secant_lines.ipynb

Kernel: Python 3 (Ubuntu Linux)

import numpy as np

f = np.exp  # The function
x_0 = 1       # The x loc of P
y_0 = f(x_0)  # The y loc of P


increments = [1,1/2,1/4,1/8,1/16]

a_pos = x_0 + np.array(increments)
a_neg = np.flip(x_0 - np.array(increments),axis=0)
a = np.concatenate((a_pos,a_neg))


Q = [(_a,f(_a)) for _a in a]

import pandas as pd

data = {'P':[(x_0,y_0)]*len(a),'Q':Q,'slope of secant':(f(a)-y_0)/(a-x_0)}
df = pd.DataFrame(data=data)
df

	P	Q	slope of secant
0	(1, 2.718281828459045)	(2.0, 7.38905609893065)	4.670774
1	(1, 2.718281828459045)	(1.5, 4.4816890703380645)	3.526814
2	(1, 2.718281828459045)	(1.25, 3.4903429574618414)	3.088245
3	(1, 2.718281828459045)	(1.125, 3.080216848918031)	2.895480
4	(1, 2.718281828459045)	(1.0625, 2.893595944171761)	2.805026
5	(1, 2.718281828459045)	(0.9375, 2.553589458062927)	2.635078
6	(1, 2.718281828459045)	(0.875, 2.398875293967098)	2.555252
7	(1, 2.718281828459045)	(0.75, 2.117000016612675)	2.405127
8	(1, 2.718281828459045)	(0.5, 1.6487212707001282)	2.139121
9	(1, 2.718281828459045)	(0.0, 1.0)	1.718282

%matplotlib inline
import matplotlib.pyplot as plt
fig = plt.gcf()
fig.set_size_inches(18.5, 10.5)


Qx = [q[0] for q in Q]
Qy = [q[1] for q in Q]
x = np.arange(min(Qx),1.3*max(Qx),0.1)
plt.plot(x,f(x))
plt.scatter(Qx,Qy)
plt.title(r"Secant approximations of the tangent line")
for index, row in df.iterrows():
   plt.plot([row['P'][0], row['Q'][0]],[row['P'][1], row['Q'][1]])
    

fig.savefig('secants_arctan.png', dpi=100)