Reference no: EM132362063
Questions -
Q1. Suppose f(x) has a root at x = 1. Suppose that we use the starting interval [-1, 2] for the bisection method and the starting point x0 = 2 for Newton's method.
(a) Provide equations for two functions where
- The first should be a continuous function that meets the above criteria where Newton's method succeeds in finding the root, but the bisection method fails.
- The second should be a continuous function that meets the above criteria where the bisection method succeeds in finding the root, but Newton's method fails.
(b) Use Matlab to plot your both of your functions on a single figure. Label your plot well - include a descriptive title, label both axis, and include a legend to clearly distinguish the two functions, and include this plot.
Q2. Consider the system of equations
ax + y = 1 (1)
x + y = 2. (2)
(a) Show that the solution is x = 1/(1- a) and y = 2 - 1/(1- a).
(b) Set up and solve the system using gaussian elimination without pivoting with a = 10-n, for n = 4, 8, 12, 16. How big are the errors in each case compared to the true solution?
(c) Now reverse the order of the two equations and repeat. How big are the errors in each case compared to the true solution?
(d) Which method is more accurate, (a) or (b)? Why? Explain?
(e) Finally, repeat using Matlab's backslash command, and compare. Is this any better?
Q3. (a) Find the third degree Taylor polynomial for f(x) = ln(x) expanded about c = 2. Plot both f(x) and the Taylor approximation. Label your plot well - include a descriptive title, label both axis, and include a legend to clearly distinguish the two.
(b) Approximate 2∫3 ln(x)dx by integrating the polynomial found in part (a).
(c) Derive an error bound for your result in part (b) by integrating and manipulating the remainder term that would be associated with your polynomial in part (a).
(d) What is the minimum degree Taylor polynomial does your error bound say is required to approximate 2∫3 ln(x) dx to machine precision? In practice, what degree is required?