Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
a. lim(x→∞) tanh x
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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 7.4.1a
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)
Verified step by step guidance1
Identify the given differential equation: \(2y' + 3y = e^{-x}\).
Given the function \(y = e^{-x}\), find its derivative \(y'\) with respect to \(x\). Recall that the derivative of \(e^{-x}\) is \(-e^{-x}\), so \(y' = -e^{-x}\).
Substitute \(y = e^{-x}\) and \(y' = -e^{-x}\) into the left-hand side of the differential equation: \(2y' + 3y = 2(-e^{-x}) + 3(e^{-x})\).
Simplify the expression: \(2(-e^{-x}) + 3(e^{-x}) = -2e^{-x} + 3e^{-x} = ( -2 + 3 ) e^{-x} = e^{-x}\).
Since the left-hand side simplifies to \(e^{-x}\), which matches the right-hand side of the differential equation, this shows that \(y = e^{-x}\) is indeed a solution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differential Equations
A differential equation relates a function with its derivatives. Solving it means finding a function that satisfies this relationship. In this problem, verifying a solution involves substituting the given function and its derivative into the equation to check if both sides are equal.
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Classifying Differential Equations
Derivative of Exponential Functions
The derivative of an exponential function like e^(-x) is found using the chain rule. Specifically, d/dx[e^(-x)] = -e^(-x). Understanding this derivative is essential to correctly substitute y' in the differential equation.
Recommended video:
04:50Derivatives of General Exponential Functions
Verification of Solutions
To verify a proposed solution to a differential equation, substitute the function and its derivative into the equation. If the left-hand side equals the right-hand side for all x in the domain, the function is a valid solution.
Recommended video:
04:00Solutions to Basic Differential Equations
Related Practice
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + √x
Textbook Question
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?
Textbook Question
89. Use limits to find horizontal asymptotes for each function.
a. y = x tan(1/x)
Textbook Question
88. Given that x>0, find the maximum value, if any, of
a. x^(1/x)
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)