4. Applications of Derivatives
Differentials
Problem 4.8.43
Textbook Question
{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.
Where is the first local minimum of f(x) = (cos x)/x on the interval (0,∞) located?
Verified step by step guidance1
Identify the function f(x) = (cos x)/x and its domain, which is (0, ∞).
Calculate the first derivative f'(x) using the quotient rule, which states that if f(x) = g(x)/h(x), then f'(x) = (g'h - gh')/h².
Set the first derivative f'(x) equal to zero to find critical points, as these points will help identify local minima.
Use Newton's method, which involves iterating the formula x_{n+1} = x_n - f(x_n)/f'(x_n), starting from an initial guess close to where you suspect the local minimum might be.
Continue iterating until the values converge to a stable point, which will give you an approximate location of the first local minimum.
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