함수 그래프: 임계점 표시

다음 그래프들은 전자책 파이썬과 함께하는 미분적분의 5.2 장에 소개된 그래프와 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 5.2.2
x=np.linspace(-.5, 6, 100)
f=(x-2)**4+1
g=(x-2)**3+1
plt.figure(figsize=(4, 3))
plt.plot(x, f, color="g", label="f(x)")
plt.plot(x, g, color="b", label="g(x)")
plt.hlines(1, -0.5, 6, color="k", ls="--")
plt.scatter(2, 1, s=50, c="r", label="crtical point")
plt.xlabel("x", fontsize="11")
plt.ylabel("y", rotation="horizontal", fontsize="11")
plt.ylim((-11, 11))
plt.legend(loc="best", labelcolor=['g','b','r'], frameon=False)
plt.show()

#그림 5.2.3
a=symbols("a", real=True)
f=3*a**5-20*a**2-1
df=f.diff(a)
c=[float(i) for i in solve(df)]
ddf=df.diff(a)
x=np.linspace(-.5, 4, 100)
fy=[f.subs(a, i) for i in x]
dfy=[df.subs(a, i) for i in x]
ddfy=[ddf.subs(a, i) for i in x]
plt.figure(figsize=(4, 3))
plt.plot(x, fy, color="g", label="f(x)")
plt.plot(x, dfy, color="b", label=r"$\frac{df(x)}{dx}$")
plt.plot(x, ddfy, color="r", label=r"$\frac{d}{dx}\,\frac{df(x)}{dx}$")
plt.scatter(0, -1, s=50, c="brown", label="f''(A)<0")
plt.scatter(1.39, -24.8, s=50, c="k", label="f''(B)>0")
plt.scatter(0.87, 0.06, s=50, c="r", label="f''(C)=0")
plt.scatter(0.87, -14.75, s=50, c="orange", label="D")
plt.xlabel("x", fontsize="11")
plt.ylabel("y", rotation="horizontal",  fontsize="11")
plt.ylim((-60,60))
plt.legend(loc='best', labelcolor=['g','b','r', "brown","k","r","orange"], frameon=False)
plt.show()

#그림 5.2.4
a=symbols('a', real=True)
f=a**5-20*a**3
df=f.diff(a)
ddf=f.diff(a, 2)
ip=solve(ddf)
x=np.linspace(-5, 5, 100)
fy=[f.subs(a, i) for i in x]
dfy=[df.subs(a, i).evalf() for i in x]
ddfy=[arg(ddf.subs(a, i)).evalf() for i in x]
plt.figure(figsize=(4, 3))
plt.plot(x, fy, color="g", label=r"$f(x)$")
plt.plot(x, dfy, color="b", ls="--", alpha=0.6, label=r"$\frac{df(x)}{dx}$")
plt.plot(x, ddfy, color="r",ls="--", alpha=0.6,  label=r"$\frac{d}{dx}\frac{df(x)}{dx}$")
p=((2.45, -205.76), (-2.45, 205.76),(0,0))
nme=("A", "B","C")
for i in range(3):
    plt.scatter(p[i][0], p[i][1], s=50, c="brown")
    plt.text(p[i][0], p[i][1]+40, nme[i], fontweight="bold", color="brown")
    plt.vlines(p[i][0], -500, 500, color="brown", alpha=0.5, ls="-.")
plt.xlabel("x", fontsize="11")
plt.ylabel("y", rotation="horizontal", fontsize="11")
plt.ylim((-500, 500))
plt.legend(bbox_to_anchor=(1,1), labelcolor=['g','b',"r"], frameon=False)
plt.show()

#그림 5.2.5
a=symbols('a', real=True)
f=a**3-3*a**2+2
df=f.diff(a)
ddf=f.diff(a, 2)
critical=solve(df)
inflec=solve(ddf)
x=np.linspace(-3, 4, 100)
fy=[f.subs(a, i) for i in x]
dfy=[df.subs(a, i) for i in x]
ddfy=[ddf.subs(a, i) for i in x]
plt.figure(figsize=(4, 3))
plt.plot(x, fy, color="g", label=r"$f(x)$")
plt.plot(x, dfy, color="b", ls="--", alpha=0.6, label=r"$\frac{df(x)}{dx}$")
plt.plot(x, ddfy, color="r",ls="--", alpha=0.6,  label=r"$\frac{d}{dx}\frac{df(x)}{dx}$")
p=((1,0), (0,0), (2,0))
nme=("inflection","critical", "critical")
col=("k","brown","orange")
for i in range(len(p)):
    plt.scatter(p[i][0], p[i][1], s=50, c=col[i],label=nme[i])
plt.xlabel("x", fontsize="11")
plt.ylabel("y", rotation="horizontal", fontsize="11")
plt.ylim((-10, 10))
plt.legend(bbox_to_anchor=(1,1), labelcolor=['g','b',"r"], frameon=False)
plt.show()