Find an approximation to the root with this degree of

Question

1. Use the Bisection method to find solutions accurate to within 10^-2 for x³ - 7x² + 14x - 6 = 0 on each interval.

a. [0,1] b. [1, 3.2] c. [3.2, 4]

2. Let f (x) = (x + 2)(x + 1)²x(x -1)³(x - 2). To which zero of f does the Bisection method converge when applied on the following intervals?

a. [-1.5, 2.5] b. [ -0.5, 2.4] c. [-0.5, 3] d. [-3, -0.5]

3. Use Theorem 2.1 to find a bound for the number of iterations needed to achieve an approximation with accuracy 10-3 to the solution of x³ + x - 4 = 0 lying in the interval [1, 4]. Find an approximation to the root with this degree of accuracy.

4. Let f (x) = (x - 1)¹⁰, p = 1, and p_n = 1 + 1/n. Show that |(f(p_n)| < 10^-3 whenever n > 1 but that |p - p_n| < 10^-3 requires that n > 1000.