함수의 그래프: 극값

다음 그림들은 전자책 파이썬과 함께하는 미분적분의 5.4장에 수록된 그래프들과 코드들입니다.

import numpy as np 
import pandas as pd
from sympy import *
import matplotlib.pyplot as plt
import seaborn as sns
sns.set_style("darkgrid")

#그림 5.4.1
a=symbols('a')
f=a**3-3*a+2
x=np.linspace(-3, 3, 100)
y=[f.subs(a, i) for i in x]
df=f.diff(a)
dy=[df.subs(a, i) for i in x]
plt.figure(figsize=(4, 3))
plt.plot(x, y, color="g", label="f(x)")
plt.plot(x, dy, ls="dashed", color="b", alpha=0.3, label=r"$\frac{df(x)}{dx}$")
col=[("b", "f(-2): Lower Int"), ("r", "f(-1): critical)"),('r', "f(1): critical"), ('b', "f(1.5): Upper Int")]
for i, j in enumerate([-2, -1, 1, 1.5]):
    plt.scatter(j, f.subs(a, j), s=50, c=col[i][0])
    if j<0:
        plt.text(j+0.2, f.subs(a, j)+0.2, col[i][1], c=col[i][0])
    else:
        plt.text(j+0.2, f.subs(a, j)-0.3, col[i][1], c=col[i][0])
plt.xlabel("x",fontsize="11")
plt.ylabel("y", rotation="horizontal",  fontsize="11")
plt.ylim((-4, 5))
plt.legend(loc="best", labelcolor=['g',"b"], frameon=False)
plt.show()

#그림 5.4.2
a=symbols('a')
f=4*a-a**4
df=f.diff(a)
x=np.linspace(-2, 3, 100)
y=[f.subs(a, i) for i in x]
dy=[df.subs(a, i) for i in x]
plt.figure(figsize=(4, 3))
plt.plot(x, y, color="g", label="f(x)")
plt.plot(x, dy, color="b", ls="dashed", alpha=0.3, label=r"$\frac{df(x)}{dx}$")
plt.scatter(1, f.subs(a, 1), s=50, c="r")
plt.text(1+0.2, f.subs(a, 1)+0.2, 'f(1)', c="r")
plt.xlabel("x", fontsize="11")
plt.ylabel("y", rotation="horizontal", fontsize="11")
plt.ylim((-20, 20))
plt.legend(loc="best", labelcolor=['g', "b"], frameon=False)
plt.show()

#그림 5.4.3
x=symbols('x', real=True)
f=5*x+35/x
df=f.diff(x)
a1, a2=np.linspace(-5, -0.01, 50), np.linspace(0.01, 5, 50)
y1, y2=[f.subs(x, i) for i in a1], [f.subs(x, i) for i in a2]
dy1, dy2=[df.subs(x, i) for i in a1], [df.subs(x, i) for i in a2]
plt.figure(figsize=(4,3))
plt.plot(a1, y1, color="g", label="f(x)")
plt.plot(a2, y2, color="g")
plt.plot(a1, dy1, ls="dashed", color="b", alpha=0.3, label=r"$\frac{df(x)}{dx}$")
plt.plot(a2, dy2, ls="dashed", color="b", alpha=0.3)
plt.scatter(np.sqrt(7), f.subs(x, np.sqrt(7)), s=30, c="brown", label="critical point")
plt.ylim(-400, 400)
plt.xlabel("x", fontsize="11")
plt.ylabel("y", rotation="horizontal", fontsize="11")
plt.legend(loc="best", labelcolor=['g','b','brown'])
plt.show()