EXERCISE SET 3.1
1. Verify that the function f(x) = x^2 sin x + 2x - 3 has exactly one root in (0, 2). Find this root by using the bisection method with an error of no more than 10^-5. Count the number of iterations needed to find the root and compare your answer with the result obtained by using the error formula given by Theorem 1 of Section 3.1.
EXERCISE SET 3.2
1. Verify that the function f(x) = x^2 sin x + 2x - 3 has exactly one root in (0, 2). Find this root using the method of false position with an error of no more than 10^-3.
EXERCISE SET 3.4
1. Approximate to within 10^-6 the root of the equation e^-2x - 7x = 0 in [1/9, 2/3] by the secant method.
EXERCISE SET 3.5
1. Find the roots of the following equations correct to six decimal places using Newton’s method.
(a) x^5 - 3x^3 - x + 4 = 0, x0 = 0, (b) 3x^3 - x + 4 = 0, x0 = -1, (c) x^3 - x^2 + ex = 0, x0 = -1, (d) ex + x sin x = 0, x0 = 0.
EXERCISE SET 3.8
1. Solve the nonlinear system of equations
ln(x^2 + y) - 1 + y = 0
√x + xy = 0
using Newton’s method with x0 = 2.4 and y0 = -0.6.