x)} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +凑成基本积分公式。 + +\end_layout + +\begin_layout Example +\begin_inset Formula $3\cos(3x)dx$ +\end_inset + + +\end_layout + +\begin_layout Example +积分的线性性质允许我们把 3 提出： + +\end_layout + +\begin_layout Example +\begin_inset Formula +\[ +I=3\int\cos(3x)\,dx +\] + +\end_inset + + +\end_layout + +\begin_layout Example +换元 +\end_layout + +\begin_layout Example +令： + +\begin_inset Formula $u=3x$ +\end_inset + + +\end_layout + +\begin_layout Example +对两边求导： + +\begin_inset Formula $du=3dx$ +\end_inset + + +\end_layout + +\begin_layout Example +所以： + +\begin_inset Formula $dx=\frac{du}{3}$ +\end_inset + + +\end_layout + +\begin_layout Example +代换积分 +\begin_inset Formula $I=3\int\cos u\cdot\frac{du}{3}=\int\cos udu$ +\end_inset + + +\end_layout + +\begin_layout Example +\begin_inset Formula +\[ +\int\cos udu=\sin u+C +\] + +\end_inset + + +\end_layout + +\begin_layout Example +\begin_inset Formula +\[ +\int3\cos(3x)dx=\sin(3x)+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\int3\cos(3x)dx=\int\cos(3x)d(3x)=\sin(3x)+C$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Example +求 +\begin_inset Formula $\int\frac{1}{3+2x}dx$ +\end_inset + + +\end_layout + +\begin_layout Solution +设 +\begin_inset Formula $\frac{1}{3+2x}=\frac{1}{u}$ +\end_inset + +, + +\begin_inset Formula $u=3+2x$ +\end_inset + + +\end_layout + +\begin_layout Solution +\begin_inset Formula $du=2dx$ +\end_inset + +, +\begin_inset Formula $dx=\frac{du}{2}$ +\end_inset + + +\end_layout + +\begin_layout Solution +\begin_inset Formula $I=\int\frac{1}{u}\cdot\frac{du}{2}=\frac{1}{2}\int\frac{du}{u}=\frac{1}{2}\ln|u|+C$ +\end_inset + + +\end_layout + +\begin_layout Solution +\begin_inset Formula $\int\frac{1}{3+2x}dx=\frac{1}{2}\int\frac{1}{3+2x}\cdot2dx=\frac{1}{2}\int\frac{1}{3+2x}\cdot2dx=\frac{1}{2}\int\frac{1}{3+2x}d(2x+3)$ +\end_inset + + +\end_layout + +\begin_layout Solution +\begin_inset Formula $=\frac{1}{2}\int\frac{1}{u}d(u)$ +\end_inset + + +\end_layout + +\begin_layout Solution +\begin_inset Formula $d$ +\end_inset + +内部的项可任意 +\begin_inset Formula $\pm C$ +\end_inset + + +\end_layout + +\begin_layout Problem +计算 +\begin_inset Formula $\int xe^{x^{2}}\,dx$ +\end_inset + + +\end_layout + +\begin_layout Standard +令： + +\begin_inset Formula $u=x^{2}$ +\end_inset + +, + 则 +\begin_inset Formula $du=2xdx$ +\end_inset + +, + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +xdx=\frac{du}{2} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +代入积分： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +I=\int xe^{x^{2}}\,dx=\int e^{u}\cdot\frac{du}{2} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +=\frac{1}{2}\int e^{u}\,du +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +=\frac{1}{2}e^{u}+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +换回 +\begin_inset Formula $u=x^{2}$ +\end_inset + +： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +I=\frac{1}{2}e^{x^{2}}+C +\] + +\end_inset + + +\end_layout + +\begin_layout Example +\begin_inset Formula $\int\frac{dx}{a^{2}+x^{2}}=\int\frac{1}{a^{2}}\frac{1}{1+\frac{x}{a}^{2}}d\frac{x}{a}=\frac{1}{a}\arctan\frac{x}{a}+C$ +\end_inset + + +\end_layout + +\begin_layout Example +\begin_inset Formula +\[ +\int\frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Separator plain +\end_inset + + +\end_layout + +\begin_layout Example +\begin_inset Formula +\begin{align*} +\int\sin^{3}xdx= & \int\sin^{2}x\sin xd(\cos x)\\ += & -\int(1-\cos^{2}x)d\cos x\\ += & -\cos x+\frac{1}{3}\cos^{3}x+C +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +第二换元法 +\end_layout + +\begin_layout Standard +第一换元法是将被积表达式“凑”成 +\begin_inset Formula $f(\varphi(x))d\varphi(x)$ +\end_inset + +的形式, + 从而找到新的积分变量 +\begin_inset Formula $u=\varphi(x)$ +\end_inset + +,但是对于有些被积函数需要作另一种方式的换元,即令 +\begin_inset Formula $x=\varphi(t)$ +\end_inset + +,这样 +\begin_inset Formula $dx=\varphi'(t)dt$ +\end_inset + +, + 被积表达式 +\begin_inset Formula $f(x)dx$ +\end_inset + + 改写为变量 +\begin_inset Formula $t$ +\end_inset + +的形式 +\begin_inset Formula $f[\varphi(t)]\varphi'(t)dt$ +\end_inset + +,此时,为新的积分变量且容易积出结果. + 这种代换方法称为第二换元法. + 对于第二换元法, + +\end_layout + +\begin_layout Standard +设 +\begin_inset Formula $x=\psi(t)$ +\end_inset + + 是单调的可导函数， +并且 +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $\psi'(t)\neq0$ +\end_inset + + +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +. + 又设 +\begin_inset Formula $f[\psi(t)]\psi'(t)$ +\end_inset + + 具有原函数， +则有换元公式 +\begin_inset Formula +\[ +\int f(x)dx=\left[\int f[\varphi(t)]\varphi'(t)dt\right]_{t=\varphi^{-1}(x)} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +其中 +\begin_inset Formula $\varphi^{-1}(x)$ +\end_inset + +是 +\begin_inset Formula $x=\varphi(t)$ +\end_inset + +的反函数. +\end_layout + +\begin_layout Paragraph +简单根式换元 +\end_layout + +\begin_layout Standard +一般地,我们将形如 +\begin_inset Formula $\sqrt[n]{ax+b}\,(n\geqslant2)$ +\end_inset + +的根式称为简单根式. + 当被积函数含有简单根式时,可作代换 +\begin_inset Formula $t=\sqrt[n]{ax+b}$ +\end_inset + +,即 +\begin_inset Formula $x=\frac{t^{n}-b}{a}\,(a\neq0)$ +\end_inset + +, + 从而去掉根式, + 再用直接积分法或凑微分法处理积分即可.这种代换方法通常称为 +\series bold +简单根式换元 +\series default +或 +\series bold +根式换元 +\series default +。 + +\end_layout + +\begin_layout Example +求 +\begin_inset Formula $\int\frac{\sqrt{x+1}}{x}dx$ +\end_inset + + +\end_layout + +\begin_layout Solution +设 +\begin_inset Formula $t=\sqrt{x+1}$ +\end_inset + +, + 则 +\begin_inset Formula $x=t^{2}-1$ +\end_inset + +,且 +\begin_inset Formula $d(x)=d(t^{2}-1)=2tdt$ +\end_inset + +, + 于是 +\end_layout + +\begin_layout Solution +\begin_inset Formula +\begin{align*} +\int\frac{\sqrt{x+1}}{x}dx & =\int\frac{t}{t^{2}-1}2tdt=\int\frac{2t^{2}-2+2}{t^{2}-1}dt=\int(2+\frac{2}{t^{2}-1})dt\\ + & =2t+\int(\frac{1}{t-1}-\frac{1}{t+1})dt\\ + & =2t+|[\ln(t-1)-\ln(t+1)]|+C=2t+\ln\left|\frac{t-1}{t+1}\right|+C\\ + & \overset{t=\sqrt{x+1}}{\implies}=2\sqrt{x+1}+\ln\left|\frac{\sqrt{x+1}-1}{\sqrt{x+1}+1}\right|+C +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Example +求 +\begin_inset Formula $\int\frac{dx}{\sqrt{x}+\sqrt[3]{x}}$ +\end_inset + + +\end_layout + +\begin_layout Solution +被积函数有两个简单根式, + 为引入一个变量能同时换掉两个根式,我们令 +\begin_inset Formula $t=\sqrt[6]{x}$ +\end_inset + +, +\end_layout + +\begin_layout Solution +则 +\begin_inset Formula $x=t^{6}$ +\end_inset + +,且 +\begin_inset Formula $dx=6t^{5}dt$ +\end_inset + +,于是 +\end_layout + +\begin_layout Solution +\begin_inset Formula +\begin{align*} +\int\frac{dx}{\sqrt{x}+\sqrt[3]{x}} & =\int\frac{6t^{5}dt}{t^{3}+t^{2}}=6\int\frac{t^{3}dt}{t+1}\\ + & =6\int\frac{t^{3}dt}{t+1}=6\int\frac{t^{3}+1-1}{t+1}dt\\ + & =6\int(\frac{(t^{2}-t+1)(t+1)-1}{t+1})dt\\ + & =6\int(t^{2}-t+1-\frac{1}{t+1})dt\\ + & =6\times\frac{1}{3}t^{3}+6\times\frac{1}{2}t^{2}+6t-6\ln|t+1|+C\\ +\overset{t=\sqrt[6]{x}}{\implies} & = +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Standard +引入简单根式换元实际上是将被积函数中的根式化为有理式, + 但有时候直接用根式换元并不奏效, + 形如 +\begin_inset Formula $\sqrt{a^{2}-x^{2}}$ +\end_inset + +, +\begin_inset Formula $\sqrt{a^{2}+x^{2}}$ +\end_inset + +的二次根式, + 如果直接用根式换元并不起作用. +\end_layout + +\begin_layout Standard +此时,我们可借助于三角函数, + 分别用 +\begin_inset Formula $x=a\sin t$ +\end_inset + +, + +\begin_inset Formula $x=a\tan t$ +\end_inset + +, +\begin_inset Formula $x=a\sec t$ +\end_inset + +将这些表达式的二次根式去掉, +\end_layout + +\begin_layout Standard +这种代换我们通常称为 +\series bold +三角换元 +\series default +, + 由于计算不定积分实际上只要求出被积函数的一个原函数再加上积分常数 +\begin_inset Formula $C$ +\end_inset + +即可, + 因而在三角换元中,我们总将三角函数的角度设定为 +\series bold +锐角 +\series default +, + 也就是说在开方运算时 +\series bold +不必加绝对值号 +\series default +. +\end_layout + +\begin_layout Example +求 +\begin_inset Formula $\int\sqrt{a^{2}-x^{2}}dx$ +\end_inset + + +\begin_inset Formula $(a>0)$ +\end_inset + + +\end_layout + +\begin_layout Solution +设 +\begin_inset Formula $x=a\sin t$ +\end_inset + + ,则 +\begin_inset Formula $dx=a\cos tdt$ +\end_inset + + , + 于是( +\begin_inset Formula $\sin^{2}x+\cos^{2}x=1$ +\end_inset + + ) +\end_layout + +\begin_layout Solution +\begin_inset Formula +\begin{align*} +\int\sqrt{a^{2}-x^{2}}dx & =\int\sqrt{a^{2}-a^{2}\sin^{2}t}\cdot a\cos tdt=\int a\sqrt{(1-\sin^{2}t)}\cdot a\cos tdt\\ + & =\int a\cos t\cdot a\cos tdt=a^{2}\int\cos^{2}tdt\\ + & =a^{2}\int\frac{1+\cos2t}{2}dt\\ + & =\frac{1}{2}a^{2}t+\frac{1}{4}a^{2}\sin2t+C\\ + & =\frac{1}{2}a^{2}t+\frac{1}{2}a^{2}\sin t\cos t+C +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Solution +为能方便进行变量回代, + 可根据 +\begin_inset Formula $x=a\sin t$ +\end_inset + + 作一辅助直角三角形, + 利用边角关系实现回代(这种回代方法, + 我们通常称为画直角三角形回代法)。 + +\end_layout + +\begin_layout Solution +\begin_inset Float figure +placement document +alignment document +wide false +sideways false +status open + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout +\begin_inset Graphics + filename image/不定积分-三角回代法.jpg + lyxscale 25 + scale 25 + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +三角形回代法 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Solution +由图 +\begin_inset Formula $\sin t=\frac{x}{a}$ +\end_inset + + , +\begin_inset Formula $\cos t=\frac{\sqrt{a^{2}-x^{2}}}{a}$ +\end_inset + +, + 于是 +\begin_inset Formula $\int\sqrt{a^{2}-x^{2}}dx=\frac{1}{2}a^{2}\arcsin\frac{x}{a}+\frac{x\sqrt{a^{2}-x^{2}}}{2}+C$ +\end_inset + + +\end_layout + +\begin_layout Solution +需要指出的是,两种代换的本质是用换元的思想将被积函数中 +\series bold +比较难处理的项代换掉 +\series default +, + 我们不能拘泥于两种代换使用时对被积函数形式的规定. +\end_layout + +\begin_layout Subsection +分部积分法 +\end_layout + +\begin_layout Standard +当被积函数是两个不同类型的函数的乘积时,如 +\begin_inset Formula $\int xe^{x}dx$ +\end_inset + +等, + 往往用下面的分部积分法解決. + +\end_layout + +\begin_layout Standard +分部积分法是与两个函数乘积相对应的,也是一种基本积分方法 +\end_layout + +\begin_layout Standard +设函数 +\begin_inset Formula $u=u(x)$ +\end_inset + +及 +\begin_inset Formula $v=v(x)$ +\end_inset + +具有连续导数, + 则两个函数乘积的导数公式为 +\begin_inset Formula +\[ +(uv)'=u'v+uv' +\] + +\end_inset + +移项， +得 +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +(uv)' +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit +- +\begin_inset Formula +\[ +uv'=(uv)'-u'v +\] + +\end_inset + + 对这个等式两边求不定积分, + 得 +\begin_inset Formula +\[ +\int uv'dx=uv-\int u'vdx +\] + +\end_inset + + +\end_layout + +\begin_layout Example +求 +\begin_inset Formula $\int x\cos xdx$ +\end_inset + + +\end_layout + +\begin_layout Solution +\begin_inset Formula $\int x\cos xdx=\int xd(\sin x)$ +\end_inset + +,设 +\begin_inset Formula $u=x$ +\end_inset + +, +\begin_inset Formula $v=\sin x$ +\end_inset + +, + 则 +\begin_inset Formula +\[ +\int x\cos xdx=\int xd(\sin x)=x\sin x-\int\sin xdx=x\sin x+\cos x+C +\] + \end_inset \end_layout -\begin_layout Quotation -其中 -\end_layout +\begin_layout Standard +本题如这样变化: + +\begin_inset Formula $\int x\cos xdx=\frac{1}{2}\int\cos xdx^{2}$ +\end_inset -\begin_layout Quotation -\begin_inset Formula ${\displaystyle |R_{n}(x)|\leq M_{n}\frac{r^{n+1}}{(n+1)!}}$ +,即 +\begin_inset Formula $u=\cos x$ \end_inset -。 -这个上界估计对区间 -\begin_inset Formula $(a−r,a+r)$ +, +\begin_inset Formula $v=x^{2}$ \end_inset - 里的任意 -\begin_inset Formula $x$ +. + 则由分部积分公式有 +\begin_inset Formula +\[ +\int x\cos xdx=\frac{1}{2}\int\cos xdx^{2}=\frac{1}{2}(x^{2}\cos x-\int x^{2}\sin xdx) +\] + \end_inset - 都成立， -是一个一致估计。 \end_layout -\begin_layout Quotation -如果当n 趋向于无穷大时， -还有 -\begin_inset Formula ${\displaystyle M_{n}\frac{r^{n+1}}{(n+1)!}\rightarrow0}$ +\begin_layout Standard +反而比原来的积分更难求了, + 因此, + 在使用分部积分法时,确定 +\begin_inset Formula $u$ \end_inset -， -那么可以推出 -\begin_inset Formula ${\displaystyle R_{n}(x)\rightarrow0}$ +和 +\begin_inset Formula $v$ \end_inset -， +非常关键. + 一般地, + +\begin_inset Formula $v$ +\end_inset -\begin_inset Formula $f$ +要容易求得(用凑微分法), + 并且 +\begin_inset Formula $\int vdu$ \end_inset -是区间 -\begin_inset Formula $(a−r,a+r)$ +要比 +\begin_inset Formula $\int udv$ \end_inset -上解析函数。 +容易积分 +\end_layout -\begin_inset Formula $f$ +\begin_layout Standard +为更便捷准确地选取恰当的 +\begin_inset Formula $u$ \end_inset -在区间 -\begin_inset Formula $(a−r,a+r)$ +和 +\begin_inset Formula $v$ \end_inset -上任一点的值都等于在这一点的泰勒展开式的极限。 - +,我们指出: \end_layout -\begin_layout Example -\begin_inset Formula $f(x)=e^{x}$ +\begin_layout Standard + +\series bold +当被积函数为幂函数与三角函数(或指数函数)的乘积时, + 选取幂函数作为 +\begin_inset Formula $u$ \end_inset - 的马克劳林公式 +,将三角函数(或指数函数)凑到“d”的后面选作v. + \end_layout -\begin_layout Solution -\begin_inset Formula $f'(x)=f''(x)=\cdots=f^{(n)}(x)=e^{x}$ +\begin_layout Example +求 +\begin_inset Formula $\int x^{2}e^{x}dx$ \end_inset \end_layout \begin_layout Solution -\begin_inset Formula $f'(0)=f''(0)=\cdots=f^{(n)}(0)=1$ +\begin_inset Formula $\int x^{2}e^{x}dx=\int x^{2}de^{x}$ \end_inset - -\begin_inset Formula $e^{x}=1+\frac{1}{1!}x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots+\frac{x^{n}}{n!}+\frac{e^{\theta x}(x)^{n+1}}{(n+1)!},(0<\theta<1)$ +此时取 +\begin_inset Formula $u=x^{2}$ \end_inset - -\end_layout - -\begin_layout Solution -\begin_inset Formula $R_{n}(x)<\frac{e^{|x|}(x)^{n+1}}{(n+1)!}$ +, +\begin_inset Formula $v=e^{x}$ \end_inset - + ,于是 \end_layout \begin_layout Solution -\begin_inset Formula $e^{x}\approx1+\frac{1}{1!}x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots+\frac{x^{n}}{n!}$ -\end_inset - - -\end_layout +\begin_inset Formula +\begin{align*} +\int x^{2}e^{x}dx & =\int x^{2}de^{x}=x^{2}e^{x}-\int e^{x}dxx^{2}\\ + & =x^{2}e^{x}-2\int xe^{x}dx\\ + & =x^{2}e^{x}-2(xe^{x}-\int e^{x}dx)\\ + & =x^{2}e^{x}-2xe^{x}+2e^{x}+C\\ + & =(x^{2}-2x+2)e^{x}+C +\end{align*} -\begin_layout Example -\begin_inset Formula $f(x)=\sin x$ \end_inset \end_layout -\begin_layout Example -\begin_inset Formula $\sin x=x-\frac{1}{3!}x^{3}+\frac{1}{5!}x^{5}-\cdots+\frac{(-1)^{n-1}}{(2n-1)!}x^{2n-1}+R_{2n}(x)$ -\end_inset - - +\begin_layout Standard +该例表明,有时要多次使用分部积分法才能求出结果,一般地,幂函数最高次有几次,就要作几次分部积分. \end_layout -\begin_layout Example -\begin_inset Formula $\sin x\approx x-\frac{1}{3!}x^{3}$ +\begin_layout Standard + +\series bold +当被积函数为幂函数与对数函数(或反三角函数)的乘积时, + 选取对数函数(或反三角函 数)作为u,将幂函数凑到“ +\begin_inset Formula $d$ \end_inset +”的后面选作 +\begin_inset Formula $v$ +\end_inset +. \end_layout \begin_layout Example -\begin_inset Formula $\sin x\approx x-\frac{1}{3!}x^{3}+\frac{1}{5!}x^{5}$ +求 +\begin_inset Formula $\int x\ln xdx$ \end_inset \end_layout -\begin_layout Subsection -二元函数的泰勒公式 -\end_layout +\begin_layout Solution +\begin_inset Formula +\begin{align*} +\int x\ln xdx & =\frac{1}{2}\int\ln xdx^{2}=\frac{1}{2}(x^{2}\ln x-\int x^{2}d\ln x)\\ + & =\frac{1}{2}(x^{2}\ln x-\int x^{2}d\ln x)=\frac{1}{2}(x^{2}\ln xe-\int xdx)\\ + & =\frac{1}{2}(x^{2}\ln x-\frac{1}{2}x^{2})+C +\end{align*} + +\end_inset -\begin_layout Chapter -不定积分 -\end_layout -\begin_layout Section -原函数 \end_layout \begin_layout Standard -设 -\begin_inset Formula $f(x)$ +有些不定积分如 +\begin_inset Formula $\int e^{-x^{2}}dx$ \end_inset -是定义在某区间 -\begin_inset Formula $I$ +, +\begin_inset Formula $\int\frac{\sin x}{x}dx$ \end_inset -上的已知函数, - 若存在函数 -\begin_inset Formula $F(x)$ +, +\begin_inset Formula $\int\frac{\ln x}{x}dx$ \end_inset -,使得 -\begin_inset Formula $F'(x)=f(x)$ -\end_inset +等,虽然这些不定积分都存在, + 但是不能用初等函数表示这些被积函数的原函数, + 我们称这种情况为“不可积”. +\end_layout -或 -\begin_inset Formula $dF(x)=f(x)dx$ -\end_inset +\begin_layout Section +有理函数的积分 +\end_layout -,则称 -\begin_inset Formula $F(x)$ +\begin_layout Standard +两个 +\series bold +多项式 +\series default +的商 +\begin_inset Formula $\frac{P(x)}{Q(x)}$ \end_inset -是 -\begin_inset Formula $f(x)$ +称为 +\series bold +有理函数 +\series default +, + 又称 +\series bold +有理分式 +\series default +. + 我们总假定分子多项式 +\begin_inset Formula $P(x)$ \end_inset -在区间 -\begin_inset Formula $I$ +与分母多项式 +\begin_inset Formula $Q(x)$ \end_inset -上的一个原函数. - +之间 +\series bold +没有公因式 +\series default +. \end_layout \begin_layout Standard -如果函数 -\begin_inset Formula $f(x)$ +当分子多项式Ρ(x)的次数小于分母多项式 +\begin_inset Formula $Q(x)$ \end_inset -在区间 -\begin_inset Formula $I$ -\end_inset +的次数时， +称这个有理函数为 +\series bold +真分式 +\series default +， +否则称为 +\series bold +假分式 +\series default +. +\end_layout -上连续， -那么在区间 -\begin_inset Formula $I$ -\end_inset +\begin_layout Subsection +多项式除法 +\end_layout -上存在可 导函数F(x), - 使对任一 -\begin_inset Formula $x\in I$ -\end_inset +\begin_layout Standard +利用多项式的除法， +可以将一个假分式化成一个多项式与一个真分式之和的形式. +\end_layout -都有 -\begin_inset Formula $F'(x)=f(x)$ +\begin_layout Standard +对于真分式 +\begin_inset Formula $\frac{P(x)}{Q(x)}$ \end_inset , - 连续函数一定有原函数. -\end_layout + 如果分母可以分解为两个多相式的乘积 +\begin_inset Formula +\[ +Q(x)=Q_{1}(x)Q_{2}(x) +\] -\begin_layout Standard +\end_inset -\series bold -如果一个函数有原函数, - 则原函数不是唯一的, - 而是有无穷多个. -\end_layout -\begin_layout Section -不定积分的定义 \end_layout \begin_layout Standard -如果 -\begin_inset Formula $F(x)$ +且 +\begin_inset Formula $Q_{1}(x)$ \end_inset -是 -\begin_inset Formula $f(x)$ +与 +\begin_inset Formula $Q_{2}(x)$ \end_inset -在区间 -\begin_inset Formula $I$ -\end_inset +没有公因式, + 那么它可以拆分为两个真分式之和 +\begin_inset Formula +\[ +\frac{P(x)}{Q(x)}=\frac{P_{1}(x)}{Q_{1}(x)}+\frac{P_{2}(x)}{Q_{2}(x)} +\] -上的一个原函数, - 那么 -\begin_inset Formula $f(x)$ \end_inset -的所有原函数的全体 -\begin_inset Formula $F(x)+C(C是任意常数)$ + +\end_layout + +\begin_layout Standard +上述步骤称为把 +\series bold +真分式化成部分分式之和 +\series default +.如果 +\begin_inset Formula $Q_{1}(x)$ \end_inset -称为 -\begin_inset Formula $f(x)$ +或 +\begin_inset Formula $Q_{2}(x)$ \end_inset -在区间 -\begin_inset Formula $I$ +还能再分解成两个没有公因式的多项式的乘积， +那么就可再分拆成更简单的部分分式.最后， +有理函数的分解式中只出现 +\series bold +多项式, +\begin_inset Formula $\frac{P_{1}(x)}{(x-a)^{k}}$ \end_inset -上的不定积分, - 记为 -\begin_inset Formula $\int f(x)dx$ +, +\begin_inset Formula $\frac{P_{2}(x)}{(x^{2}+px+q)^{l}}$ \end_inset -,即 -\begin_inset Formula -\[ -\int f(x)dx=F(x)+C -\] +\series default +等三类函数(这里 +\begin_inset Formula $p^{2}-4q<0$ \end_inset -,其中, - -\begin_inset Formula $"\int"$ +, +\begin_inset Formula $P_{1}(x)$ \end_inset -为不定积分符号, - -\begin_inset Formula $f(x)$ +为小于 +\begin_inset Formula $k$ \end_inset -称为被积函数, +次的多项式, -\begin_inset Formula $f(x)dx$ +\begin_inset Formula $P_{2}(x)$ \end_inset -称为被积表达式, - -\begin_inset Formula $x$ +为小于 +\begin_inset Formula $2l$ \end_inset -称为积分变量, - -\begin_inset Formula $C$ -\end_inset +次的多项式). +\end_layout -称为积分常数. +\begin_layout Standard +多项式除法类似于我们在小学学过的长除法， +但它用于**两个多项式之间的除法**。 + \end_layout \begin_layout Standard -不定积分 -\begin_inset Formula $\int f(x)dx$ +如果分子多项式 +\begin_inset Formula $P(x)$ \end_inset - 实际上是求被积函数 -\begin_inset Formula $f(x)$ + 的次数大于或等于分母多项式 +\begin_inset Formula $Q(x)$ \end_inset -的所有原函数的全体, - 也即只要求出 -\begin_inset Formula $f(x)$ -\end_inset + 的次数， +我们可以使用多项式除法， +将其拆分成： + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\frac{P(x)}{Q(x)}=H(x)+\frac{R(x)}{Q(x)} +\] -的一个原函数再加上积分常数 -\begin_inset Formula $C$ \end_inset -即可. + \end_layout -\begin_layout Section -不定积分的性质 +\begin_layout Standard +其中： + \end_layout -\begin_layout Corollary -设函数 -\begin_inset Formula $f(x)$ +\begin_layout Enumerate +\begin_inset Formula $H(x)$ \end_inset -及 -\begin_inset Formula $g(x)$ -\end_inset + 是商（一个多项式）。 -的原函数存在, - 则 \end_layout -\begin_layout Corollary -\begin_inset Formula $\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$ +\begin_layout Enumerate +\begin_inset Formula $R(x)$ +\end_inset + + 是余数， +使得 +\begin_inset Formula $\deg(R)<\deg(Q)$ \end_inset +。 \end_layout -\begin_layout Proof -将上式右端求导， -得 +\begin_layout Enumerate +余数的次数必须小于分母的次数， +否则说明除法还没完成。 + \end_layout \begin_layout Standard +然后我们可以分别对 +\begin_inset Formula $H(x)$ +\end_inset -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none -\begin_inset Formula $[f(x)dx+g(x)dx]'=[f(x)dx]'+[g(x)dx]'=f(x)+g(x)$ + 和 +\begin_inset Formula $\frac{R(x)}{Q(x)}$ \end_inset + 进行积分。 \end_layout -\begin_layout Corollary -设函数 -\begin_inset Formula $f(x)$ -\end_inset - -的原函数存在, - -\begin_inset Formula $k$ -\end_inset - -为非零常数， -则 -\begin_inset Formula -\[ -\int kf(x)dx=k\int f(x)dx -\] +\begin_layout Example +简单多项式除法 求解： +\begin_inset Formula $\int\frac{x^{3}+2x^{2}-x+3}{x^{2}+1}\,dx$ \end_inset \end_layout -\begin_layout Standard -检验积分结果是否正确， -只要对结果求导， -看它的导数是否等于被积 函数， -相等时结果是正确的， -否则结果是错误的 +\begin_layout Solution +进行多项式除法 \end_layout -\begin_layout Section -不定积分的计算 -\end_layout +\begin_layout Solution +被除式（分子）： -\begin_layout Subsection -基本积分公式 -\end_layout +\begin_inset Formula $\ensuremath{x^{3}+2x^{2}-x+3}$ +\end_inset -\begin_layout Standard -微分和积分是互逆关系,可以由基本初导函数的求导公式推导. -\end_layout -\begin_layout Subsubsection -积分表 \end_layout -\begin_layout Subsection -不定积分的直接积分法 -\end_layout +\begin_layout Solution +除式（分母）： + +\begin_inset Formula $x^{2}+1$ +\end_inset -\begin_layout Subsection -换元积分法 -\end_layout -\begin_layout Standard -根据复合函数的微分法则推导. \end_layout -\begin_layout Subsubsection -第一换元法(凑微分法) +\begin_layout Solution +第一步： +最高次项相除 \end_layout -\begin_layout Standard -设 -\begin_inset Formula $f(u)$ +\begin_layout Solution +用 +\begin_inset Formula $x^{3}$ \end_inset -具有原函数 -\begin_inset Formula $F(u)$ + 除以 +\begin_inset Formula $x^{2}$ \end_inset -, - 即 -\begin_inset Formula -\[ -F'(u)=f(u),\int f(u)du=F(u)+C -\] - +， +得到商 +\begin_inset Formula $x$ \end_inset +。 \end_layout -\begin_layout Standard -如果 -\begin_inset Formula $u$ +\begin_layout Solution +然后用 +\begin_inset Formula $x$ \end_inset -是中间变量且 -\begin_inset Formula $u=\varphi(x)$ + 乘以 +\begin_inset Formula $x^{2}+1$ \end_inset -,且设 -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none +： -\begin_inset Formula $\varphi(x)$ -\end_inset +\end_layout -可微, - 那么, - 根据复合函数微分法, - 有 +\begin_layout Solution \begin_inset Formula \[ -dF[\varphi(x)]=f[\varphi(x)]\varphi'(x)dx +(x^{2}+1)\cdot x=x^{3}+x \] \end_inset @@ -50384,11 +52791,15 @@ dF[\varphi(x)]=f[\varphi(x)]\varphi'(x)dx \end_layout -\begin_layout Standard -从而根据不定积分的定义就得 +\begin_layout Solution +将它们相减： + +\end_layout + +\begin_layout Solution \begin_inset Formula \[ -\int f[\varphi(x)]\varphi'(x)dx=F[\varphi(x)]+C=[\int f(u)du]_{u=\varphi(x)} +(x^{3}+2x^{2}-x+3)-(x^{3}+x)=2x^{2}-2x+3 \] \end_inset @@ -50396,28 +52807,46 @@ dF[\varphi(x)]=f[\varphi(x)]\varphi'(x)dx \end_layout -\begin_layout Example -求 -\begin_inset Formula $\int\frac{1}{3+2x}dx$ +\begin_layout Solution +第二步： +继续除法 +\end_layout + +\begin_layout Solution +用 +\begin_inset Formula $2x^{2}$ +\end_inset + + 除以 +\begin_inset Formula $x^{2}$ +\end_inset + +， +得到商 +\begin_inset Formula $2$ \end_inset +。 \end_layout \begin_layout Solution -设 -\begin_inset Formula $\frac{1}{3+2x}=\frac{1}{u}$ +然后用 +\begin_inset Formula $2$ \end_inset -, - -\begin_inset Formula $u=3+2x$ + 乘以 +\begin_inset Formula $x^{2}+1$ \end_inset +： + +\end_layout +\begin_layout Solution \begin_inset Formula \[ -\frac{1}{3+2x}=\frac{1}{2}\cdot\frac{1}{3+2x}\cdot2=\frac{1}{2}\cdot\frac{1}{3+2x}(3+2x)' +(x^{2}+1)\cdot2=2x^{2}+2 \] \end_inset @@ -50426,104 +52855,60 @@ dF[\varphi(x)]=f[\varphi(x)]\varphi'(x)dx \end_layout \begin_layout Solution -\begin_inset Formula -\begin{align*} -\int\frac{1}{3+2x}dx & =\int\frac{1}{2}\cdot\frac{1}{3+2x}(3+2x)'dx\\ - & =\int\frac{1}{2}\cdot\frac{1}{u}du\\ - & =\frac{1}{2}\ln|u|+C -\end{align*} - -\end_inset - +相减： \end_layout -\begin_layout Subsubsection -第二换元法 -\end_layout +\begin_layout Solution +\begin_inset Formula +\[ +(2x^{2}-2x+3)-(2x^{2}+2)=-2x+1 +\] -\begin_layout Standard -第一换元法是将被积表达式“凑”成 -\begin_inset Formula $f(\varphi(x))d\varphi(x)$ \end_inset -的形式, - 从而找到新的积分变量 -\begin_inset Formula $u=\varphi(x)$ -\end_inset -,但是对于有些被积函数需要作另一种方式的换元,即令 -\begin_inset Formula $x=\varphi(t)$ -\end_inset +\end_layout -,这样 -\begin_inset Formula $dx=\varphi'(t)dt$ +\begin_layout Solution +此时， +余数 +\begin_inset Formula $-2x+1$ \end_inset -, - 被积表达式 -\begin_inset Formula $f(x)dx$ + 的次数比分母 +\begin_inset Formula $x^{2}+1$ \end_inset - 改写为变量 -\begin_inset Formula $t$ -\end_inset + 低， +所以多项式除法完成。 -的形式 -\begin_inset Formula $f[\varphi(t)]\varphi'(t)dt$ -\end_inset +\end_layout -,此时,为新的积分变量且容易积出结果. - 这种代换方法称为第二换元法. - 对于第二换元法,这里介绍和三角换元 两种情形. +\begin_layout Solution +拆分积分 \end_layout -\begin_layout Standard -设 -\begin_inset Formula $x=\psi(t)$ +\begin_layout Solution +\begin_inset Formula +\[ +\frac{x^{3}+2x^{2}-x+3}{x^{2}+1}=x+2+\frac{-2x+1}{x^{2}+1} +\] + \end_inset - 是单调的可导函数， -并且 -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none -\begin_inset Formula $\psi'(t)\neq0$ -\end_inset +\end_layout +\begin_layout Solution +所以原积分变为： -\family default -\series default -\shape default -\size default -\emph default -\bar default -\strikeout default -\xout default -\uuline default -\uwave default -\noun default -\color inherit -. - 又设 -\begin_inset Formula $f[\psi(t)]\psi'(t)$ -\end_inset +\end_layout - 具有原函数， -则有换元公式 +\begin_layout Solution \begin_inset Formula \[ -\int f(x)dx=\left[\int f[\varphi(t)]\varphi'(t)dt\right]_{t=\varphi^{-1}(x)} +\int(x+2)\,dx+\int\frac{-2x+1}{x^{2}+1}\,dx \] \end_inset @@ -50531,421 +52916,346 @@ dF[\varphi(x)]=f[\varphi(x)]\varphi'(x)dx \end_layout -\begin_layout Standard -其中 -\begin_inset Formula $\varphi^{-1}(x)$ -\end_inset +\begin_layout Solution +计算每个部分 +\end_layout + +\begin_layout Solution +\begin_inset Formula +\[ +\int x\,dx=\frac{x^{2}}{2},\quad\int2\,dx=2x +\] -是 -\begin_inset Formula $x=\varphi(t)$ \end_inset -的反函数. + \end_layout -\begin_layout Paragraph -简单根式换元 +\begin_layout Solution +对于第二个积分： + \end_layout -\begin_layout Standard -一般地,我们将形如 -\begin_inset Formula $\sqrt[n]{ax+b}\,(n\geqslant2)$ -\end_inset +\begin_layout Solution +\begin_inset Formula +\[ +\int\frac{-2x}{x^{2}+1}\,dx+\int\frac{1}{x^{2}+1}\,dx +\] -的根式称为简单根式. - 当被积函数含有简单根式时,可作代换 -\begin_inset Formula $t=\sqrt[n]{ax+b}$ \end_inset -,即 -\begin_inset Formula $x=\frac{t^{n}-b}{a}\,(a\neq0)$ -\end_inset -, - 从而去掉根式, - 再用直接积分法或凑微分法处理积分即可.这种代换方法通常称为 -\series bold -简单根式换元 -\series default -或 -\series bold -根式换元 -\series default -, \end_layout -\begin_layout Example -求 -\begin_inset Formula $\int\frac{\sqrt{x+1}}{x}dx$ -\end_inset - +\begin_layout Solution +第一个积分： \end_layout \begin_layout Solution -设 -\begin_inset Formula $t=\sqrt{x+1}$ +令 +\begin_inset Formula $u=x^{2}+1$ \end_inset -, - 则 -\begin_inset Formula $x=t^{2}-1$ +， +则 +\begin_inset Formula $du=2x\,dx$ \end_inset -,且 -\begin_inset Formula $d(x)=d(t^{2}-1)=2tdt$ -\end_inset +， +所以： -, - 于是 \end_layout \begin_layout Solution \begin_inset Formula -\begin{align*} -\int\frac{\sqrt{x+1}}{x}dx & =\int\frac{t}{t^{2}-1}2tdt=\int\frac{2t^{2}-2+2}{t^{2}-1}dt=\int(2+\frac{2}{t^{2}-1})dt\\ - & =2t+\int(\frac{1}{t-1}-\frac{1}{t+1})dt\\ - & =2t+|[\ln(t-1)-\ln(t+1)]|+C=2t+\ln\left|\frac{t-1}{t+1}\right|+C\\ - & \overset{t=\sqrt{x+1}}{\implies}=2\sqrt{x+1}+\ln\left|\frac{\sqrt{x+1}-1}{\sqrt{x+1}+1}\right|+C -\end{align*} +\[ +\int\frac{-2x}{x^{2}+1}\,dx=-\ln|x^{2}+1| +\] \end_inset \end_layout -\begin_layout Example -求 -\begin_inset Formula $\int\frac{dx}{\sqrt{x}+\sqrt[3]{x}}$ -\end_inset - +\begin_layout Solution +第二个积分是标准的： \end_layout \begin_layout Solution -被积函数有两个简单根式, - 为引入一个变量能同时换掉两个根式,我们令 -\begin_inset Formula $t=\sqrt[6]{x}$ +\begin_inset Formula +\[ +\int\frac{1}{x^{2}+1}\,dx=\tan^{-1}x +\] + \end_inset -, + \end_layout \begin_layout Solution -则 -\begin_inset Formula $x=t^{6}$ -\end_inset - -,且 -\begin_inset Formula $dx=6t^{5}dt$ -\end_inset - -,于是 +合并结果 \end_layout \begin_layout Solution \begin_inset Formula -\begin{align*} -\int\frac{dx}{\sqrt{x}+\sqrt[3]{x}} & =\int\frac{6t^{5}dt}{t^{3}+t^{2}}=6\int\frac{t^{3}dt}{t+1}\\ - & =6\int\frac{t^{3}dt}{t+1}=6\int\frac{t^{3}+1-1}{t+1}dt\\ - & =6\int(\frac{(t^{2}-t+1)(t+1)-1}{t+1})dt\\ - & =6\int(t^{2}-t+1-\frac{1}{t+1})dt\\ - & =6\times\frac{1}{3}t^{3}+6\times\frac{1}{2}t^{2}+6t-6\ln|t+1|+C\\ -\overset{t=\sqrt[6]{x}}{\implies} & = -\end{align*} +\[ +I=\frac{x^{2}}{2}+2x-\ln|x^{2}+1|+\tan^{-1}x+C +\] \end_inset \end_layout -\begin_layout Standard -引入简单根式换元实际上是将被积函数中的根式化为有理式, - 但有时候直接用根式换元并不奏效, - 形如 -\begin_inset Formula $\sqrt{a^{2}-x^{2}}$ -\end_inset +\begin_layout Example +高次多项式除法, + 求： -, -\begin_inset Formula $\sqrt{a^{2}+x^{2}}$ +\begin_inset Formula $\int\frac{x^{4}-3x^{2}+2x+5}{x^{2}-x+1}\,dx$ \end_inset -的二次根式, - 如果直接用根式换元并不起作用. + \end_layout -\begin_layout Standard -此时,我们可借助于三角函数, - 分别用 -\begin_inset Formula $x=a\sin t$ +\begin_layout Solution +用 +\begin_inset Formula $x^{4}$ \end_inset - -, - -\begin_inset Formula $x=a\tan t$ + + 除以 +\begin_inset Formula $x^{2}$ \end_inset -, -\begin_inset Formula $x=a\sec t$ +， +得到商 +\begin_inset Formula $x^{2}$ \end_inset -将这些表达式的二次根式去掉, +。 + \end_layout -\begin_layout Standard -这种代换我们通常称为 -\series bold -三角换元 -\series default -, - 由于计算不定积分实际上只要求出被积函数的一个原函数再加上积分常数 -\begin_inset Formula $C$ +\begin_layout Solution +乘回去： + +\begin_inset Formula $(x^{2}-x+1)\cdot x^{2}=x^{4}-x^{3}+x^{2}$ \end_inset -即可, - 因而在三角换元中,我们总将三角函数的角度设定为 -\series bold -锐角 -\series default -, - 也就是说在开方运算时 -\series bold -不必加绝对值号 -\series default -. -\end_layout +。 -\begin_layout Example -求 -\begin_inset Formula $\int\sqrt{a^{2}-x^{2}}dx$ -\end_inset +\end_layout +\begin_layout Solution +相减： -\begin_inset Formula $(a>0)$ +\begin_inset Formula $(x^{4}-3x^{2}+2x+5)-(x^{4}-x^{3}+x^{2})=x^{3}-4x^{2}+2x+5$ \end_inset +。 \end_layout \begin_layout Solution -设 -\begin_inset Formula $x=a\sin t$ +用 +\begin_inset Formula $x^{3}$ \end_inset - ,则 -\begin_inset Formula $dx=a\cos tdt$ + 除以 +\begin_inset Formula $x^{2}$ \end_inset - , - 于是( -\begin_inset Formula $\sin^{2}x+\cos^{2}x=1$ +， +得到商 +\begin_inset Formula $x$ \end_inset - ) +。 + \end_layout \begin_layout Solution -\begin_inset Formula -\begin{align*} -\int\sqrt{a^{2}-x^{2}}dx & =\int\sqrt{a^{2}-a^{2}\sin^{2}t}\cdot a\cos tdt=\int a\sqrt{(1-\sin^{2}t)}\cdot a\cos tdt\\ - & =\int a\cos t\cdot a\cos tdt=a^{2}\int\cos^{2}tdt\\ - & =a^{2}\int\frac{1+\cos2t}{2}dt\\ - & =\frac{1}{2}a^{2}t+\frac{1}{4}a^{2}\sin2t+C\\ - & =\frac{1}{2}a^{2}t+\frac{1}{2}a^{2}\sin t\cos t+C -\end{align*} +5. + 乘回去： +\begin_inset Formula $(x^{2}-x+1)\cdot x=x^{3}-x^{2}+x$ \end_inset +。 \end_layout \begin_layout Solution -为能方便进行变量回代, - 可根据 -\begin_inset Formula $x=a\sin t$ +6. + 相减： + +\begin_inset Formula $(x^{3}-4x^{2}+2x+5)-(x^{3}-x^{2}+x)=-3x^{2}+x+5$ \end_inset - 作一辅助直角三角形, - 利用边角关系实现回代(这种回代方法, - 我们通常称为画直角三角形回代法)。 +。 \end_layout \begin_layout Solution -\begin_inset Float figure -placement document -alignment document -wide false -sideways false -status open +7. + 用 +\begin_inset Formula $-3x^{2}$ +\end_inset -\begin_layout Plain Layout + 除以 +\begin_inset Formula $x^{2}$ +\end_inset + +， +得到商 +\begin_inset Formula $-3$ +\end_inset + +。 \end_layout -\begin_layout Plain Layout -\begin_inset Graphics - filename image/不定积分-三角回代法.jpg - lyxscale 25 - scale 25 +\begin_layout Solution +8. + 乘回去： +\begin_inset Formula $(x^{2}-x+1)\cdot(-3)=-3x^{2}+3x-3$ \end_inset +。 -\begin_inset Caption Standard - -\begin_layout Plain Layout -三角形回代法 \end_layout +\begin_layout Solution +9. + 相减： + +\begin_inset Formula $(-3x^{2}+x+5)-(-3x^{2}+3x-3)=-2x+8$ \end_inset +。 \end_layout -\begin_layout Plain Layout +\begin_layout Solution +所以： \end_layout +\begin_layout Solution +\begin_inset Formula +\[ +\frac{x^{4}-3x^{2}+2x+5}{x^{2}-x+1}=x^{2}+x-3+\frac{-2x+8}{x^{2}-x+1} +\] + \end_inset \end_layout \begin_layout Solution -由图 -\begin_inset Formula $\sin t=\frac{x}{a}$ -\end_inset +积分 +\end_layout - , -\begin_inset Formula $\cos t=\frac{\sqrt{a^{2}-x^{2}}}{a}$ -\end_inset +\begin_layout Solution +\begin_inset Formula +\[ +I=\int(x^{2}+x-3)\,dx+\int\frac{-2x+8}{x^{2}-x+1}\,dx +\] -, - 于是 -\begin_inset Formula $\int\sqrt{a^{2}-x^{2}}dx=\frac{1}{2}a^{2}\arcsin\frac{x}{a}+\frac{x\sqrt{a^{2}-x^{2}}}{2}+C$ \end_inset \end_layout \begin_layout Solution -需要指出的是,两种代换的本质是用换元的思想将被积函数中 -\series bold -比较难处理的项代换掉 -\series default -, - 我们不能拘泥于两种代换使用时对被积函数形式的规定. -\end_layout +第一部分： -\begin_layout Subsection -分部积分法 \end_layout -\begin_layout Standard -当被积函数是两个不同类型的函数的乘积时,如 -\begin_inset Formula $\int xe^{x}dx$ +\begin_layout Solution +\begin_inset Formula +\[ +\int x^{2}\,dx=\frac{x^{3}}{3},\quad\int x\,dx=\frac{x^{2}}{2},\quad\int-3\,dx=-3x +\] + \end_inset -等, - 往往用下面的分部积分法解決. - + \end_layout -\begin_layout Standard -分部积分法是与两个函数乘积相对应的,也是一种基本积分方法 +\begin_layout Solution +第二部分： + \end_layout -\begin_layout Standard -设函数 -\begin_inset Formula $u=u(x)$ +\begin_layout Solution +令 +\begin_inset Formula $u=x^{2}-x+1$ \end_inset -及 -\begin_inset Formula $v=v(x)$ +， +则 +\begin_inset Formula $du=(2x-1)dx$ \end_inset -具有连续导数, - 则两个函数乘积的导数公式为 -\begin_inset Formula -\[ -(uv)'=u'v+uv' -\] -\end_inset +\end_layout -移项， -得 -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none -(uv)' -\family default -\series default -\shape default -\size default -\emph default -\bar default -\strikeout default -\xout default -\uuline default -\uwave default -\noun default -\color inherit -- +\begin_layout Solution \begin_inset Formula \[ -uv'=(uv)'-u'v +\int\frac{-2x+8}{x^{2}-x+1}\,dx \] \end_inset - 对这个等式两边求不定积分, - 得 -\begin_inset Formula -\[ -\int uv'dx=uv-\int u'vdx -\] +\end_layout + +\begin_layout Solution +可以拆分并用换元法计算。 + +\end_layout + +\begin_layout Solution +\begin_inset Formula $-2x+8=-2(x-\frac{1}{2})+8-1=-2x+1+7=-2(x-\frac{1}{2})+7$ \end_inset \end_layout -\begin_layout Example -求 -\begin_inset Formula $\int x\cos xdx$ +\begin_layout Solution +\begin_inset Formula $I=\int\frac{-2(x-\frac{1}{2})+7}{x^{2}-x+1}\,dx=\int\frac{-2(x-\frac{1}{2})}{x^{2}-x+1}\,dx+\int\frac{7}{x^{2}-x+1}\,dx$ \end_inset \end_layout \begin_layout Solution -\begin_inset Formula $\int x\cos xdx=\int xd(\sin x)$ +令 +\begin_inset Formula $u=x^{2}-x+1$ \end_inset -,设 -\begin_inset Formula $u=x$ +， +则 +\begin_inset Formula $du=(2x-1)dx$ \end_inset -, -\begin_inset Formula $v=\sin x$ +， +我们需要凑 +\begin_inset Formula $2x-1$ \end_inset -, - 则 +： + +\end_layout + +\begin_layout Solution \begin_inset Formula \[ -\int x\cos xdx=\int xd(\sin x)=x\sin x-\int\sin xdx=x\sin x+\cos x+C +\int\frac{-2(x-\frac{1}{2})}{x^{2}-x+1}\,dx \] \end_inset @@ -50953,25 +53263,39 @@ uv'=(uv)'-u'v \end_layout -\begin_layout Standard -本题如这样变化: - -\begin_inset Formula $\int x\cos xdx=\frac{1}{2}\int\cos xdx^{2}$ +\begin_layout Solution +注意 +\begin_inset Formula $-2(x-\frac{1}{2})=-2x+1$ +\end_inset + +， +所以： + +\end_layout + +\begin_layout Solution +\begin_inset Formula $\int\frac{-2(x-\frac{1}{2})}{x^{2}-x+1}\,dx=-\int\frac{(2x-1)dx}{x^{2}-x+1}$ +\end_inset + + +\end_layout + +\begin_layout Solution +\begin_inset Formula $-\int\frac{du}{u}=-\ln|u|=-\ln|x^{2}-x+1|$ \end_inset -,即 -\begin_inset Formula $u=\cos x$ -\end_inset -, -\begin_inset Formula $v=x^{2}$ -\end_inset +\end_layout + +\begin_layout Solution +最终答案： + +\end_layout -. - 则由分部积分公式有 +\begin_layout Solution \begin_inset Formula \[ -\int x\cos xdx=\frac{1}{2}\int\cos xdx^{2}=\frac{1}{2}(x^{2}\cos x-\int x^{2}\sin xdx) +I=\frac{x^{3}}{3}+\frac{x^{2}}{2}-3x-2\ln|x^{2}-x+1|+C \] \end_inset @@ -50979,299 +53303,222 @@ uv'=(uv)'-u'v \end_layout -\begin_layout Standard -反而比原来的积分更难求了, - 因此, - 在使用分部积分法时,确定 -\begin_inset Formula $u$ -\end_inset +\begin_layout Paragraph +待定系数法 +\end_layout -和 -\begin_inset Formula $v$ +\begin_layout Example +\begin_inset Formula $\int\frac{x^{3}}{(x+1)^{4}}dx$ \end_inset -非常关键. - 一般地, - -\begin_inset Formula $v$ -\end_inset -要容易求得(用凑微分法), - 并且 -\begin_inset Formula $\int vdu$ -\end_inset +\end_layout -要比 -\begin_inset Formula $\int udv$ +\begin_layout Example +\begin_inset Formula $\int(\frac{A_{1}}{(x+1)^{4}}+\frac{A_{2}}{(x+1)^{3}}+\frac{A_{3}}{(x+1)^{2}}+\frac{A_{4}}{(x+1)})dx$ \end_inset -容易积分 + \end_layout -\begin_layout Standard -为更便捷准确地选取恰当的 -\begin_inset Formula $u$ +\begin_layout Example +\begin_inset Formula $x^{3}=A_{1}+A_{2}(x+1)+A_{3}(x+1)^{2}+A_{4}(x+1)^{3}$ \end_inset -和 -\begin_inset Formula $v$ -\end_inset -,我们指出: \end_layout -\begin_layout Standard - -\series bold -当被积函数为幂函数与三角函数(或指数函数)的乘积时, - 选取幂函数作为 -\begin_inset Formula $u$ +\begin_layout Example +\begin_inset Formula $x^{3}=(A_{4})x^{3}+(A_{3}+3A_{4})x^{2}+(A_{2}+2A_{3}+3A_{4})x+(A_{1}+A_{2}+A_{3}+A_{4})$ \end_inset -,将三角函数(或指数函数)凑到“d”的后面选作v. - + \end_layout \begin_layout Example -求 -\begin_inset Formula $\int x^{2}e^{x}dx$ +\begin_inset Formula $\ensuremath{x^{3}}$ +\end_inset + +项： + +\begin_inset Formula $1=A_{4}$ +\end_inset + +， +所以 +\begin_inset Formula $A_{4}=1$ \end_inset \end_layout -\begin_layout Solution -\begin_inset Formula $\int x^{2}e^{x}dx=\int x^{2}de^{x}$ +\begin_layout Example +\begin_inset Formula $A_{3}=-3$ \end_inset -此时取 -\begin_inset Formula $u=x^{2}$ +, +\begin_inset Formula $\ensuremath{A_{2}+2(-3)+3(1)=0\Rightarrow A_{2}-6+3=0\Rightarrow A_{2}=3}$ \end_inset , -\begin_inset Formula $v=e^{x}$ +\begin_inset Formula $A_{1}=-1$ \end_inset - ,于是 -\end_layout -\begin_layout Solution -\begin_inset Formula -\begin{align*} -\int x^{2}e^{x}dx & =\int x^{2}de^{x}=x^{2}e^{x}-\int e^{x}dxx^{2}\\ - & =x^{2}e^{x}-2\int xe^{x}dx\\ - & =x^{2}e^{x}-2(xe^{x}-\int e^{x}dx)\\ - & =x^{2}e^{x}-2xe^{x}+2e^{x}+C\\ - & =(x^{2}-2x+2)e^{x}+C -\end{align*} +\end_layout +\begin_layout Example +\begin_inset Formula $I=\int\left(\frac{-1}{(x+1)^{4}}+\frac{3}{(x+1)^{3}}-\frac{3}{(x+1)^{2}}+\frac{1}{x+1}\right)dx$ \end_inset \end_layout -\begin_layout Standard -该例表明,有时要多次使用分部积分法才能求出结果,一般地,幂函数最高次有几次,就要作几次分部积分. +\begin_layout Example +分别积分： + \end_layout -\begin_layout Standard +\begin_layout Example +\begin_inset Formula $\int\frac{-1}{(x+1)^{4}}dx=\int-(x+1)^{-4}dx=\frac{(x+1)^{-3}}{3}=-\frac{1}{3(x+1)^{3}}$ +\end_inset -\series bold -当被积函数为幂函数与对数函数(或反三角函数)的乘积时, - 选取对数函数(或反三角函 数)作为u,将幂函数凑到“ -\begin_inset Formula $d$ + +\end_layout + +\begin_layout Example +\begin_inset Formula $\int\frac{3}{(x+1)^{3}}dx=\int3(x+1)^{-3}dx=\frac{3(x+1)^{-2}}{-2}=-\frac{3}{2(x+1)^{2}}$ \end_inset -”的后面选作 -\begin_inset Formula $v$ + +\end_layout + +\begin_layout Example +\begin_inset Formula $\int\frac{-3}{(x+1)^{2}}dx=\int-3(x+1)^{-2}dx=\frac{-3(x+1)^{-1}}{-1}=\frac{3}{x+1}$ \end_inset -. + \end_layout \begin_layout Example -求 -\begin_inset Formula $\int x\ln xdx$ +\begin_inset Formula $\int\frac{1}{x+1}dx=\ln|x+1|$ \end_inset \end_layout -\begin_layout Solution +\begin_layout Example \begin_inset Formula -\begin{align*} -\int x\ln xdx & =\frac{1}{2}\int\ln xdx^{2}=\frac{1}{2}(x^{2}\ln x-\int x^{2}d\ln x)\\ - & =\frac{1}{2}(x^{2}\ln x-\int x^{2}d\ln x)=\frac{1}{2}(x^{2}\ln xe-\int xdx)\\ - & =\frac{1}{2}(x^{2}\ln x-\frac{1}{2}x^{2})+C -\end{align*} +\[ +I=-\frac{1}{3(x+1)^{3}}-\frac{3}{2(x+1)^{2}}+\frac{3}{x+1}+\ln|x+1|+C +\] \end_inset \end_layout -\begin_layout Standard -有些不定积分如 -\begin_inset Formula $\int e^{-x^{2}}dx$ -\end_inset +\begin_layout Subsection +部分分式分解 +\end_layout -, -\begin_inset Formula $\int\frac{\sin x}{x}dx$ -\end_inset +\begin_layout Subsection +特殊代换 +\end_layout -, -\begin_inset Formula $\int\frac{\ln x}{x}dx$ -\end_inset +\begin_layout Standard +某些有理函数的积分可以通过三角代换或其他换元简化。 -等,虽然这些不定积分都存在, - 但是不能用初等函数表示这些被积函数的原函数, - 我们称这种情况为“不可积”. \end_layout -\begin_layout Section -有理函数的积分 +\begin_layout Standard +常见的特殊代换 \end_layout \begin_layout Standard -两个多项式的商 -\begin_inset Formula $\frac{P(x)}{Q(x)}$ +\begin_inset Formula $x=a\tan\theta$ \end_inset -称为 -\series bold -有理函数 -\series default -, - 又称 -\series bold -有理分式 -\series default -. - 我们总假定分子多项式 -\begin_inset Formula $P(x)$ + 适用于 +\begin_inset Formula $\frac{dx}{a^{2}+x^{2}}$ \end_inset -与分母多项式 -\begin_inset Formula $Q(x)$ -\end_inset -之间 -\series bold -没有公因式 -\series default -. \end_layout \begin_layout Standard -当分子多项式Ρ(x)的次数小于分母多项式 -\begin_inset Formula $Q(x)$ +\begin_inset Formula $x=a\sinh t$ +\end_inset + + 适用于 +\begin_inset Formula $\frac{dx}{\sqrt{x^{2}+a^{2}}}$ \end_inset -的次数时， -称这个有理函数为 -\series bold -真分式 -\series default -， -否则称为 -\series bold -假分式 -\series default -. -\end_layout -\begin_layout Standard -利用多项式的除法， -可以将一个假分式化成一个多项式与一个真分式之和的形式. \end_layout \begin_layout Standard -对于真分式 -\begin_inset Formula $\frac{P(x)}{Q(x)}$ +\begin_inset Formula $x=a\cosh t$ \end_inset -, - 如果分母可以分解为两个多相式的乘积 -\begin_inset Formula -\[ -Q(x)=Q_{1}(x)Q_{2}(x) -\] - + 适用于 +\begin_inset Formula $\frac{dx}{\sqrt{x^{2}-a^{2}}}$ \end_inset \end_layout -\begin_layout Standard -且 -\begin_inset Formula $Q_{1}(x)$ +\begin_layout Example +\begin_inset Formula $I=\int\frac{dx}{x^{2}+4}$ \end_inset -与 -\begin_inset Formula $Q_{2}(x)$ -\end_inset -没有公因式, - 那么它可以拆分为两个真分式之和 -\begin_inset Formula -\[ -\frac{P(x)}{Q(x)}=\frac{P_{1}(x)}{Q_{1}(x)}+\frac{P_{2}(x)}{Q_{2}(x)} -\] +\end_layout +\begin_layout Solution +令 +\begin_inset Formula $x=2\tan\theta$ \end_inset +， +则： \end_layout -\begin_layout Standard -上述步骤称为把 -\series bold -真分式化成部分分式之和 -\series default -.如果 -\begin_inset Formula $Q_{1}(x)$ +\begin_layout Solution +\begin_inset Formula $dx=2\sec^{2}\theta\,d\theta$ \end_inset -或 -\begin_inset Formula $Q_{2}(x)$ -\end_inset -还能再分解成两个没有公因式的多项式的乘积， -那么就可再分拆成更简单的部分分式.最后， -有理函数的分解式中只出现 -\series bold -多项式, -\begin_inset Formula $\frac{P_{1}(x)}{(x-a)^{k}}$ -\end_inset +\end_layout -, -\begin_inset Formula $\frac{P_{2}(x)}{(x^{2}+px+q)^{l}}$ -\end_inset +\begin_layout Solution +积分化简为： +\end_layout -\series default -等三类函数(这里 -\begin_inset Formula $p^{2}-4q<0$ +\begin_layout Solution +\begin_inset Formula $\int\frac{2\sec^{2}\theta\,d\theta}{4(1+\tan^{2}\theta)}=\frac{1}{2}\int\frac{d\theta}{1+\tan^{2}\theta}=\frac{1}{2}\int d\theta=\frac{1}{2}\theta+C$ \end_inset -, -\begin_inset Formula $P_{1}(x)$ -\end_inset -为小于 -\begin_inset Formula $k$ -\end_inset +\end_layout -次的多项式, - -\begin_inset Formula $P_{2}(x)$ +\begin_layout Solution +换回 +\begin_inset Formula $\theta=\tan^{-1}(x/2)$ \end_inset -为小于 -\begin_inset Formula $2l$ +， +最终结果： + +\end_layout + +\begin_layout Solution +\begin_inset Formula $\frac{1}{2}\tan^{-1}\frac{x}{2}+C$ \end_inset -次的多项式). + \end_layout \begin_layout Chapter diff --git 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