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

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

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

Problem 7.3.83

Chapter 7, Problem 7.3.83

Points of inflection Find the x-coordinate of the point(s) of inflection of f(x) = tanh² x.

Verified step by step guidance

Recall that points of inflection occur where the second derivative of the function changes sign, which typically happens where the second derivative is zero or undefined.

Start by finding the first derivative of the function \(f(x) = \tanh^2 x\). Use the chain rule: if \(f(x) = (g(x))^2\), then \(f'(x) = 2 g(x) g'(x)\). Here, \(g(x) = \tanh x\).

Calculate \(g'(x)\), the derivative of \(\tanh x\). Recall that \(\frac{d}{dx} \tanh x = \operatorname{sech}^2 x\).

Combine these results to write the first derivative: \(f'(x) = 2 \tanh x \cdot \operatorname{sech}^2 x\).

Next, find the second derivative \(f''(x)\) by differentiating \(f'(x)\). Use the product rule on \(2 \tanh x \cdot \operatorname{sech}^2 x\), then set \(f''(x) = 0\) and solve for \(x\) to find potential inflection points.

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Key Concepts

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

Points of Inflection

Points of inflection occur where the concavity of a function changes, which means the second derivative changes sign. To find these points, we identify where the second derivative is zero or undefined and verify a sign change around those points.

Second Derivative

The second derivative of a function measures the curvature or concavity of the graph. It is found by differentiating the first derivative. Analyzing the second derivative helps determine where the function is concave up or down and locate inflection points.

Derivative of Hyperbolic Functions

Hyperbolic functions like tanh(x) have specific derivatives: the derivative of tanh(x) is sech²(x). Understanding these derivatives is essential for differentiating f(x) = tanh²(x) correctly and finding the first and second derivatives needed for inflection points.

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Related Practice

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∫₀ ˡⁿ ² tanh x dx

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57–58. Two ways

Evaluate the following integrals two ways.

a. Simplify the integrand first and then integrate.

b. Change variables (let u = ln x), integrate, and then simplify your answer. Verify that both methods give the same answer.

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