复习用，终究还是画了图标记出来比较清楚。参考张宇老师的30讲。其中公式类似MathJax，部分可参考但略有不同，详细内容见：
matplotlib 官网
https://matplotlib.org/stable/tutorials/text/mathtext.html
基本函数图像(p6)：
幂函数：y = xᵃ
1.定义域：定义域取决于指数a。当 | a | < 1，x ＜ 0 在实数范围无定义。
import mathimport numpy as npimport matplotlib.pyplot as pltimport mpl_toolkits.axisartist as axisartistdef draw(X):fig = plt.figure(figsize=(10, 10))ax = axisartist.Subplot(fig, 111)fig.add_axes(ax)ax.axis[“x”] = ax.new_floating_axis(0, 0, axis_direction=“bottom”)ax.axis[“y”] = ax.new_floating_axis(1, 0, axis_direction=“bottom”)ax.set_yticks([-5,-4,-3,-2,-1.5,-1,1,1.5,2,3,4,5])ax.set_xticks([-5,-4,-3,-2,-1.5,-1,0,1,1.5,2,3,4,5])ax.axis[“x”].set_axisline_style(”->“, size=1.0)ax.axis[“y”].set_axisline_style(”->“, size=1.0)ax.axis[‘top’,‘right’,‘left’,‘bottom’].set_visible(False)plt.xlim(-max(X), max(X))plt.ylim(-max(X), max(X))ax.text(-0.5, max(x), ‘y’, fontsize=20)ax.annotate(text=’ x’, xy=(max(X), 0), fontsize=20)ax.grid(True, linestyle=’-.’)return axif name==”main“:x = np.linspace(-5, 5, 100)ax = draw(x)y = xax.plot(x, y,label = ‘y=x’)y = x2ax.annotate(r’y=x²’,xy=(2, 4), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=-0.3”))ax.annotate(r’y=x²’,xy=(-2, 4), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=-0.3”))ax.plot(x, y,label = ‘y=x²’)y = x3ax.annotate(r’y=x³’,xy=(math.pow(3, 1/3), 3), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=-0.3”))ax.annotate(r’y=x³’,xy=(-math.pow(3, 1/3), -3), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=-0.3”))ax.plot(x, y,label = ‘y=x³’)y = 1/xax.annotate(r’y=\(\frac{1}{x}\)‘,xy=(0.5, 2), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=-0.3”))ax.annotate(r’y=\(\frac{1}{x}\)‘,xy=(-(1/1.2), -1.2), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=0”))ax.plot(x, y,label = ‘y=1/x (y=x⁻¹)’)y = 1/x2y = x-2ax.annotate(r’y=\(\frac{1}{x²}\)‘,xy=(2, 0.25), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=-0.3”))ax.annotate(r’y=\(\frac{1}{x²}\)‘,xy=(-2, 0.25), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=-0.3”))ax.plot(x, y,label = ‘y=1/x² (y=x⁻²)’)x = np.linspace(0, 5, 100)y = list(map(lambda x:eval(“math.sqrt(x)”), x))label = ‘y=x⁻²’ax.annotate(r’y=\(\sqrt{x}\)‘,xy=(4, 2), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=-0.3”))ax.plot(x, y, label=label)y = list(map(lambda x:eval(“math.pow(x, 1⁄3)”), x))label = ‘y=x⁻³’ax.annotate(r’y=\(\sqrt[3]{x}\)‘,xy=(3, math.pow(3, 1/3)), xycoords=‘data’, xytext=(+10, +30), textcoords=‘offset points’, fontsize=16,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=-0.3”))ax.plot(x, y, label=label)plt.legend()plt.show()
讯享网
指数函数：y = aˣ
3.单调性：底数 a > 1 单增，0 < a< 1 单减
4.极限：当 x -> -∞，eˣ -> 0；x -> +∞：eˣ -> +∞
讯享网x = np.linspace(-5, 5, 100)ax = draw(x)y = list(map(lambda x:math.exp(x), x))label = ‘y=eˣ’ax.plot(x, y, label=label)y = list(map(lambda x:1/math.exp(x), x))label = ‘y=e⁻ˣ’ax.plot(x, y, label=label)y = 2xlabel = ‘y=2ˣ’ax.plot(x, y, label=label)y = 0.5xlabel = r’y=(\(\frac{1}{2}\))ˣ’ax.plot(x, y, label=label)plt.legend()plt.show()
对数函数：y = logₐx
3.单调性：底数 a > 1 单增，0 < a< 1 单减
4.特殊值：lnx = logₑx = 2.71828…，logₐ1 = 0，logₐa = 1，lne=1
5.极限：当 x -> 0⁺，lnx -> -∞ (∵e>1)；x -> +∞：lnx -> +∞
6.代码：
x = np.linspace(0.001, 5, 100)ax = draw(x)y = np.log(x)label = ‘y=lnx’ax.plot(x, y, label=label)y = list(map(lambda x:math.log(x,0.2), x))label = r’y=log\(_\frac{1}{5}\)x’ax.plot(x, y, label=label)y = list(map(lambda x:math.log(x,5), x))label = r’y=log\(_5\)x’ax.plot(x, y, label=label)plt.legend()plt.show()
三角函数：
①正弦、余弦
2.值域：[-1，+1]
3.奇偶性：y = sin(x) 是奇函数，y = cos(x) 是偶函数
4.周期性：2π 为最小正周期
5.有界性：| sin(x) | ≤ 1，| cos(x) | ≤ 1
6.特殊值：sin0=0，sin(π/6)=0.5…
讯享网def show_value(ax, show,coordinate,xytext):ax.annotate(show,xy=coordinate, xycoords=‘data’,xytext=xytext, textcoords=‘offset points’, fontsize=12,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=.1”))if name==“main”:x = np.linspace(-5, 5, 100)ax = draw(x)y = np.sin(x)label = ‘y=sinx’plt.xticks([-np.pi, -np.pi/2, 0, np.pi/2, np.pi],[r’\(-\pi\)‘, r’\(-\pi/2\)‘, r’\(0\)‘, r’\(+\pi/2\)‘, r’\(+\pi\)‘])ax.plot(x, y, label=label)y = np.cos(x)label = ‘y=cosx’plt.xticks([-np.pi, -np.pi/2, 0, np.pi/2, np.pi],[r’\(-\pi\)‘, r’\(-\pi/2\)‘, r’\(0\)‘, r’\(+\pi/2\)‘, r’\(+\pi\)‘])ax.plot(x, y, label=label)t = -4*np.pi/3ax.plot([t,t],[0,np.cos(t)], color =‘blue’, linewidth=2.5, linestyle=”–”)ax.scatter([t,],[np.cos(t),], 50, color =‘blue’)ax.plot([t,t],[0,np.sin(t)], color =‘red’, linewidth=2.5, linestyle=”–”)ax.scatter([t,],[np.sin(t),], 50, color =‘red’)show_value(ax,r’\(\sin(\frac{-4\pi}{3})=\frac{\sqrt{3}}{2}\)‘,(t, np.sin(t)),(-30, +50))show_value(ax,r’\(\cos(\frac{-4\pi}{3})=-\frac{1}{2}\)‘,(t, np.cos(t)),(-30, -70))show_value(ax,r’\(\sin(\frac{2\pi}{3})=\frac{\sqrt{3}}{2}\)‘,(2*np.pi/3, np.sqrt(3)/2),(+50, +20))show_value(ax,r’\(\cos(\frac{2\pi}{3})=-\frac{1}{2}\)‘,(2*np.pi/3, -1/2),(-30, -70))show_value(ax,r’sin0=0’,(0,0),(-70, -70))show_value(ax,r’\(\sin(\frac{\pi}{6})=\frac{1}{2}\)‘,(np.pi/6,0.5),(-120, -20))show_value(ax,r’\(\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\)‘,(np.pi/4,np.sqrt(2)/2),(-40, -75))show_value(ax,r’\(\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}\)‘,(np.pi/3, np.sqrt(3)/2),(-10, +50))show_value(ax,r’\(\sin(\frac{\pi}{2})=1\)‘,(np.pi/2, 1),(+60, +50))show_value(ax,r’\(\sin\pi(\sin2\pi)=0\)‘,(np.pi, 0),(+20, +30))show_value(ax,r’\(\sin(\frac{3\pi}{2})=-1\)‘,(3*np.pi/2, -1),(+20, -40))show_value(ax,r’cos0=1’,(0,1),(-60, +40))show_value(ax,r’\(\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}\)‘,(np.pi/6, np.sqrt(3)/2),(-130, -5))show_value(ax,r’\(\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\)‘,(np.pi/4,np.sqrt(2)/2),(-30, +100))show_value(ax,r’\(\cos(\frac{\pi}{3})=\frac{1}{2}\)‘,(np.pi/3,0.5),(-5, -90))show_value(ax,r’\(\cos(\frac{\pi}{2})=1\)‘,(np.pi/2, 1),(-10, -40))show_value(ax,r’\(\cos(\frac{3\pi}{2})=0\)‘,(3*np.pi/2, 0),(+10, -35))show_value(ax,r’\(\cos\pi(\cos2\pi)=-1\)‘,(np.pi, -1),(-20, -70))ax.plot([np.pi/6,np.pi/6],[0.5, np.sqrt(3)/2], color =‘orange’, linewidth=0.5, linestyle=”–”)ax.plot([np.pi/3,np.pi/3],[0.5, np.sqrt(3)/2], color =‘orange’, linewidth=0.5, linestyle=”–”)ax.plot([2*np.pi/3,2*np.pi/3],[0, np.sqrt(3)/2], color =‘orange’, linewidth=0.5, linestyle=”–”)ax.plot([2*np.pi/3,2*np.pi/3],[0, -1/2], color =‘cyan’, linewidth=0.5, linestyle=”–”)ax.plot([3*np.pi/2,3*np.pi/2],[0, -1], color =‘cyan’, linewidth=0.5, linestyle=”–”)plt.legend() plt.show()
② 正切，余切
1.定义域：y = tan(x) 的定义域 x ≠ kπ + π/2(k∈Ζ)的实数x；
y = cot(x) 的定义域 x ≠ kπ (k∈Ζ)的实数x。
3.周期性：均以 π 为最小正周期
4.代码(由于除法原因有些位置不好调，不想浪费时间，所以代码乱点)：
# 参考了 stackoverflow 上的一个问题def draw_point_by_point(plt,x_largest,y_largest,expr):n = 1000X = []Y = []Z = []for i in range(0,2*n):x = -x_largest + i*x_largest/ny = eval(expr)if -y_largest < y and y < y_largest:X.append(x)Y.append(y)else:if len(X) > 0 and len(Y) > 0:Z.append([X,Y])plt.plot(X[0],Y[0])del X,YX = []Y = []for i in range(0, len(Z)):plt.plot(Z[i][0],Z[i][1])x = np.linspace(-5, 5, 100)ax = draw(x)y_largest = 5x_largest = 5n = 1000X = []Y = []Z = []for i in range(0,2*n):x = 3.14 + i*x_largest/ntanx = np.tan(x)if tanx == 0:continuey = 1/tanxif -y_largest < y and y < y_largest and x < x_largest:X.append(x)Y.append(y)else:if len(X) > 0 and len(Y) > 0:Z.append([X,Y])plt.plot(X[0],Y[0])del X,YX = []Y = []for i in range(0, len(Z)):plt.plot(Z[i][0],Z[i][1])pass# y = np.tan(x)label = ‘y=tanx’plt.xticks([-np.pi, -np.pi/2, 0, np.pi/2, np.pi],[r’\(-\pi\)‘, r’\(-\pi/2\)‘, r’\(0\)‘, r’\(+\pi/2\)‘, r’\(+\pi\)‘])draw_point_by_point(plt,3*np.pi/2,5,“np.tan(x)”)show_value(ax,r’y=tan(x)’,(math.atan(3),3),(+20, +20))show_value(ax,r’y=tan(x)’,(math.atan(3)-np.pi,3),(+20, +20))show_value(ax,r’y=tan(x)’,(math.atan(3)+np.pi,3),(+20, +20))draw_point_by_point(plt,3*np.pi/2,5,“1/np.tan(x)”)show_value(ax,r’y=cot(x)’,(math.atan(-1/3),-3),(+20, +20))# x = 1/math.atan(-1⁄3)# print(x)# x = 1/math.atan(-1⁄3) + np.pi# show_value(ax,r’y=cot(x)‘,(x,-3),(+20, +20))# x = 1/math.atan(-1⁄3) + 2*np.pi# print(x)# show_value(ax,r’y=cot(x)‘,(x,-3),(+20, +20))show_value(ax,r’y=cot(x)’,(2.9,-1/math.atan(3)-np.pi),(+20, +20))show_value(ax,r’y=cot(x)’,(2.9-2*np.pi,-1/math.atan(3)-np.pi),(+50, +20))show_value(ax,r’tan(0)=0’,(0,0),(-70, -70))show_value(ax,r’\(\tan(\frac{\pi}{6})=\frac{\sqrt{3}}{3}\)‘,(np.pi/6,np.sqrt(3)/3),(-120, -10))show_value(ax,r’\(\tan(\frac{\pi}{4})=1\)‘,(np.pi/4, 1),(+60, +10))show_value(ax,r’\(\tan(\frac{\pi}{3})=\sqrt{3}\)‘,(np.pi/3, np.sqrt(3)),(+40, +20))show_value(ax,r’\(\tan(\pi)=0 [\tan(2\pi)=0]\)‘,(np.pi, 0),(+20, -30))# show_value(ax,r’\(\tan2\pi=0\)’,(2*np.pi, 0),(+20, +30))show_value(ax,r’\(\cot(\frac{\pi}{6})=\sqrt{3}\)‘,(np.pi/6, np.sqrt(3)),(0, +40))show_value(ax,r’\(\cot(\frac{\pi}{4})=1\)‘,(np.pi/4, 1),(+60, -30))show_value(ax,r’\(\cot(\frac{\pi}{3})=\frac{\sqrt{3}}{3}\)‘,(np.pi/3,np.sqrt(3)/3),(-150, +30))show_value(ax,r’\(\cot(\frac{\pi}{2})=0\)‘,(np.pi/2, 0),(-60, -60))show_value(ax,r’\(\cot(\frac{3\pi}{2})=0\)‘,(3*np.pi/2, 0),(-10, +40))show_value(ax,r’\(\lim_{x\to \frac{\pi}{2}}\tan(x)=∞\)‘,(np.pi/2, 5),(+20, +30))show_value(ax,r’\(\lim_{x\to \frac{3\pi}{2}}\tan(x)=∞\)‘,(3*np.pi/2, 5),(+50, +30))show_value(ax,r’\(\lim_{x\to 0}\cot(x)=∞\)‘,(0, 5),(+20, +30))show_value(ax,r’\(\lim_{x\to \pi(2\pi)}\cot(x)=∞\)‘,(np.pi, 5),(+20, +30))ax.scatter([np.pi/4,],[1,], 50, color =‘blue’)plt.show()# 也可以用插入 nan 值不让它自动连线的方法，但是图很丑# pos = np.where(np.abs(np.diff(y)) >= np.pi/2)[0]+1# x[pos] = np.nan# y[pos] = np.nan# x = np.insert(x, pos, np.nan)# y = np.insert(y, pos, np.nan)
③ 正割、余割
1.定义域：y = sec(x) 的定义域 x ≠ kπ + π/2(k∈Ζ)的实数x；
y = csc(x) 的定义域 x ≠ kπ (k∈Ζ)的实数x。
值域：(-∞，-1] ∪ [1，+∞)。
2.奇偶性：y = sec(x) 是偶函数；y = csc(x) 奇函数。
3.周期性：都以 2π 为最小正周期。
讯享网x = np.linspace(-7, 7, 100)ax = draw(x,5) # plt.ylim(-y, y)plt.xticks([-2*np.pi, -3*np.pi/2, -np.pi, -np.pi/2, 0, np.pi/2, np.pi, 3*np.pi/2, 2*np.pi],[r’\(-2\pi\)‘, r’\(-3\pi/2\)‘, r’\(-\pi\)‘, r’\(-\pi/2\)‘, r’\(0\)‘, r’\(+\pi/2\)‘, r’\(+\pi\)‘, r’\(+3\pi/2\)‘, r’\(+2\pi\)‘])draw_point_by_point(plt,3*np.pi,5,“1/np.cos(x)”)ax.plot([-7,7],[1, 1], color =‘red’, linewidth=1, linestyle=”–”)ax.plot([-7,7],[-1, -1], color =‘red’, linewidth=1, linestyle=”–”)ax.plot([-3*np.pi/2,-3*np.pi/2],[-5, 5], color =‘orange’, linewidth=1, linestyle=”–”)ax.plot([-np.pi/2,-np.pi/2],[-5, 5], color =‘orange’, linewidth=1, linestyle=”–”)ax.plot([3*np.pi/2,3*np.pi/2],[-5, 5], color =‘orange’, linewidth=1, linestyle=”–”)ax.plot([np.pi/2,np.pi/2],[-5, 5], color =‘orange’, linewidth=1, linestyle=”–”)ax.annotate(r’\(\sec(x)=\frac{1}{\cos(x)}\)‘,xy=(-1.5,5), xycoords=‘data’,xytext=(0,+50), textcoords=‘offset points’, fontsize=25)plt.show()x = np.linspace(-7, 7, 100)ax = draw(x,5)plt.xticks([-2*np.pi, -3*np.pi/2, -np.pi, -np.pi/2, 0, np.pi/2, np.pi, 3*np.pi/2, 2*np.pi],[r’\(-2\pi\)‘, r’\(-3\pi/2\)‘, r’\(-\pi\)‘, r’\(-\pi/2\)‘, r’\(0\)‘, r’\(+\pi/2\)‘, r’\(+\pi\)‘, r’\(+3\pi/2\)‘, r’\(+2\pi\)‘])draw_point_by_point(plt,3*np.pi,7,“1/np.sin(x)”)ax.plot([-7,7],[1, 1], color =‘red’, linewidth=1, linestyle=”–”)ax.plot([-7,7],[-1, -1], color =‘red’, linewidth=1, linestyle=”–”)ax.plot([-2*np.pi,-2*np.pi],[-5, 5], color =‘orange’, linewidth=1, linestyle=”–”)ax.plot([-np.pi,-np.pi],[-5, 5], color =‘orange’, linewidth=1, linestyle=”–”)ax.plot([2*np.pi,2*np.pi],[-5, 5], color =‘orange’, linewidth=1, linestyle=”–”)ax.plot([np.pi,np.pi],[-5, 5], color =‘orange’, linewidth=1, linestyle=”–”)ax.annotate(r’\(\csc(x)=\frac{1}{\sin(x)}\)‘,xy=(-1.5,5), xycoords=‘data’,xytext=(0,+50), textcoords=‘offset points’, fontsize=25)plt.show()
反三角函数：
① 反正弦函数、反余弦函数：
1.定义域：[-1，1]
2.单调性：arcsin 单增，arccos 单减
3.奇偶性：arcsin 奇函数
4.有界性：定义域内有界
-π/2 ≤ arcsin(x) ≤ π/2
0 ≤ arccos(x) ≤ π
def draw(X,x,y,ax):ax.axis[“x”] = ax.new_floating_axis(0, 0, axis_direction=“bottom”)ax.axis[“y”] = ax.new_floating_axis(1, 0, axis_direction=“bottom”)ax.set_yticks([-5,-4,-3,-2,-1.5,-1,1,1.5,2,3,4,5])ax.set_xticks([-5,-4,-3,-2,-1.5,-1,0,1,1.5,2,3,4,5])ax.axis[“x”].set_axisline_style(”->“, size=1.0)ax.axis[“y”].set_axisline_style(”->“, size=1.0)ax.axis[‘top’,‘right’,‘left’,‘bottom’].set_visible(False)plt.xlim(-x, x)plt.ylim(-y, y)ax.text(-0.5, x, ‘y’, fontsize=20)ax.annotate(text=’ x’, xy=(max(X), 0), fontsize=20)ax.grid(True, linestyle=’-.’) return axdef show_value(ax, show,coordinate,xytext):ax.annotate(show,xy=coordinate, xycoords=‘data’,xytext=xytext, textcoords=‘offset points’, fontsize=12,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=.1”))if name==“main”:x = np.linspace(-1, 1, 10000)fig = plt.figure(12,figsize=(15, 10))ax = axisartist.Subplot(fig, 121)fig.add_axes(ax)ax = draw(x,1.5,2,ax)plt.yticks([ -np.pi/2, 0, np.pi/2],[r’\(-\pi/2\)‘, r’\(0\)‘, r’\(+\pi/2\)‘])y = np.arcsin(x)#list(map(lambda x:math.asin(x), x))ax.plot(x, y,label = ‘y=arcsin(x)’)ax.annotate(r’\(y=arcsin(x)\)‘,xy=(-1.4,.5), xycoords=‘data’,xytext=(0,0), textcoords=‘offset points’, fontsize=25)show_value(ax,r’arcsin(0)=0’,(0,0),(+70, -70))show_value(ax,r’\(\arcsin(\frac{1}{2})=\frac{\pi}{6}\)‘,(0.5,np.pi/6),(+50, -20))show_value(ax,r’\(\arcsin(\frac{\sqrt{2}}{2})=\frac{\pi}{4}\)‘,(np.sqrt(2)/2,np.pi/4),(-75, +50))show_value(ax,r’\(\arcsin(\frac{\sqrt{3}}{2})=\frac{\pi}{3}\)‘,(np.sqrt(3)/2,np.pi/3),(+60, +30))show_value(ax,r’\(\arcsin(1)=\frac{\pi}{2}\)‘,(1,np.pi/2),(-10, +50))# ——————————————————————————————x = np.linspace(-1, 1, 10000)ax1 = axisartist.Subplot(fig, 122)fig.add_axes(ax1)ax1 = draw(x,1.5,2,ax1)plt.yticks([ -.5, 0, np.pi/2, np.pi],[r’\(0.5\)‘, r’\(0\)‘, r’\(+\pi/2\)‘, r’\(+\pi\)‘])ax1.annotate(r’\(y=arccos(x)\)‘,xy=(-1.4,.5), xycoords=‘data’,xytext=(0,+50), textcoords=‘offset points’, fontsize=25)y = np.arccos(x)#list(map(lambda x:math.asin(x), x))ax1.plot(x, y,label = ‘y=arccos(x)’)show_value(ax1,r’arccos(1)=0’,(1,0),(-10, -70))show_value(ax1,r’\(\arccos(\frac{\sqrt{3}}{2})=\frac{\pi}{6}\)‘,(np.sqrt(3)/2,np.pi/6),(+60, +5))show_value(ax1,r’\(\arccos(\frac{\sqrt{2}}{2})=\frac{\pi}{4}\)‘,(np.sqrt(2)/2,np.pi/4),(+20, +70))show_value(ax1,r’\(\arccos(\frac{1}{2})=\frac{\pi}{3}\)‘,(1/2,np.pi/3),(-10, +90))show_value(ax1,r’\(\arccos(0)=\frac{\pi}{2}\)‘,(0,np.pi/2),(-90, +100))plt.show()
②反正切、反余切
4.有界性：定义域内都有界
arctan(x) ∈ (-π/2，π/2)
arccot(x) ∈ (0，π)
5.性质：arctan(x) + arccot(x) = π/2 (-∞ ≤ x ≤ +∞)
讯享网def draw(X,x,y):fig = plt.figure(figsize=(2*x, 2*y))ax = axisartist.Subplot(fig, 111)fig.add_axes(ax)ax.axis[“x”] = ax.new_floating_axis(0, 0, axis_direction=“bottom”) ax.axis[“y”] = ax.new_floating_axis(1, 0, axis_direction=“bottom”)ax.set_xticks([-5,-4,-3,-2,-1.5,-1,0,1,1.5,2,3,4,5])ax.axis[“x”].set_axisline_style(”->“, size=1.0)ax.axis[“y”].set_axisline_style(”->“, size=1.0)ax.axis[‘top’,‘right’,‘left’,‘bottom’].set_visible(False)plt.xlim(-x, x)plt.ylim(-y, y)ax.text(-0.5, y, ‘y’, fontsize=20)ax.annotate(text=’ x’, xy=(max(X), 0), fontsize=20)ax.grid(True, linestyle=’-.’)return axdef show_value(ax, show,coordinate,xytext):ax.annotate(show,xy=coordinate, xycoords=‘data’,xytext=xytext, textcoords=‘offset points’, fontsize=12,arrowprops=dict(arrowstyle=”->“, connectionstyle=“arc3,rad=.1”))def acot(x): r = []for one in x:if one > 0:r.append(np.arctan(1/one))elif one < 0:r.append(np.pi + np.arctan(1/one))else:r.append(np.pi/2) passpassreturn rif name==“main”:x = np.linspace(-5, 5, 10000)ax = draw(x,5,np.pi/2)plt.yticks([ -np.pi/2, -1, 1, np.pi/2],[r’\(-\pi/2\)‘, ’-1’, ‘1’, r’\(+\pi/2\)‘])y = np.arctan(x)ax.plot(x, y, label = ‘y=arctan(x)’)ax.annotate(r’\(y = arctan(x)\)‘,xy=(-1.5,np.pi/2), xycoords=‘data’,xytext=(0,+30), textcoords=‘offset points’, fontsize=25)show_value(ax,r’arctan(0)=0’,(0,0),(+50, -50))show_value(ax,r’\(\arctan(\frac{\sqrt{3}}{3})=\frac{\pi}{6}\)‘,(np.sqrt(3)/3,np.pi/6),(+60, -15))show_value(ax,r’\(\arctan(1)=\frac{\pi}{4}\)‘,(1,np.pi/4),(-10, +50))show_value(ax,r’\(\arctan(\sqrt{3})=\frac{\pi}{3}\)‘,(np.sqrt(3),np.pi/3),(+60, +30))show_value(ax,r’\(\lim_{x\to -∞}\arctan(x)=-\frac{\pi}{2}\)‘,(-5, -np.pi/2),(-90, -40))show_value(ax,r’\(\lim_{x\to +∞}\arctan(x)=\frac{\pi}{2}\)‘,(5, np.pi/2),(+20, +30))ax.plot([0,np.sqrt(3)/3],[np.pi/6, np.pi/6], color =‘orange’, linewidth=1, linestyle=”–”)ax.plot([np.sqrt(3)/3,np.sqrt(3)/3],[0, np.pi/6], color =‘violet’, linewidth=1, linestyle=”–”)plt.show()# ——————————————————————————————x = np.linspace(-5, 5, 10000)ax = draw(x,5,np.pi)plt.yticks([1, np.pi/2, np.pi],[‘1’, r’\(+\pi/2\)‘, r’\(+\pi\)‘])y = acot(x)ax.plot(x, y, label = ‘y=arccot(x)’)ax.annotate(r’\(y = arccot(x)\)‘,xy=(-1.5,np.pi), xycoords=‘data’,xytext=(0,+30), textcoords=‘offset points’, fontsize=25)show_value(ax,r’\(arccot(0)=\frac{\pi}{2}\)‘,(0,np.pi/2),(-120, +20))show_value(ax,r’\(arccot(\sqrt{3})=\frac{\pi}{6}\)‘,(np.sqrt(3),np.pi/6),(+60, +5))show_value(ax,r’\(arccot(1)=\frac{\pi}{4}\)‘,(1,np.pi/4),(+50, +26))show_value(ax,r’\(arccot(\frac{\sqrt{3}}{3})=\frac{\pi}{3}\)‘,(np.sqrt(3)/3,np.pi/3),(+30, +50))show_value(ax,r’\(\lim_{x\to -∞}arccot(x)=\pi\)‘,(-5, np.pi),(-90, +30))show_value(ax,r’\(\lim_{x\to +∞}arccot(x)=0\)‘,(5, 0),(+30, -40))ax.plot([0,np.sqrt(3)],[np.pi/6, np.pi/6], color =‘orange’, linewidth=1, linestyle=”–”)ax.plot([np.sqrt(3),np.sqrt(3)],[0, np.pi/6], color =‘violet’, linewidth=1, linestyle=”–”)plt.show()
初等函数：
基本初等函数经过有限次四则运算，以及有限次复合步骤构成，并可由一个式子表示
1.初等函数可以是一个区间，也可以是几个区间的并集，甚至可以是一些孤立的点。如 y = (cosπ - 1)⁰·⁵，定义域 x = 0, ±2, ±4 …。这个图就不画了，matplotlib 默认会自己给点连起来。
2.幂指函数 u(x)ᵛ⁽ˣ⁾ = eᵛ⁽ˣ⁾ ᴵⁿ ᵘ⁽ˣ⁾ 也是初等函数。还得找俩函数，而且本身就是各种组合，也不画了。
三个重要的分段函数：
绝对值函数和符号函数：
取整函数：
1.定义域 R，值域 Z。在 x 为整数值处图形发生跳跃。
def roundX(x, plt, ax):plt.scatter(x,x, color=‘blue’, marker=‘o’, edgecolors=‘g’, linewidth=2, s=100)plt.scatter(x+1,x, color=“, marker=‘o’, edgecolors=‘r’, linewidth=2, s=100)ax.plot([x,x+0.9],[x,x], color =‘blue’, linewidth=2, linestyle=”-”)if name==“main”:x = np.linspace(-5, 5, 10000)fig = plt.figure(12,figsize=(20, 10))ax = axisartist.Subplot(fig, 121)fig.add_axes(ax)ax = draw(x,5,5,ax)y = np.fabs(x)ax.plot(x, y, label = ‘y =|x|’)ax.annotate(r’\(y = |x| = \{ \genfrac{}{}{0}{}{ x , x ≥ 0}{ -x , x < 0}\)‘,xy=(-4.5,-2), xycoords=‘data’,xytext=(-20,+30), textcoords=‘offset points’, fontsize=30)ax1 = axisartist.Subplot(fig, 122)fig.add_axes(ax1)ax1 = draw(x,5,5,ax1)ax1.annotate(r’\(y = sign(x) = \{ \genfrac{}{}{0}{}{ 0 , x = 0 }{ \genfrac{}{}{0}{}{ 1 , x > 0}{ -1 , x < 0}} : x = |x|sign(x)\)‘,xy=(-5,2), xycoords=‘data’,xytext=(-40,+30), textcoords=‘offset points’, fontsize=30)plt.scatter(0,0, color=‘blue’, marker=‘o’, edgecolors=‘g’, linewidth=2, s=100)plt.scatter(0,1, color=“, marker=‘o’, edgecolors=‘r’, linewidth=2, s=100)ax1.plot([0.1,6],[1,1], color =‘blue’, linewidth=2, linestyle=”-”)plt.scatter(0,-1, color=“, marker=‘o’, edgecolors=‘r’, linewidth=2, s=100)ax1.plot([-0.1,-6],[-1,-1], color =‘blue’, linewidth=2, linestyle=”-”)# ax.scatter([0,0],[0,0], 50, color =‘blue’)plt.show()x = np.linspace(-5, 5, 10000)fig = plt.figure(11,figsize=(10, 10))ax = axisartist.Subplot(fig, 111)fig.add_axes(ax)ax = draw(x,5,5,ax)ax.annotate(r’\(y = [x] : (x-1 < [x] <≤ x)\)‘,xy=(-4.5,2), xycoords=‘data’,xytext=(-100,+30), textcoords=‘offset points’, fontsize=25)roundX(0, plt, ax)roundX(1, plt, ax)roundX(-1, plt, ax)roundX(2, plt, ax)roundX(-2, plt, ax)roundX(3, plt, ax)roundX(-3, plt, ax)roundX(4, plt, ax)roundX(-4, plt, ax)show_value(ax,r’[0.9]=0’,(0.9,0),(+50, +25))show_value(ax,r’\([\pi]=3\)‘,(np.pi,3),(+50, -50))show_value(ax,r’[-1]=-1’,(-1,-1),(-110, +10))show_value(ax,r’[-1.99]=-2’,(-1.9,-2),(-60, +30))show_value(ax,r’\(\lim_{x\to 0^{+}}[x]=0\)‘,(0, 0),(+50, -90))show_value(ax,r’\(\lim_{x\to 0^{-}}[x]=-1\)‘,(0, -1),(-120, +90))plt.show()
常见图像就这样了，篇幅有点长了。如果后续画图像变换和极坐标图像的话就再说了。
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