Domanda del libro di testo
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
a. y = e^(-x)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.5.88a
88. Given that x>0, find the maximum value, if any, of
a. x^(1/x)
Guida verificata passo dopo passo1
Identify the function to maximize: \(f(x) = x^{\frac{1}{x}}\) with the domain \(x > 0\).
Rewrite the function using the natural logarithm to simplify differentiation: consider \(y = x^{\frac{1}{x}}\), then take the natural log to get \(\ln y = \frac{1}{x} \ln x\).
Differentiate both sides with respect to \(x\) using implicit differentiation: \(\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left( \frac{\ln x}{x} \right)\).
Apply the quotient rule to differentiate \(\frac{\ln x}{x}\): \(\frac{d}{dx} \left( \frac{\ln x}{x} \right) = \frac{(\frac{1}{x}) \cdot x - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2}\).
Set the derivative \(\frac{dy}{dx}\) equal to zero to find critical points: since \(\frac{dy}{dx} = y \cdot \frac{1 - \ln x}{x^2}\), solve \(1 - \ln x = 0\) to find \(x = e\). Then analyze the behavior around \(x = e\) to determine if it is a maximum.
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Function Analysis and Domain
Understanding the behavior of the function f(x) = x^(1/x) for x > 0 is essential. This involves recognizing the domain restrictions and how the function behaves as x approaches 0 and infinity, which helps in identifying potential maxima or minima.
Video consigliato:
Percorso guidato
5:10
Finding the Domain and Range of a Graph
Differentiation and Critical Points
To find the maximum value of the function, we use differentiation to find critical points where the derivative equals zero or is undefined. These points are candidates for local maxima or minima and are crucial for optimization problems.
Video consigliato:
04:50Critical Points
Logarithmic Differentiation
Since the function involves a variable exponent, logarithmic differentiation simplifies the process. Taking the natural logarithm of f(x) = x^(1/x) transforms it into a product, making it easier to differentiate and solve for critical points.
Video consigliato:
06:30Logarithmic Differentiation
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31. The incidence of a disease (Continuation of Example 4.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.
a. How long will it take to reduce the number of cases to 1000?
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Find the inverse of the function f(x)=mx, where m is a constant different from zero.
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3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + 4x
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a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.
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[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
141. a. y=arccos(cos x)
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