a.

Use zero- through third-order Taylor series expansions to predict f(3) for

f (x) = 25x³ 6x² + 7x 88

using a base point at x = 1. Compute the true percent relative error for each approximation.

b.Use forward difference approximations of O(h) to estimate the first derivative of the function examined in Prob. 5. 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.

c,

The following infinite series can be used to approximate e^x:

e^x = 1 + x + x²/ 2! + x³/3! + + xⁿ/n!

Prove that this Maclaurin series expansion is a special case of the Taylor series expansion with xi = 0 and h = x.