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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.R.28
Capitolo 7, Problema 7.R.28

27–28. Curve sketching Use the graphing techniques of Section 4.4 to graph the following functions on their domains. Identify local extreme points, inflection points, concavity, and end behavior. Use a graphing utility only to check your work.
f(x) = ln x – ln² x

Guida verificata passo dopo passo
1
Identify the domain of the function \(f(x) = \ln x - (\ln x)^2\). Since \(\ln x\) is defined only for \(x > 0\), the domain is \(x > 0\).
Find the first derivative \(f'(x)\) to determine critical points and intervals of increase or decrease. Use the chain rule and derivative of \(\ln x\): \(f'(x) = \frac{1}{x} - 2 \ln x \cdot \frac{1}{x} = \frac{1 - 2 \ln x}{x}\).
Set the first derivative equal to zero to find critical points: \(\frac{1 - 2 \ln x}{x} = 0 \implies 1 - 2 \ln x = 0 \implies \ln x = \frac{1}{2} \implies x = e^{\frac{1}{2}}\). Also check where \(f'(x)\) is undefined, but since \(x > 0\), \(f'(x)\) is defined everywhere in the domain.
Find the second derivative \(f''(x)\) to analyze concavity and inflection points. Differentiate \(f'(x)\): \(f''(x) = \frac{d}{dx} \left( \frac{1 - 2 \ln x}{x} \right)\). Use the quotient or product rule carefully to express \(f''(x)\) in terms of \(x\) and \(\ln x\).
Set \(f''(x) = 0\) to find possible inflection points and analyze the sign of \(f''(x)\) on intervals to determine concavity: - If \(f''(x) > 0\), the graph is concave up. - If \(f''(x) < 0\), the graph is concave down. Finally, analyze the end behavior by considering the limits as \(x \to 0^+\) and \(x \to \infty\).

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Domain and Logarithmic Functions

Understanding the domain of the function is crucial, especially since it involves natural logarithms (ln x). The function is defined only for x > 0 because the logarithm of non-positive numbers is undefined. Recognizing this domain restriction helps avoid errors in graphing and analysis.
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Graphs of Logarithmic Functions

Critical Points and Local Extrema

Finding local maxima and minima requires computing the first derivative and solving for points where it equals zero or is undefined. These critical points indicate potential local extreme values. Analyzing the sign changes of the derivative around these points confirms whether they are maxima, minima, or neither.
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Critical Points

Concavity and Inflection Points

The second derivative reveals the concavity of the function—whether it curves upward or downward. Points where the second derivative changes sign are inflection points, indicating a change in concavity. Identifying these helps in sketching the curve accurately and understanding the function's behavior.
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Determining Concavity Given a Function
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