How is the finite difference formulation for the first derivative related to the Taylor series expansion of the solution function?

A Taylor series expansion of a function f(x) about a point x=a is given by: f(x) = f(a) + f'(a)(x-a) + (1/2!)f''(a)(x-a)^2 + (1/3!)f'''(a)(x-a)^3 + ... Now, let's focus on the first few terms of the Taylor series expansion of a function f(x) about the point x=x0. f(x) = f(x0) + f'(x0)(x-x0) + (1/2)f''(x0)(x-x0)^2 + O((x-x0)^3)

The finite difference formulation for the first derivative can be obtained using the forward difference method, where we approximate the derivative by considering the function values at some distance h forward from the point x0. Therefore, for the forward difference method, f'(x0) can be approximated by: f'(x0) ≈ (f(x0 + h) - f(x0))/h

Now, let's express f(x0 + h) in terms of the Taylor series expansion: f(x0 + h) = f(x0) + f'(x0)(h) + (1/2)f''(x0)(h)^2 + O(h^3) Now, let's subtract f(x0) from both sides of the equation: f(x0 + h) - f(x0) = f'(x0)(h) + (1/2)f''(x0)(h)^2 + O(h^3) By dividing both sides by h, we get: (f(x0 + h) - f(x0))/h = f'(x0) + (1/2)f''(x0)(h) + O(h^2) Comparing this expression with the finite difference approximation for the first derivative, we can deduce the following relationship: f'(x0) ≈ (f(x0 + h) - f(x0))/h, with an error term of (1/2)f''(x0)(h) + O(h^2) So, the finite difference formulation for the first derivative is related to the Taylor series expansion of the solution function in terms of its approximation and the error terms.