함수의 그래프: 평균값 정리

다음 그림들은 전자책 파이썬과 함께하는 미분적분의 5.5장에 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")

def tgline(slope, x0, y0, x):
    b=y0-slope*x0
    re=slope*x+b
    return(re)

#그림 5.5.1
x=symbols('x', real=True)
f=x**3+2*x**2
df=f.diff(x)
cp=solve(df)
a=np.linspace(-1.618, 0.618, 100)
y=[f.subs(x, i) for i in a]
dy=[df.subs(x, i) for i in a]
plt.figure(figsize=(4,3))
plt.plot(a, y, color='g', label="f(x)")
plt.plot(a, dy, color='b', ls="dashed", alpha=0.3, label=r"$\frac{df(x)}{dx}$")
idx=[(a[0],"r","A"), (cp[0], "b","D"), (cp[1], "k","C"), (a[-1], "brown","B")]
for i in idx:
    plt.scatter(i[0], f.subs(x, i[0]), s=30, c=i[1], label=f"{i[2]}")
plt.ylim(-0.5, 1.5)
plt.xlabel("x")
plt.ylabel("y", rotation="horizontal")
plt.legend(bbox_to_anchor=(1, 1), labelcolor='linecolor')
plt.show()

#그림 5.5.2
x=symbols('x')
f=x**3-x
f0=f.subs(x,0)
f2=f.subs(x, 2)
x=symbols('x')
f=x**3-x
avg=(f2-f0)/(2-0)
df=f.diff(x)
c=solve(Eq(df, avg))

a=np.linspace(0, 2, 100)
y=[f.subs(x, i) for i in a]
plt.figure(figsize=(4,3))
k=tgline(3, c[1], f.subs(x, c[1]), x)
ky=[k.subs(x, i) for i in a]
plt.plot(a, y, color="g", label="f(x)")
plt.plot(a, ky, ls="dashed",color="b", label="tangent")
plt.plot([0,2], [f0, f2], ls="dashed", color="r", label="average")
plt.xlabel("x")
plt.ylabel("y", rotation="horizontal")
plt.legend(loc="best", labelcolor='linecolor')
plt.show()