MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
The function h(x)=x5−3x−1 is continuous on [1,2]. A numerical approximation of a zero of h will be obtained by repeatedly halving an initial interval that brackets the zero.
(a) Show that h(1) and h(2) have opposite signs and hence that h has at least one zero in (1,2).
(b) After n successive halvings of the interval [1,2], the length of the bracketing interval is 2n1. Determine the least integer n such that this length is less than 10−4. Show your reasoning clearly.
(c) State the maximum possible absolute error of any approximation produced after the number of halvings found in part (b).