Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.3.94

Chapter 7, Problem 7.3.94

Newton’s method Use Newton’s method to find all local extreme values of ƒ(x) = x sech x.

Verified step by step guidance

Start by recalling that local extreme values occur where the first derivative of the function is zero and the second derivative test can be applied to classify the critical points.

Given the function \(f(x) = x \, \text{sech} \; x\), first find the derivative \(f'(x)\) using the product rule: \(f'(x) = \frac{d}{dx}[x] \cdot \text{sech} \; x + x \cdot \frac{d}{dx}[\text{sech} \; x]\).

Calculate the derivative of \(\text{sech} \; x\), which is \(\frac{d}{dx}[\text{sech} \; x] = -\text{sech} \; x \tanh x\), and substitute it back into the expression for \(f'(x)\) to get \(f'(x) = \text{sech} \; x - x \, \text{sech} \; x \tanh x\).

Set \(f'(x) = 0\) to find critical points: \(\text{sech} \; x - x \text{sech} \; x \tanh x = 0\). Simplify this to \(1 - x \tanh x = 0\), or equivalently \(x \tanh x = 1\).

Use Newton's method to solve \(g(x) = x \tanh x - 1 = 0\). Compute \(g'(x) = \tanh x + x \, \text{sech}^2 x\), then iterate using the formula \(x_{n+1} = x_n - \frac{g(x_n)}{g'(x_n)}\) starting from an initial guess to approximate the roots where \(f'(x) = 0\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton’s Method

Newton’s method is an iterative numerical technique used to approximate roots of a real-valued function. Starting from an initial guess, it uses the function and its derivative to generate successively better approximations. The formula is x_{n+1} = x_n - f(x_n)/f'(x_n). It is especially useful when analytical solutions are difficult.

Local Extreme Values

Local extreme values of a function are points where the function attains a local maximum or minimum. These occur where the first derivative is zero or undefined, and the second derivative test or other methods determine the nature of these points. Finding these values helps understand the function’s behavior.

Derivative of the Function f(x) = x sech x

To find local extrema, you need the first derivative of f(x) = x sech x. This requires applying the product rule and the derivative of sech x, which is -sech x tanh x. Correctly computing f'(x) is essential for setting up the equation f'(x) = 0 to find critical points.

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