Homework SourseConsider the equation 1 2x sin x a Use the Intermediate Va
Consider the equation 1 ? 2x = sin x.
(a) Use the Intermediate Value Theorem to prove that this equation has a
root in the interval (0, pie/4). Check all the criterions in the IVT.
(b) Use a graphing calculator to find an interval of length 0.01 that contains
this root. Explain how to obtain this interval.
(a) Use the Intermediate Value Theorem to prove that this equation has a
root in the interval (0, pie/4). Check all the criterions in the IVT.
(b) Use a graphing calculator to find an interval of length 0.01 that contains
this root. Explain how to obtain this interval.
Solution
(a)f(x)=sinx+2x-1 Notice that, f(0) < 0 and f(pi/4) > 0 Therefore, by the Intermediate Value theorem there exists a zero in the interval [0, pi/4]. Now, is this the only root? f\'(x) = cos x+2 for all x. Therefore, f(x) is strictly increasing and therefore, the root we found above is the ONLY root.
