Create a user-defined function \(f(x)\) that returns the value \(1+\frac{1}{2} \tanh 2 x\), then use a central difference to calculate the derivative of the function in the range \(-2 \leq x \leq 2\). Calculate an analytic formula for the derivative and make a graph with your numerical result and the analytic answer on the same plot. It may help to plot the exact answer as lines and the numerical one as dots. (Hint: In Python the tanh function is found in the math package, and it's called simply tanh.)

The Central Difference Method is a numerical technique to approximate the derivative of a function. This method is very useful when you don't have the exact form of the derivative or when dealing with discretized data.
Here's how it works. Suppose you have a function \( f(x) \). The central difference method formula to approximate \( f'(x) \) is:
\[ f'(x) \approx \frac{f(x+h) - f(x-h)}{2h} \]
This formula gives us the slope of the function at a point \( x \) by looking at the points just to its left and right, separated by a small interval \( h \). The central difference method is more accurate than forward or backward difference methods because it takes into account the change in the function on both sides of \( x \).
To implement this in the Python programming language, we can use the numpy library for handling array operations efficiently. Here, \( h \) can be defined as a small value, like 0.01.

The analytic derivative means determining the derivative of a function using calculus rather than numerical methods. For the function given in the exercise, \( f(x) = 1 + \frac{1}{2} \tanh(2x) \), we need to find its derivative analytically.
First, recall that the derivative of \( \tanh(x) \) is given by:
\[ \frac{d}{dx} \tanh(x) = 1 - \tanh^2(x) \]
Given the function \( f(x) \), its derivative can be computed step by step:
1. The derivative of \(1\) is \(0\).
2. The derivative of \( \frac{1}{2} \tanh(2x) \) requires the chain rule:
\[ \frac{d}{dx} \left( \frac{1}{2} \tanh(2x) \right) = \frac{1}{2} \cdot \frac{d}{dx} \tanh(2x) \cdot \frac{d}{dx} (2x) \]
3. Using the chain rule, we get:
\[ \frac{1}{2} \cdot (1 - \tanh^2(2x)) \cdot 2 = 1 - \tanh^2(2x) \]
Thus, the analytic derivative is:
\[ f'(x) = 1 - \tanh^2(2x) \]
This exact expression can be used to compare against the numerical approximation.

Python is a versatile programming language that is very useful for scientific computing. By utilizing libraries such as math, numpy, and matplotlib, you can perform complex numerical methods and visualize results effortlessly.
Here's a quick rundown to solve our problem in Python:
1. **Define the function**:
def f(x):
     return 1 + 1/2 * math.tanh(2 * x)
2. **Import necessary libraries**:
import math
import numpy as np
import matplotlib.pyplot as plt
3. **Central Difference Method**:
Create an array of x values using numpy:
x = np.linspace(-2, 2, 400)
Define h as a small step:
h = 0.01
Compute the numerical derivative:
numerical_derivative = (f(x + h) - f(x - h)) / (2 * h)
4. **Analytic Derivative**:
Using numpy for vectorized operations, define the exact derivative:
analytic_derivative = 1 - np.tanh(2 * x)**2
5. **Plot the Results**:
Use matplotlib to create the plot:
plt.plot(x, analytic_derivative, label='Analytic Derivative')
plt.scatter(x, numerical_derivative, label='Numerical Derivative', color='red', marker='o')
plt.legend()
plt.show()
This code will produce a graph comparing the numerical and analytical results, helping you visualize the accuracy of the Central Difference Method.