Textbook Question
86. This exercise explores the difference between
lim(x→∞)(1 + 1/x²)^x
and
lim(x→∞)(1 + 1/x)^x = e
c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.
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 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
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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07:39
Classifying Differential Equations
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.
Recommended video:
04:50Derivatives of General Exponential Functions
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.
Recommended video:
05:11Integrals of General Exponential Functions
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
7. c. arcsec(-2)
Textbook Question
155. Which is bigger, πᵉ or e^π?
Calculators have taken some of the mystery out of this once-challenging question.
(Go ahead and check; you will see that it is a very close call.)
You can answer the question without a calculator, though.
d. Conclude that
xᵉ < eˣfor all positivex ≠ e.
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. c. arccos(√3/2)
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
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
c. 3ln ∛(t² - 1) - ln(t+1)