Textbook Question
1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
c. √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 7.4.1c
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
c. y = e^(-x) + Ce^(-(3/2)x)
Verified step by step guidance1
Identify the given differential equation: \(2y' + 3y = e^{-x}\) and the proposed solution: \(y = e^{-x} + Ce^{-\frac{3}{2}x}\).
Compute the derivative \(y'\) of the given function \(y\). Since \(y = e^{-x} + Ce^{-\frac{3}{2}x}\), use the chain rule to find \(y' = -e^{-x} - \frac{3}{2}Ce^{-\frac{3}{2}x}\).
Substitute \(y\) and \(y'\) into the left-hand side of the differential equation: calculate \(2y' + 3y = 2\left(-e^{-x} - \frac{3}{2}Ce^{-\frac{3}{2}x}\right) + 3\left(e^{-x} + Ce^{-\frac{3}{2}x}\right)\).
Simplify the expression by distributing and combining like terms carefully, paying attention to coefficients and exponents.
Verify that after simplification, the expression equals the right-hand side of the differential equation, \(e^{-x}\), confirming that the given function 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 all functions that satisfy this relationship. In this problem, verifying a solution involves substituting the given function into the equation to check if it holds true.
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Derivative of Exponential Functions
The derivative of an exponential function e^(kx) is ke^(kx). Understanding this rule is essential to compute y' when y involves exponential terms, which is necessary to substitute into the differential equation.
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General Solution and Constant of Integration
The general solution to a linear differential equation often includes an arbitrary constant C, representing infinitely many solutions. Recognizing how this constant appears and affects the solution helps verify that the given function satisfies the equation for all values of C.
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Related Practice
Textbook Question
10. True, or false? As x→∞,
c. 1/x - 1/x² = o(1/x)
Textbook Question
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
c. 1/√x
Textbook Question
5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
c. ln(√x)
Textbook Question
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
c. f(x) = x³/ (3 - 4x), g(x) = x², (a, b) = (0, 3)
Textbook Question
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
c. √(1+x^4)
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