## Description

Problem 1. Consider the bisection method starting with the interval [1.5, 3.5]

(a) What is the width of the interval at the nth step of this method?

(b) What is the maximum distance possible between the root r and the midpoint of this

interval?

Problem 2. Write and test a program to implement the bisection method. Test the program

on the following functions and intervals

(a) x

−1 − tan(x) on [0, π/2].

(b) x

−1 − 2

x on [0, 1].

(c) Find a root of

x

8 − 36x

7 + 546x

6 − 4536x

5 + 22449x

4 − 67284x

3 + 118124x

2 − 109584x + 40320 = 0

in the interval [5.5,6.5].

(d) In (c), change 36 to 36.001 and repeat the test.

Problem 3. Write a program for Newton’s method and test it on the following problems

(a) Find the roots of x − tan x = 0 near 4.5 and 7.7.

(b) Solve the equation x

3 + 3x = 5x

2 + 7, starting at x0 = 5.

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