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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.83
Capitolo 7, Problema 7.3.83

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

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1
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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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.
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Critical 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.
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The Second Derivative Test: Finding Local Extrema

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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Percorso guidato
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Asymptotes of Hyperbolas
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