From 42a8d20f838fef393838eb80bfa90d6729091ca8 Mon Sep 17 00:00:00 2001 From: Didnelpsun <48906416+Didnelpsun@users.noreply.github.com> Date: Fri, 15 Jan 2021 21:59:22 +0800 Subject: [PATCH] Update perpare.tex --- 1.1-perpare/perpare.tex | 635 +++++++++++++++++++++++++++++++++++----- 1 file changed, 557 insertions(+), 78 deletions(-) diff --git a/1.1-perpare/perpare.tex b/1.1-perpare/perpare.tex index c33f188..1d9cf59 100644 --- a/1.1-perpare/perpare.tex +++ b/1.1-perpare/perpare.tex @@ -131,7 +131,7 @@ $$ \textbf{例题3：}设$ f(x)=\left\{ - \begin{array}{rcl} + \begin{array}{lcl} \ln\sqrt{x}, & & x\geqslant 1 \\ 2x-1, & & x< 1 \end{array} @@ -140,7 +140,7 @@ $，求$f[f(x)]$ 首先广义化：$ f[f(x)]=\left\{ - \begin{array}{rcl} + \begin{array}{lcl} \ln\sqrt{f(x)}, & & f(x)\geqslant 1 \\ 2f(x)-1, & & x<1 \end{array} @@ -153,20 +153,21 @@ $ \draw[-latex](-1.5,0) -- (9.5,0) node[below]{$x$}; \draw[-latex](0,-1.5) -- (0, 1.5) node[above]{$y$}; \draw[very thin, gray, densely dashed](-1.5,1.5)grid(9.5,-1.5); - \draw [black, thick](-0.25,-1.5) -- (1,1); + \draw[black, thick](-0.25,-1.5) -- (1,1); \draw[black, thick,domain=1:9.5] plot (\x, {ln(sqrt(\x))}); - \draw [red, densely dashed](-1.5,1) -- (9.5,1) node[below]{$x=1$}; - \filldraw [black] (1,1) circle (2pt) node[above]{$(1,1)$}; - \filldraw [black] (e^2,1) circle (2pt) node[above]{$(e^2,1)$}; + \draw[blue, densely dashed](-1.5,1) -- (9.5,1) node[below]{$x=1$}; + \filldraw[black] (1,1) circle (2pt) node[above]{$(1,1)$}; + \filldraw[black] (e^2,1) circle (2pt) node[above]{$(e^2,1)$}; \draw[densely dashed](1,1) -- (1, 0) node[below]{$1$}; \draw[densely dashed](e^2,1) -- (e^2,0) node[below]{$e^2$}; + \filldraw[black] (0,0) node[below]{$O$}; \end{tikzpicture} 所以将定义域分为三段：$[-\infty ,1],[1,e^2],[e^2, +\infty]$，然后根据不同定义域对应的不同函数再代回$f[f(x)]$： $$ f[f(x)]=\left\{ - \begin{array}{rcl} + \begin{array}{lcl} \ln\sqrt{\ln\sqrt{x}}, & & x\geqslant e^2 \\ \ln x-2, & & 1\geqslant x0 & \Rightarrow & (x_1-x_2)[f(x_1)-f(x_2)]>0 & \Rightarrow & f(x)\nearrow \\ - \frac{\rm{d}y}{\rm{d}x}<0 & \Rightarrow & (x_1-x_2)[f(x_1)-f(x_2)]<0 & \Rightarrow &f(x)\searrow + \frac{\rm{d}y}{\rm{d}x}>0 & \Rightarrow & (x_1-x_2)[f(x_1)-f(x_2)]>0 & \Rightarrow & f(x)\nearrow \\ + \frac{\rm{d}y}{\rm{d}x}<0 & \Rightarrow & (x_1-x_2)[f(x_1)-f(x_2)]<0 & \Rightarrow & f(x)\searrow \end{matrix} $ @@ -238,6 +239,8 @@ $y=A$，A为常数，图像平行于x轴： \draw[-latex](-1,0) -- (5,0) node[below]{$x$}; \draw[-latex](0,-0.5) -- (0, 1.5) node[above]{$y$}; \draw[black, thick](-1,1) -- (5,1) node[right]{$y=A$}; + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (0,1) circle (2pt) node at(0.75,0.5){$(0,A)$}; \end{tikzpicture} \subparagraph{幂函数} \leavevmode \bigskip @@ -247,11 +250,13 @@ $y=x^{\mu}$，$\mu$为实数，当$x>0$，$y=x^{\mu}$都有定义： \begin{tikzpicture}[scale=0.9] \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; \draw[-latex](0,-2) -- (0,4) node[above]{$y$}; - \draw[black, thick, domain=0.3:2] plot (\x,1/\x) node[right]{$\mu =-1$}; - \draw[black, thick, domain=-2:-0.5] plot (\x,1/\x) node[right]{$\mu =-1$}; - \draw[black, thick, domain=0.01:2] plot (\x, {sqrt(\x)}) node[right]{$\mu =\frac{1}{2}$}; - \draw[black, thick, domain=-2:2] plot (\x,\x) node[right]{$\mu =1$}; - \draw[black, thick, domain=-2:2] plot (\x, {\x*\x}) node[right]{$\mu =2$}; + \draw[black, thick, smooth, domain=0.3:2] plot (\x,1/\x) node[right]{$\mu =-1$}; + \draw[black, thick, smooth, domain=-2:-0.5] plot (\x,1/\x) node[right]{$\mu =-1$}; + \draw[black, thick, smooth, domain=0.01:2] plot (\x, {sqrt(\x)}) node[right]{$\mu =\frac{1}{2}$}; + \draw[black, thick, smooth, domain=-2:2] plot (\x,\x) node[right]{$\mu =1$}; + \draw[black, thick, smooth, domain=-2:2] plot (\x, {\x*\x}) node[right]{$\mu =2$}; + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (1,1) circle (2pt) node at(1.25,0.5){$(1,1)$}; \end{tikzpicture} 所以对于幂函数，可以根据不同幂下相同单调性来研究最值： @@ -272,6 +277,7 @@ $y=a^x(a>0,a\neq 1)$： \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; \draw[black, thick, domain=-2:2] plot (\x,{pow(1/2,\x)}) node[right]{$01$}; + \filldraw[black] (0,0) node[below]{$O$}; \end{tikzpicture} 指数函数具有如下性质： @@ -293,6 +299,7 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数： \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; \draw[black, thick, domain=0.2:4] plot (\x,{ln(1/\x)}) node[right]{$01$}; + \filldraw[black] (0,0) node[below]{$O$}; \end{tikzpicture} 对数函数具有如下性质： @@ -313,15 +320,15 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数： \begin{tikzpicture}[scale=0.9] \draw[-latex](-5,0) -- (5,0) node[below]{$x$}; \draw[-latex](0,-1.5) -- (0,2) node[above]{$y$}; - \draw[black, thick, domain=-5:5] plot (\x,{sin(\x r)}) node at (0,1.5){$\sin(x)$}; - \draw [black, densely dashed](-5,1) -- (5,1) node[right]{$x=1$}; - \draw [black, densely dashed](-5,-1) -- (5,-1) node[right]{$x=-1$}; - \draw [black, densely dashed](-pi/2*3,1) -- (-pi/2*3,0) node[below]{$-\frac{3\pi}{2}$}; - \draw [black](-pi,0) -- (-pi,0) node[below]{$-\pi$}; - \draw [black, densely dashed](-pi/2,-1) -- (-pi/2,0) node[above]{$-\frac{\pi}{2}$}; - \draw [black](0,0) -- (0,0) node[above]{$0$}; - \draw [black, densely dashed](pi/2,1) -- (pi/2,0) node[below]{$\frac{\pi}{2}$}; - \draw [black](pi,0) -- (pi,0) node[below]{$\pi$}; + \draw[black, thick, smooth, domain=-5:5] plot (\x,{sin(\x r)}) node at (0,1.5){$\sin(x)$}; + \draw[black, densely dashed](-5,1) -- (5,1) node[right]{$x=1$}; + \draw[black, densely dashed](-5,-1) -- (5,-1) node[right]{$x=-1$}; + \draw[black, densely dashed](-pi/2*3,1) -- (-pi/2*3,0) node[below]{$-\frac{3\pi}{2}$}; + \draw[black, densely dashed](-pi/2,-1) -- (-pi/2,0) node[above]{$-\frac{\pi}{2}$}; + \draw[black, densely dashed](pi/2,1) -- (pi/2,0) node[below]{$\frac{\pi}{2}$}; + \draw[black](0,0) -- (0,0) node[above]{$O$}; + \filldraw[black] (-pi-0.1,0) node[below]{$-\pi$}; + \filldraw[black] (pi,0) node[below]{$\pi$}; \end{tikzpicture} 余弦函数： @@ -329,15 +336,15 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数： \begin{tikzpicture}[scale=0.9] \draw[-latex](-5,0) -- (5,0) node[below]{$x$}; \draw[-latex](0,-1.5) -- (0,2) node[above]{$y$}; - \draw[black, thick, domain=-5:5] plot (\x,{cos(\x r)}) node at (0,1.5){$\cos(x)$}; - \draw [black, densely dashed](-5,1) -- (5,1) node[right]{$x=1$}; - \draw [black, densely dashed](-5,-1) -- (5,-1) node[right]{$x=-1$}; - \draw [black](-pi/2*3,0) -- (-pi/2*3,0) node[below]{$-\frac{3\pi}{2}$}; - \draw [black, densely dashed](-pi,-1) -- (-pi,0) node[above]{$-\pi$}; - \draw [black](-pi/2,0) -- (-pi/2,0) node[below]{$-\frac{\pi}{2}$}; - \draw [black, densely dashed](0,1) -- (0,0) node[below]{$0$}; - \draw [black](pi/2,0) -- (pi/2,0) node[below]{$\frac{\pi}{2}$}; - \draw [black, densely dashed](pi,-1) -- (pi,0) node[above]{$\pi$}; + \draw[black, thick, smooth, domain=-5:5] plot (\x,{cos(\x r)}) node at (0,1.5){$\cos(x)$}; + \draw[black, densely dashed](-5,1) -- (5,1) node[right]{$x=1$}; + \draw[black, densely dashed](-5,-1) -- (5,-1) node[right]{$x=-1$}; + \draw[black, densely dashed](-pi,-1) -- (-pi,0) node[above]{$-\pi$}; + \draw[black, densely dashed](pi,-1) -- (pi,0) node[above]{$\pi$}; + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (-pi/2*3-0.25,0) node[below]{$-\frac{3\pi}{2}$}; + \filldraw[black] (-pi/2,0) node[below]{$-\frac{\pi}{2}$}; + \filldraw[black] (pi/2,0) node[below]{$\frac{\pi}{2}$}; \end{tikzpicture} 弦函数有如下特征： @@ -356,33 +363,33 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数： \draw[-latex](-6,0) -- (6,0) node[below]{$x$}; \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; \draw[black, thick, domain=-pi/2+0.5:pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$}; - \draw [black](0,0) -- (0,0) node[below]{$0$}; \draw[black, densely dashed](pi/2,2) -- (pi/2,-2); - \draw[black](pi/2,0) -- (pi/2,0) node at (pi/2+0.5,-0.5){$\frac{\pi}{2}$}; \draw[black, densely dashed](-pi/2,2) -- (-pi/2,-2); - \draw[black](-pi/2,0) -- (-pi/2,0) node at (-pi/2-0.5,-0.5){$-\frac{\pi}{2}$}; \draw[black, thick, domain=-pi/2*3+0.5:-pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$}; \draw[black, densely dashed](pi/2*3,2) -- (pi/2*3,-2); - \draw[black](pi/2*3,0) -- (pi/2*3,0) node at (pi/2*3+0.5,-0.5){$\frac{3\pi}{2}$}; \draw[black, thick, domain=pi/2+0.5:pi/2*3-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$}; \draw[black, densely dashed](-pi/2*3,2) -- (-pi/2*3,-2); - \draw[black](-pi/2*3,0) -- (-pi/2*3,0) node at (-pi/2*3-0.5,-0.5){$-\frac{3\pi}{2}$}; + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (pi/2+0.5,-0.5) node{$\frac{\pi}{2}$}; + \filldraw[black] (-pi/2-0.75,-0.5) node{$-\frac{\pi}{2}$}; + \filldraw[black] (pi/2*3+0.5,-0.5) node{$\frac{3\pi}{2}$}; + \filldraw[black] (-pi/2*3-0.75,-0.5) node{$-\frac{3\pi}{2}$}; \end{tikzpicture} 余切函数： -\begin{tikzpicture}[scale=0.7] +\begin{tikzpicture}[scale=0.7] \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; \draw[black, thick, domain=0.5:pi-0.5] plot (\x,{cot(\x r)}) node at(pi-1,2){$\cot(x)$}; - \draw [black](0,0) -- (0,0) node[below]{$0$}; - \draw[black](pi/2,0) -- (pi/2,0) node[below]{$\frac{\pi}{2}$}; \draw[black, densely dashed](pi,2) -- (pi,-2); - \draw[black](pi,0) -- (pi,0) node at (pi+0.5,-0.5){$\pi$}; \draw[black, thick, domain=-0.5:-pi+0.5] plot (\x,{cot(\x r)}) node at(-1,2){$\cot(x)$}; - \draw[black](-pi/2,0) -- (-pi/2,0) node[below]{$-\frac{\pi}{2}$}; \draw[black, densely dashed](-pi,2) -- (-pi,-2); - \draw[black](-pi,0) -- (-pi,0) node at (-pi-0.5,-0.5){$-\pi$}; + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (pi/2,0) node[below]{$\frac{\pi}{2}$}; + \filldraw[black] (pi+0.5,-0.5) node{$\pi$}; + \filldraw[black] (-pi/2-0.25,0) node[below]{$-\frac{\pi}{2}$}; + \filldraw[black] (-pi-0.5,-0.5) node{$-\pi$}; \end{tikzpicture} 切函数有如下特征： @@ -395,7 +402,7 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数： \end{enumerate} $$ -\sec x=\frac{1}{\cos x},\csc x=\frac{1}{\sin x} + \sec x=\frac{1}{\cos x},\csc x=\frac{1}{\sin x} $$ 正割函数： @@ -414,13 +421,13 @@ $$ \draw[black, densely dashed](-pi/2,3) -- (-pi/2,-3); \draw[black, densely dashed](pi/2,3) -- (pi/2,-3); \draw[black, densely dashed](pi/2*3,3) -- (pi/2*3,-3); - \draw[black](0,0) -- (0,0) node[below]{$0$}; - \draw[black](0,1) -- (0,1) node at(0.5,0.5){$1$}; - \draw[black](0,-1) -- (0,-1) node at(0.5,-1.5){$-1$}; - \draw[black](-pi/2*3,0) -- (-pi/2*3,0) node at(-pi/2*3-0.5,-0.5){$-\frac{3\pi}{2}$}; - \draw[black](-pi/2,0) -- (-pi/2,0) node at(-pi/2-0.5,-0.5){$-\frac{\pi}{2}$}; - \draw[black](pi/2,0) -- (pi/2,0) node at(pi/2+0.5,-0.5){$\frac{\pi}{2}$}; - \draw[black](pi/2*3,0) -- (pi/2*3,0) node at(pi/2*3+0.5,-0.5){$\frac{3\pi}{2}$}; + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (0.5,0.5) node{$1$}; + \filldraw[black] (0.5,-1.5) node{$-1$}; + \filldraw[black] (-pi/2*3-0.75,-0.5) node{$-\frac{3\pi}{2}$}; + \filldraw[black] (-pi/2-0.75,-0.5) node{$-\frac{\pi}{2}$}; + \filldraw[black] (pi/2+0.5,-0.5) node{$\frac{\pi}{2}$}; + \filldraw[black] (pi/2*3+0.5,-0.5) node{$\frac{3\pi}{2}$}; \end{tikzpicture} 余割函数： @@ -438,13 +445,13 @@ $$ \draw[black, densely dashed](-pi*2,3) -- (-pi*2,-3); \draw[black, densely dashed](pi,3) -- (pi,-3); \draw[black, densely dashed](pi*2,3) -- (pi*2,-3); - \draw[black](0,0) -- (0,0) node[below]{$0$}; - \draw[black](0,1) -- (0,1) node at(0.5,0.5){$1$}; - \draw[black](0,-1) -- (0,-1) node at(0.5,-1.5){$-1$}; - \draw[black](-pi,0) -- (-pi,0) node at(-pi-0.5,-0.5){$\pi$}; - \draw[black](-pi*2,0) -- (-pi*2,0) node at(-pi*2+0.5,-0.5){$2\pi$}; - \draw[black](pi,0) -- (pi,0) node at(pi+0.5,-0.5){$\pi$}; - \draw[black](pi*2,0) -- (pi*2,0) node at(pi*2-0.5,-0.5){$2\pi$}; + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (0.5,0.5) node{$1$}; + \filldraw[black] (0.5,-1.5) node{$-1$}; + \filldraw[black] (-pi-0.5,-0.5) node{$\pi$}; + \filldraw[black] (-pi*2+0.5,-0.5) node{$2\pi$}; + \filldraw[black] (pi+0.5,-0.5) node{$\pi$}; + \filldraw[black] (pi*2-0.5,-0.5) node{$2\pi$}; \end{tikzpicture} 割函数有如下特征： @@ -463,11 +470,11 @@ $$ \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$}; \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; \draw[black, thick, domain=-1:1] plot (\x,{rad(asin(\x))}) node[right]{$\arcsin(x)$}; - \draw [black, densely dashed](1,pi/2) -- (0,pi/2) node[left]{$\frac{\pi}{2}$}; - \draw [black](0,0) -- (0,0) node[below]{$0$}; - \draw [black, densely dashed](1,pi/2) -- (1,0) node[below]{$1$}; - \draw [black, densely dashed](-1,-pi/2) -- (0,-pi/2) node[right]{$-\frac{\pi}{2}$}; - \draw [black, densely dashed](-1,-pi/2) -- (-1,0) node[above]{$-1$}; + \draw[black, densely dashed](1,pi/2) -- (0,pi/2) node[left]{$\frac{\pi}{2}$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](1,pi/2) -- (1,0) node[below]{$1$}; + \draw[black, densely dashed](-1,-pi/2) -- (0,-pi/2) node[right]{$-\frac{\pi}{2}$}; + \draw[black, densely dashed](-1,-pi/2) -- (-1,0) node[above]{$-1$}; \end{tikzpicture} 反余弦函数： @@ -476,10 +483,11 @@ $$ \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$}; \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; \draw[black, thick, domain=-1:1] plot (\x,{rad(acos(\x)}) node at (-2, pi){$\arccos(x)$}; - \draw [black, densely dashed](0,pi/2) -- (0,pi/2) node[left]{$\frac{\pi}{2}$}; - \draw [black, densely dashed](1,0) -- (1,0) node[below]{$1$}; - \draw [black, densely dashed](-1,pi) -- (0,pi) node[right]{$\pi$}; - \draw [black, densely dashed](-1,pi) -- (-1,0) node[above]{$-1$}; + \filldraw[black] (0,pi/2) node[right]{$\frac{\pi}{2}$}; + \draw[black](1,0) -- (1,0) node[below]{$1$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](-1,pi) -- (0,pi) node[right]{$\pi$}; + \draw[black, densely dashed](-1,pi) -- (-1,0) node[below]{$-1$}; \end{tikzpicture} 反弦函数有如下特征： @@ -487,7 +495,7 @@ $$ \begin{enumerate} \item 特殊函数值：$\arcsin 0=0$，$\arcsin\frac{1}{2}=\frac{\pi}{6}$，$\arcsin\frac{\sqrt{2}}{2}=\frac{\pi}{4}$，$\arcsin\frac{\sqrt{3}}{2}=\frac{\pi}{3}$，$\arcsin 1=\frac{\pi}{2}$，$\arccos 1=0$，$\arccos\frac{\sqrt{3}}{2}=\frac{\pi}{6}$，$\arccos\frac{\sqrt{2}}{2}=\frac{\pi}{4}$，$\arccos\frac{1}{2}=\frac{\pi}{3}$，$\arccos 0=\frac{\pi}{2}$。 \item 定义域：$(-1, +1)$，值域：$\arcsin x:[-\frac{\pi}{2},+\frac{\pi}{2}]$，$\arccos x:[0,\pi]$。 - \item 单调性：$y=\arcsin x$单调增，$y=\arccos x$单调减 + \item 单调性：$y=\arcsin x$单调增，$y=\arccos x$单调减。 \item 奇偶性：$y=\arcsin x$为奇函数。 \item 有界性：$\vert\arcsin x\vert\leqslant\frac{\pi}{2}$，$0\leqslant\arccos x\leqslant\pi$。 \item 性质：$\arcsin x+\arccos x=\frac{\pi}{2}(-1\leqslant x\leqslant 1)$ @@ -505,31 +513,502 @@ $$ \draw[-latex](-3,0) -- (3,0) node[below]{$x$}; \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; \draw[black, thick, domain=-3:3] plot (\x,{rad(atan(\x))}) node[right]{$\arcsin(x)$}; - \draw [black, densely dashed](-3,pi/2) -- (3,pi/2) node[right]{$\frac{\pi}{2}$}; - % \draw [black](0,0) -- (0,0) node[below]{$0$}; - % \draw [black, densely dashed](1,pi/2) -- (1,0) node[below]{$1$}; - % \draw [black, densely dashed](-1,-pi/2) -- (0,-pi/2) node[right]{$-\frac{\pi}{2}$}; - % \draw [black, densely dashed](-1,-pi/2) -- (-1,0) node[above]{$-1$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](-3,pi/2) -- (3,pi/2); + \draw[black, densely dashed](-3,-pi/2) -- (3,-pi/2); + \filldraw[black] (0.5,pi/2-0.5) node{$\frac{\pi}{2}$}; + \filldraw[black] (0.5,-pi/2-0.5) node{$-\frac{\pi}{2}$}; +\end{tikzpicture} + +反余切函数： + +\begin{tikzpicture}[scale=0.9] + \draw[-latex](-3,0) -- (3,0) node[below]{$x$}; + \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$}; + \draw[black, thick, domain=-3:3] plot (\x,{pi/2-rad(atan(\x))}) node[right]{$\rm{arccot}(x)$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, densely dashed](-3,pi) -- (3,pi); + \filldraw[black] (-0.5,pi/2-0.5) node{$\frac{\pi}{2}$}; +\end{tikzpicture} + +反切函数有如下特征： + +\begin{enumerate} + \item 特殊函数值：$\arctan 0=0$，$\arctan\frac{\pi}{6}=\frac{\sqrt{3}}{3}=$，$\arctan 1=\frac{\pi}{4}$，$\arctan\sqrt{3}=\frac{\pi}{3}$，$\rm{arccot}0=\frac{\pi}{2}$，$\rm{arccot}\sqrt{3}=\frac{\pi}{6}$，$\rm{arccot}1=\frac{\pi}{4}$，$\rm{arccot}\frac{\sqrt{3}}{3}=\frac{\pi}{3}$。 + \item 定义域：$(-\infty, +\infty)$，值域：$\arctan x:[-\frac{\pi}{2},+\frac{\pi}{2}]$，$\rm{arccot}x:[0,\pi]$。 + \item 单调性：$y=\arctan x$单调增，$y=\rm{arccot}x$单调减。 + \item 奇偶性：$y=\arctan x$为奇函数。 + \item 有界性：$\vert\arctan x\vert\leqslant\frac{\pi}{2}$，$0\leqslant\rm{arccot}x\leqslant\pi$。 + \item 性质：$\arctan x+\rm{arccot}x=\frac{\pi}{2}(-\inftyx_0 \\ + a, & & x=x_0 \\ + \psi_2(x), & & x0 \\ + 0, & & x=0 \\ + -1, & & x<0 + \end{array} + \right. +$ + +\begin{tikzpicture} + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$}; + \draw[black, thick, domain=0:2] plot (\x,1); + \draw[black, thick, domain=-2:0] plot (\x,-1); + \filldraw[black] (-1.5,1) node{$\rm{sgn}x$}; + \filldraw[black] circle (2pt) (0,0) node[below]{$O$}; + \filldraw[white, draw=black, line width=1pt] (0,1) circle (2pt); + \filldraw[black] (0,1) node[left]{$1$}; + \filldraw[white, draw=black, line width=1pt] (0,-1) circle (2pt); + \filldraw[black] (0,-1) node[right]{$-1$}; +\end{tikzpicture} + +\subparagraph{取整函数} \leavevmode \bigskip + +$x$为实数，不超过$x$的最大整数称为其整数部分$[x]$，其定义域为$R$，值域为$Z$。 + +\begin{enumerate} + \item $x-1<[x]\leqslant x$。 + \item $\lim_{x\to 0^+}[x]=0$。 + \item $\lim_{x\to 0^-}[x]=-1$。 +\end{enumerate} + +\begin{tikzpicture}[scale=0.6] + \draw[-latex](-3.5,0) -- (4.5,0) node[below]{$x$}; + \draw[-latex](0,-3.5) -- (0,3.5) node[above]{$y$}; + \draw[black, thick, domain=1:2] plot (\x,1); + \draw[black, thick, domain=2:3] plot (\x,2); + \draw[black, thick, domain=3:4] plot (\x,3); + \draw[black, thick, domain=-1:0] plot (\x,-1); + \draw[black, thick, domain=-2:-1] plot (\x,-2); + \draw[black, thick, domain=-3:-2] plot (\x,-3); + \filldraw[black] (-2,2) node{$[x]$}; + \filldraw[black] circle (2pt) (0,0) node[below]{$O$}; + \foreach \x in {-2,...,4} + \filldraw[white, draw=black, line width=1pt] (\x,\x-1) circle (2pt); + \foreach \x in {3,...,-3} + \filldraw[black] (\x,\x) circle (2pt); + \foreach \x/\xtext in {-3,...,-1} + \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt); + \foreach \x/\xtext in {1,...,4} + \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt); + \foreach \x/\xtext in {1,...,3} + \filldraw[black] (0,\x) node[left]{\xtext} -- +(3pt, 0); + \foreach \x/\xtext in {-3,...,-1} + \filldraw[black] (0,\x) node[right]{\xtext} -- +(3pt, 0); \end{tikzpicture} -\paragraph{分段函数} \subsubsection{图像变换} \paragraph{平移变换} +\subparagraph{左右平移} \leavevmode \bigskip + +$f(x)$沿$x$轴左移$x_0$个单位长度得到$f(x+x_0)$，向右移动$x_0$个单位则得到$f(x-x_0)$： + +\begin{tikzpicture}[scale=0.9] + \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1); + \filldraw[black] (0,1.5) node{$-x^2+1$}; + \draw[black, thick, domain=0.5:3.5] plot (\x,{-pow((\x-2),2)+1}); + \filldraw[black] (2.5,1.5) node{$-(x-2)^2+1$}; + \draw[black, thick, domain=-3.5:-0.5] plot (\x,{-pow((\x+2),2)+1}); + \filldraw[black] (-2.5,1.5) node{$-(x+2)^2+1$}; + \filldraw[black] (1,0.5) node{$\rightarrow$}; + \filldraw[black] (-1,0.5) node{$\leftarrow$}; +\end{tikzpicture} + +\subparagraph{上下平移} \leavevmode \bigskip + +$f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$，向下移动$y_0$个单位则得到$f(x)-y_0$： + +\begin{tikzpicture}[scale=0.9] + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-4) -- (0,4) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1); + \filldraw[black] (0,-0.75) node{$-x^2+1$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+3}); + \filldraw[black] (0,1.5) node{$-x^2+3$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+-1}); + \filldraw[black] (0,-2.5) node{$-x^2-1$}; + \filldraw[black] (-2,2.5) node{$\uparrow $}; + \filldraw[black] (-2,-2.5) node{$\downarrow $}; +\end{tikzpicture} + \paragraph{对称变换} +\subparagraph{上下对称} \leavevmode \bigskip + +将$f(x)$关于$x$轴对称得到$-f(x)$： + +\begin{tikzpicture} + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1); + \filldraw[black] (0,1.5) node{$-x^2+1$}; + \draw[black, thick, domain=-1.5:1.5] plot (\x,\x*\x-1); + \filldraw[black] (0,-1.5) node{$x^2-1$}; +\end{tikzpicture} + +\subparagraph{左右对称} \leavevmode \bigskip + +将$f(x)$关于$y$轴对称得到$f(-x)$： + +\begin{tikzpicture}[scale=0.8] + \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)}); + \filldraw[black] (1.5,1.5) node{$\ln x$}; + \draw[black, thick, domain=-0.25:-3.5] plot (\x,{ln(-\x)}); + \filldraw[black] (-1.5,1.5) node{$\ln -x$}; +\end{tikzpicture} + +\subparagraph{原点对称} \leavevmode \bigskip + +将$f(x)$关于$x$轴$y$轴即关于原点对称得到$-f(-x)$： + +\begin{tikzpicture}[scale=0.8] + \draw[-latex](-4,0) -- (4,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)}); + \filldraw[black] (1.5,1.5) node{$\ln x$}; + \draw[black, thick, domain=-0.25:-3.5] plot (\x,{-ln(-\x)}); + \filldraw[black] (-1.5,-1.5) node{$-\ln -x$}; +\end{tikzpicture} + +\subparagraph{反函数对称} \leavevmode \bigskip + +将$f(x)$关于$y=x$轴对称得到$f^{-1}(x)$： + +\begin{tikzpicture}[scale=0.8] + \draw[-latex](-2,0) -- (e,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,e) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0.25:e] plot (\x,{ln(\x)}); + \filldraw[black] (1.5,-1.5) node{$\ln x$}; + \draw[black, thick, domain=-1:1] plot (\x,{exp(\x)}); + \filldraw[black] (-1.5,1.5) node{$e^x$}; + \draw[black, densely dashed] (-2,-2) -- (e-0.5,e-0.5) node[above]{$y=x$}; +\end{tikzpicture} + +\subparagraph{函数绝对值} \leavevmode \bigskip + +保留$f(x)$函数值在$[0,\infty]$的部分，并对$[-\infty,0]$部分进行上下对称： + +\begin{tikzpicture} + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-2) -- (0,2) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=1:1.5] plot (\x,\x*\x-1); + \draw[black, thick, densely dashed, domain=-1:1] plot (\x,\x*\x-1); + \draw[black, thick, domain=-1:1] plot (\x,-\x*\x+1); + \draw[black, thick, domain=-1.5:-1] plot (\x,\x*\x-1); + \filldraw[black] (0,1.5) node{$\vert x^2-1\vert$}; +\end{tikzpicture} + +\subparagraph{自变量绝对值} \leavevmode \bigskip + +先只保留$f(x)$定义域在$[0,\infty]$的部分，然后在$[-\infty,0]$部分使用$[0,\infty]$的部分进行左右对称： + +\begin{tikzpicture} + \draw[-latex](-2,0) -- (2,0) node[below]{$x$}; + \draw[-latex](0,-1) -- (0,3) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw[black, thick, domain=0:1.25] plot (\x,{-pow(\x,3)+1}); + \draw[black, thick, densely dashed, domain=-1.25:0] plot (\x,{-pow(\x,3)+1}); + \draw[black, thick, domain=-1.25:0] plot (\x,{-pow(-\x,3)+1}); + \filldraw[black] (1,2) node{$-\vert x\vert^3+1$}; +\end{tikzpicture} + \paragraph{伸缩变换} +\subparagraph{水平伸缩} \leavevmode \bigskip + +纵坐标不变，当$k>1$时，$y=f(kx)$是$y=f(x)$缩短k倍得到，当$00$，周期为$2\pi$。 + +在直角坐标系下表达式：$x^2+y^2+a*x=a\cdot\sqrt{x^2+y^2}$和$x^2+y^2-a\cdot x=a\cdot\sqrt{x^2+y^2}$。 + +参数方程：$x=a\cdot(2\cdot\cos(t)-cos(2\cdot t))$与$y=a\cdot(2\cdot\sin(t)-sin(2\cdot t))$ + +\begin{tikzpicture}[scale=0.8] + \draw[-latex](-5,0) -- (1,0) node[below]{$x$}; + \draw[-latex](0,-3) -- (0,3) node[above]{$y$}; + \draw[black, thick, domain=0:360,smooth,variable=\t, samples=300] plot ({\t}:{2*(1-cos(\t))}); + \filldraw[black] (0,0) node[below]{$O$}; +\end{tikzpicture} + +水平心形线对应参数： \leavevmode \bigskip + +\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} + \hline + $\theta$ & $0$ & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{\pi}{2}$ & $\frac{2\pi}{3}$ & $\frac{3\pi}{4}$ & $\frac{5\pi}{6}$ & $\pi$ \\ \hline + $r$ & $0$ & $\frac{2-\sqrt{3}}{2}a$ & $\frac{2-\sqrt{2}}{2}a$ & $\frac{1}{2}a$ & $a$ & $\frac{3}{2}a$ & $\frac{2+\sqrt{2}}{2}a$ & $\frac{2+\sqrt{3}}{2}a$ & $2a$ \\ + \hline +\end{tabular} + +\paragraph{玫瑰线} \leavevmode \bigskip + +表达式：$r=a\sin(n\theta)$，周期为$\frac{2\pi}{n}$。 + +当$n$为3时为三叶，2时为四叶，$\frac{3}{2}$为六叶。三叶时周期为$\frac{2\pi}{3}$。 + +直角坐标系下表达式：$x=a\cdot\sin(n\cdot\theta)\cdot\cos(\theta)$与$y=a\cdot\sin(n\cdot)\cdot\sin(\theta)$ + +\begin{tikzpicture}[scale=0.8] + \draw[-latex](-3,0) -- (3,0) node[below]{$x$}; + \draw[-latex](0,-pi) -- (0,pi/2) node[above]{$y$}; + \draw[domain=0:180,samples=100] plot (\x:{3*sin(\x*3)}); + \filldraw[black] (0,0) node[below]{$O$}; +\end{tikzpicture} + +三叶玫瑰线对应参数： \leavevmode \bigskip + +\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} + \hline + $\theta$ & $0$ & $\frac{\pi}{12}$ & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{5\pi}{12}$ & $\frac{\pi}{2}$ & $\frac{7\pi}{12}$ & $\frac{3\pi}{2}$ \\ \hline + $r$ & $0$ & $\frac{\sqrt{2}}{2}a$ & $a$ & $\frac{\sqrt{2}}{2}a$ & $0$ & $-frac{\sqrt{2}}{2}a$ & $-a$ & $-frac{\sqrt{2}}{2}a$ & $0$ \\ + \hline +\end{tabular} + +\paragraph{阿基米德螺线} \leavevmode \bigskip + +表达式：$r=a\theta$，其中$a>0$，$\theta\geqslant 0$由0开始增大时$r$也在不断增大。 + +\begin{tikzpicture}[scale=0.2] + \draw[-latex](-10,0) -- (15,0) node[below]{$x$}; + \draw[-latex](0,-15) -- (0,10) node[above]{$y$}; + \draw[domain=0:720,samples=100] plot (\x:{rad(\x)}); + \filldraw[black] (0,0) node[below]{$O$}; +\end{tikzpicture} + +\paragraph{伯努利双扭线} \leavevmode \bigskip + +设定线段$F_1F_2$长度为$2a$，伯努利双扭线上所有点M满足$MF_1\cdot MF_2=a^2$。 + +表达式：$r^2=2a^2\cos 2\theta$或$r^2=2a^2\sin 2\theta$。 + +直角坐标系下表达式：$(x^2+y^2)^2=2a^2(x^2-y^2)$。 + +\begin{tikzpicture}[scale=1.5] + \draw[-latex](-1.25,0) -- (1.25,0) node[below]{$x$}; + \draw[-latex](0,-1) -- (0,1) node[above]{$y$}; + \draw[domain=-45:45,samples=100] plot (\x:{sqrt(cos(\x*2))}); + \draw[domain=-45:45,samples=100] plot (\x:{-sqrt(cos(\x*2))}); + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (0,-1) node{$r^2=2a^2\cos 2\theta$}; +\end{tikzpicture} +\hspace{2.5em} +\begin{tikzpicture}[scale=1.5] + \draw[-latex](-1.25,0) -- (1.25,0) node[below]{$x$}; + \draw[-latex](0,-1) -- (0,1) node[above]{$y$}; + \draw[domain=0:90,samples=100] plot (\x:{sqrt(sin(\x*2))}); + \draw[domain=0:90,samples=100] plot (\x:{-sqrt(sin(\x*2))}); + \filldraw[black] (0,0) node[below]{$O$}; + \filldraw[black] (0,-1) node{$r^2=2a^2\sin 2\theta$}; +\end{tikzpicture} + \subsubsection{直角坐标系下画极坐标图像} + +令$\theta$为$x$，令$r$为$y$。如心形线$r=2(1-\cos\theta)$： + +\begin{tikzpicture}[scale=0.5] + \draw[-latex](-5,0) -- (5,0) node[below]{$x$}; + \draw[-latex](0,-0.5) -- (0,5) node[above]{$y$}; + \draw[black, thick, smooth, domain=-5:5] plot (\x,{2*(1-cos(\x r))}) node at (0,4){$2(1-\cos(\theta))$}; + \filldraw[black] (0,0) node[below]{$O$}; +\end{tikzpicture} + +按直角坐标系的图就可以计算出对应的$r$从而能画出对应的图像。 + \subsection{参数法} + +如果很难使用直角坐标或极坐标来表示曲线，那么可以引入一个新的变量参数来表示，即得到参数方程：$ + \left\{ + \begin{array}{lcl} + x=x(t) \\ + y=y(t) + \end{array} + \right. +$ + \subsubsection{摆线（平摆线）} + +摆线，又称旋轮线、圆滚线，是一个圆沿一条直线滚动时，圆边界上一定点所形成的轨迹。 + +令圆半径为$r$，摆点与圆心所成直线所转动夹角对应弧度为$t$，其中$t\in[0,2\pi]$，所对应参数方程为： + +$$ + \left\{ + \begin{array}{lcl} + x=r(t-\sin t) \\ + y=r(1-\cos t) + \end{array} + \right. +$$ + \subsubsection{星形线（内摆线）} + +与半径为$r$的定圆内切的半径为$\frac{r}{4}$的动圆沿定圆无滑动地滚动，动圆上一点的轨迹称为星形线。 + +令$t$表示摆点与圆心的连线所构成夹角的弧度，其中$t\in[0,2\pi]$，得对应参数方程： + +$$ + \left\{ + \begin{array}{lcl} + x=r\cos^3t \\ + y=r\sin^3t + \end{array} + \right. +$$ + +由$\cos^2t+\sin^2t=1$得到直角坐标方程：$x^{\frac{2}{3}}+y^{\frac{2}{3}}=r^{\frac{2}{3}}$ + \section{常用基础知识} \subsection{数列} +\subsubsection{等差数列} + +首项为$a_1$，公差为$d(d\neq 0)$的数列：$a_1,a_1+d,a_1+2d\cdots a_1+(n-1)d$。 + +通项公式：$a_n=a_1+(n-1)d$。 + +前$n$项和：$S_n=\frac{n}{2}[2a_1+(n-1)d]=\frac{n}{2}(a_1+a_n)$ + +\subsubsection{等比数列} + +首项为$a_1$，公比为$q(q\neq 0)$的数列：$a_1,a_1q,a_1a^2\cdots a_1q^{n-1}$。 + +通项公式：$a_n=a_1q^{n-1}$。 + +前$n$项和：$S_n= + \left\{ + \begin{array}{lcl} + na_1, & & r=1 \\ + \frac{a_1(1-r^n)}{1-r}, & & r\neq 1 + \end{array} +\right.$ + +若首项为1，则$1+r+r^2+\cdots+r^{n-1}=\frac{1-r^n}{1-r}(r\neq 1)$。 + +则对无穷的极限为$\frac{1}{1-r}$。 + +\subsubsection{常见数列前$n$项和} + +\begin{enumerate} + \item $\sum_{k=1}^nk=1+2+\cdots+n=\frac{n(n+1)}{2}$。 + \item $\sum_{k=1}^nk^2=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$。 + \item $\sum_{k=1}^n\frac{1}{k(k+1)}=\frac{1}{1\times 2}+\frac{1}{2\times 3}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}$。 +\end{enumerate} + \subsection{三角函数} + +\subsubsection{基本关系} + +$\csc\alpha=\frac{1}{\sin\alpha},\sec\alpha=\frac{1}{\cos\alpha},\cot\alpha=\frac{1}{\tan\alpha},\tan\alpha=\frac{\sin\alpha}{\cos\alpha},\cot\alpha=\frac{\cos\alpha}{\sin\alpha}$。 + +$\sin^2\alpha+\cos^2\alpha=1,1+\tan^2\alpha=\sec^2\alpha,1+\cot^2\alpha=\csc^2\alpha$。 + +\subsubsection{诱导公式} + +奇变偶不变，符号看象限。奇指前面添加的常数项是否为$\pi$的整数倍，是就需要改变函数，看象限指添加了常数后整体的符号看函数所在象限的符号。 + +$\sin(\frac{\pi}{2}\pm\alpha)=\cos\alpha$ + +$\cos(\frac{\pi}{2}\pm\alpha)=\mp\sin\alpha$ + +$\sin(\pi\pm\alpha)=\mp\sin\alpha$ + +$\cos(\pi\pm\alpha)=-\cos\alpha$ + +\subsubsection{倍角公式} + +$\sin 2\alpha=2\sin\alpha\cos\alpha,\cos 2\alpha=\cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1$。 + +$\sin 3\alpha=-4\sin^3\alpha_3\sin\alpha,\cos 3\alpha=4\cos^3\alpha-3\cos\alpha$。 + +$\tan 2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha},\cot 2\alpha=\frac{\cot^2\alpha-1}{2\cot\alpha}$。 + \subsection{指数运算法则} \subsection{对数运算法则} \subsection{一元二次方程基础}