Question

Use forward and backward difference approximations of O(h) and a centered difference approximation of O(h²) to estimate the first derivative of the following function:

f(x)=25x³-6x²+7x-88

Evaluate the derivative at x=2 using a step size of h=0.25. Compare your results with the true value of the derivative. Interpret your results on the basis of the remainder term of the Taylor series expansion.

Answer 1

First you need to evaluate the function with forward difference approximation

where you can find f'(x) = [f(x+h)-f(x)]/h

Backward Difference approximation : f'(x) = [f(x)-f(x-h)]/h

Centerd Difference Approximation : f'(x) = [f(x+h)-f(x-h)]/(2*h)

Also Calculate f'(x) by differentiating the function with respect to x this will be true value the function

Finally use Taylor series expansion

Since it is not mentioned in the problem specifically if you need to use Taylor Series expansion on differentiated function or only the function I would suggest after differentiating the function f(x) with respect to x use Taylore series of expansion(upto 5 decimal places) and afterwards compare it with the results obtained earlier.