Description

• Write your name, ID#, and Section number clearly in the very front page.
• Write all answers sequentially.
• Start answering a question (not the pat of the question) from the top of a new page.
• Write legibly and in orderly fashion maintaining all mathematical norms and rules. Prepare a single solution file.

1. A function is given by f(x) = 2x − e−6x. Now answer the following:
(a) (3 marks) Approximate the derivative of f(x) at x0 = 0.5 with step size h = 0.2 using the forward difference method up to 5 significant figures.
(b) (3 marks) Approximate the derivative of f(x) at x0 = 0.5 with step size h = 0.2 using the central difference method up to 6 significant figures.
(c) (4+4 marks) Calculate the truncation error of f(x) at x0 = 2 using h = 1, 0.1, 0.01, 0.0001 in both of the above mentioned methods.
(d) (6 marks) Compute 2 using Richardson extrapolation method up to 6 significant figures and calculate the truncation error.
2. During the class, we derived in detail the first order Richardson extrapolated derivative, by using h → h/2,
.
(a) (6 marks) Using h → h/2, derive the expression for which is the second order Richardson extrapolation.
(1)
(b) (6 marks) Now starting from the definition of Dh and using h → h/3, derive the expression for Dh .
(c) (2+2 marks) Now identify the Error Part of the expression found in the previous part, and also find the Error Bound of the expression found in the previous part.
(d) (4 marks) If f(x) = lnx, x0 = 1, h = 0.1, find the upper bound of error for .
Motto: Mathematics is NOT difficult, but what is difficult is to believe that mathematics is NOT difficult.