MATH 455 Instructions 1 Calculus Review functions i f x ex2 Homework 1 Written assignments are graded based on quality of work and fully worked out solutions Points will be deducted for incomplete reasoning and disorganized work even if your answers are correct a Find the 2nd degree T2 x and 5th degree T5 x Taylor polynomials centered at x0 0 for the following ii f x 1 1 x iii f x cos 2x iv f x sin2 x b Let f x x 2 and carry out the following calculations i Calculate the Taylor polynomial of degree 4 about the point x 1 ii Use your result in a to approximate f 0 9 and f 1 1 iii Use the Taylor remainder to nd an error formula for the Taylor polynomial Give error bounds for each of the two approximations in part b Which of the two approximations in b do you expect to be closer to the correct value iv Use a calculator to compare the actual error in each case with your error bound from part c How do the actual error and error bound compare 2 Root nding and Fixed Points a Use the Bisection Method to nd p3 for f x b Let f x 3 x 1 x 1 2 x 1 Use the Bisection Method on the following intervals to nd p3 x cos x 0 on 0 1 ii 5 4 5 2 c Find all xed points for the following functions ii g x x 6 3x 2 iii g x x2 4x 2 3 Newton s and Secant Methods a Let f x x2 6 and p0 1 Use Newton s Method to nd p2 b Let f x x3 cos x and p0 1 Use Newton s Method to nd p2 Could p0 0 be used Explain c Let f x x2 6 With p0 3 and p1 2 nd p3 using the Secant Method d Let f x x4 7x3 18x2 20x 8 Does Newton s Method converge quadratically to the root r 2 i 2 3 2 i g x 3 x e Show that Newton s Method applied to f x ax b converges in one step 4 Programming use Matlab a Write a Matlab program which performs Bisection Method and use it for the following problems i Find an approximation to within 10 5 to a value in 0 5 1 5 with ex 2 cos ex 2 ii Find an approximation to within 10 5 to the rst positive value of x with x tan x 1 MATH 455 Homework 1 b Write a Matlab program to perform Fixed Point Iteration to nd the solution to the following equations i ex x 7 ii ex sin x 4 c Write a Matlab program which performs Newton s Method Apply your program to nd both roots of the function f x 14xe x 2 12e x 2 7x3 20x2 26x 12 on the interval 0 3 For each root print the sequence of iterates the errors en xn r and both error ratios en 1 e2 n and en 1 en that converges to a nonzero limit 2