Let f(x) = (x^2) - 4*cos(x) (a) Show that f(x) has at least one zero on the inte
ID: 2966942 • Letter: L
Question
Let f(x) = (x^2) - 4*cos(x)
(a) Show that f(x) has at least one zero on the interval [1, 2]
(b) If bisection method is used to solve f(x) = 0, determine the number of iterations needed of each interval to reach an accuracy of 10^-4
(c) Perform three steps in the bisection method iteration.
What I did so far:
(a) f(1) = -1.161209223 and f(2) = 5.664587346. So, f(x) is continuous and has at least one zero on the interval [1, 2] by I.V.T.
(b) I need help with this one
(c)
Iteration 1: (1 + 2)/2 = 1.5, f(1.5) = 1.967051193
Iteration 2: (1 + 1.5)/2 = 1.25, f(1.25) = 0.3012105504
Iteration 3: (1.25 + 1)/2 = 1.125, f(1.125) = -0.4590810672
Explanation / Answer
(b)
notice that , after each iteration the interval length gets halved. So after k iterations, the interval length will be
(1/2)^k times the original.
original interval length [1 , 2] = 1
length after k iterations = (1/2)^k * 1 = (1/2)^k
accuracy = 10^-4 = (1/2)^k
k = 4/ log(2) = 13.28
so you need 14 iterations.
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