4. Applications of Derivatives
Differentials
Problem 2.R.83
Textbook Question
b. Estimate a solution to the equation in the given interval using a root finder.
x=cos x; (0,π/2)
Verified step by step guidance1
Identify the function to analyze: f(x) = x - cos(x). This is derived from rearranging the equation x = cos(x).
Determine the interval of interest, which is (0, π/2). Evaluate f(x) at the endpoints of the interval: f(0) and f(π/2).
Calculate f(0) = 0 - cos(0) = -1 and f(π/2) = π/2 - cos(π/2) = π/2. Note that f(0) is negative and f(π/2) is positive.
Since f(0) < 0 and f(π/2) > 0, by the Intermediate Value Theorem, there is at least one root in the interval (0, π/2).
Use a numerical method such as the bisection method or Newton's method to iteratively find a more precise estimate of the root within the interval.
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