Let , defined for all real . An approximation to the zero of in will be developed by…
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
Let p(x)=cosx−x, defined for all real x. An approximation to the zero of p in [0,1] will be developed by constructing a sequence of nested intervals, each obtained by halving the previous interval.
(a) Verify that p(0) and p(1) have opposite signs and conclude that a zero of p lies in (0,1).
(b) Produce the first four nested intervals produced by successive halving of [0,1] that continue to bracket the zero. For each interval, record its endpoints and midpoint, and state the sign of p at those three points.
(c) Using your work in part (b), give a decimal approximation of the zero of p that is accurate to within 0.0001. State an explicit bound on the absolute error of your approximation and justify that bound.
(d) Describe, in the context of this problem, what the error bound found in part (c) tells you about the location of the exact zero of p.