import numpy as np
import matplotlib.pyplot as plt

X = np.arange(0, 10, 0.01)
K = 100
alpha = 0.15

real_Y = K*np.exp(-alpha * X)

def approx_function_generator(delta, alpha, K):

	def f(x):
		if(x <= 0):
			return K
		return (1 - delta*alpha)*f(x - delta)

	return f

plt.plot(X, real_Y)
plt.plot(X, [approx_function_generator(2, alpha, K)(x) for x in X])
plt.plot(X, [approx_function_generator(1, alpha, K)(x) for x in X])
plt.plot(X, [approx_function_generator(0.1, alpha, K)(x) for x in X])
plt.show()