Finite Differences and Forward Difference Formula

The forward difference formula is: 𝑓 ′ (𝑥𝑖 ) = 𝑓(𝑥𝑖+1 ) − 𝑓(𝑥𝑖 ) / ℎ + 𝑂(ℎ)
It is known as forward difference because the second point after 𝑥𝑖 is selected in the positive direction of the x-axis
The term 𝑂(ℎ) indicates the error resulting from the truncation made to obtain the difference

There are three types of finite differences: forward, central, and backward
Finite differences are used to approximate derivatives
They can be applied to an array of x values within a given domain to plot derivative curves

The forward finite difference formula for the first derivative is: 𝑓 ′ (𝑥𝑖 ) = 𝑓(𝑥𝑖+1 ) − 𝑓(𝑥𝑖 ) / ℎ
The forward finite difference formula for the second derivative is: 𝑓 ′′ (𝑥𝑖 ) = 𝑓(𝑥𝑖+2 ) − 2𝑓(𝑥𝑖+1 ) + 𝑓(𝑥𝑖 ) / ℎ²

The central finite difference formula for the first derivative is: 𝑓 ′ (𝑥𝑖 ) = 𝑓(𝑥𝑖+ℎ ) − 𝑓(𝑥𝑖−ℎ ) / 2ℎ
The central finite difference formula for the second derivative is: 𝑓 ′′ (𝑥𝑖 ) = 𝑓(𝑥𝑖+ℎ ) − 2𝑓(𝑥𝑖 ) + 𝑓(𝑥𝑖−ℎ ) / ℎ²

The backward finite difference formula for the first derivative is: 𝑓 ′ (𝑥𝑖 ) = 𝑓(𝑥𝑖 ) − 𝑓(𝑥𝑖−ℎ ) / ℎ
The backward finite difference formula for the second derivative is: 𝑓 ′′ (𝑥𝑖 ) = 𝑓(𝑥𝑖 ) − 2𝑓(𝑥𝑖−ℎ ) + 𝑓(𝑥𝑖−2ℎ ) / ℎ²

Python can be used to implement finite differences to approximate derivatives
The numpy and matplotlib libraries can be used to perform calculations and plot results

Forward difference formula: f'(x_i) = (f(x_i+1) - f(x_i)) / h + O(h)
This formula is obtained by omitting the terms containing the second and higher derivatives

Forward finite differences: used to approximate the first and higher derivatives of a function
Central finite differences: used to approximate the first and higher derivatives of a function
Backward finite differences: used to approximate the first and higher derivatives of a function

In Python, use the numpy library to create arrays and the matplotlib library to plot functions
Example code: f = lambda x: 0.1*x**5 - 0.2*x**3 + 0.1*x - 0.2; h = 0.05; x = np.linspace(0,1,11)
Use central differences to approximate the first and second derivatives of a function: dfc1 = (f(x+h) - f(x-h))/(2*h); dfc2 = (f(x+h) - 2*f(x) + f(x-h))/h**2

The output of the program shows the approximate values of the first and second derivatives of the function at different points
Comparing the results with the analytical solution shows that the central differences method yields the most accurate solution (0.5% error) while the forward and backward differences methods are less accurate (3.6% and 2.6%, respectively)

Forward difference formula: f'(x_i) = (f(x_i+1) - f(x_i)) / h + O(h)
This formula is obtained by omitting the terms containing the second and higher derivatives

Forward finite differences: used to approximate the first and higher derivatives of a function
Central finite differences: used to approximate the first and higher derivatives of a function
Backward finite differences: used to approximate the first and higher derivatives of a function

In Python, use the numpy library to create arrays and the matplotlib library to plot functions
Example code: f = lambda x: 0.1*x**5 - 0.2*x**3 + 0.1*x - 0.2; h = 0.05; x = np.linspace(0,1,11)
Use central differences to approximate the first and second derivatives of a function: dfc1 = (f(x+h) - f(x-h))/(2*h); dfc2 = (f(x+h) - 2*f(x) + f(x-h))/h**2

The output of the program shows the approximate values of the first and second derivatives of the function at different points
Comparing the results with the analytical solution shows that the central differences method yields the most accurate solution (0.5% error) while the forward and backward differences methods are less accurate (3.6% and 2.6%, respectively)

The forward difference formula is: f'(x_i) = (f(x_i+1) - f(x_i)) / h + O(h)
The formula is known as forward difference because the second point after x_i is selected in the positive direction of the x-axis
The term O(h) indicates the error resulting from the truncation made to obtain the difference

There are three types of finite difference methods: forward, central, and backward differences
These methods can be used to approximate the derivative of a function at a point
Forward differences are used to estimate the derivative at a point using the function values at that point and the next point
Central differences are used to estimate the derivative at a point using the function values at that point and the previous and next points
Backward differences are used to estimate the derivative at a point using the function values at that point and the previous point

Numerical differentiation is the process of approximating the derivative of a function at a point using finite difference methods
The accuracy of the approximation depends on the step size h and the method used
The program can be written in Python using NumPy and Matplotlib libraries to plot the derivative curves

The example function is: f(x) = 0.1*x**5 - 0.2*x**3 + 0.1*x - 0.2
The step size is h = 0.05
The x values range from 0 to 1 by increment of 0.1
The program computes the first and second derivatives of the function using forward, central, and backward differences

The results obtained from the program are compared with the analytical solution
The central differences method yielded the most accurate solution in the first derivative (0.5% error)
The forward and backward differences methods were less accurate (3.6% and 2.6%, respectively)

The bisection method is a root-finding algorithm that finds the root of a function by repeatedly dividing an interval in half
The rules for the bisection method are:
- Choose two real numbers a and b such that f(a)*f(b) < 0
- The function f is continuous on the interval [a, b]
- The function f changes sign at the root