Textbook Question
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)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.5.88a
88. Given that x>0, find the maximum value, if any, of
a. x^(1/x)
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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06:30Logarithmic Differentiation
Related Practice
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Textbook Question
[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)