Textbook Question
90. Find f'(0) for
f(x) = e^(-1/x²), x≠0
= 0, x = 0.
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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 7.2.87
Solve the initial value problems in Exercises 87 and 88.
87. dy/dx = 1 + 1/x, y(1) = 3
Verified step by step guidance1
Identify the given differential equation: \(\frac{dy}{dx} = 1 + \frac{1}{x}\) with the initial condition \(y(1) = 3\).
Rewrite the differential equation to separate variables or recognize it as a direct integration problem since the right side is expressed explicitly in terms of \(x\).
Integrate both sides with respect to \(x\): \(y = \int \left(1 + \frac{1}{x}\right) \, dx\).
Compute the integral by splitting it into two simpler integrals: \(\int 1 \, dx\) and \(\int \frac{1}{x} \, dx\).
Use the initial condition \(y(1) = 3\) to solve for the constant of integration after finding the general solution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
First-Order Differential Equations
A first-order differential equation involves the first derivative of a function and can often be solved by direct integration or separation of variables. Understanding how to manipulate and integrate such equations is essential for finding the general solution.
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07:39
Classifying Differential Equations
Initial Value Problems (IVP)
An initial value problem specifies the value of the unknown function at a particular point, allowing determination of the unique solution from the family of general solutions. Applying the initial condition helps find the constant of integration.
Recommended video:
05:03Initial Value Problems
Integration of Rational Functions
Solving dy/dx = 1 + 1/x requires integrating terms like 1 and 1/x separately. Knowing the integral of 1/x is ln|x| and the integral of 1 is x is crucial for correctly finding the antiderivative.
Recommended video:
6:04Intro to Rational Functions
Related Practice
Textbook Question
Solve the differential equation in Exercises 9–22.
12. (dy/dx) = 3x²e^(-y)
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Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = cos(e^(-θ^2))
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Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
29. y=arcsec(1/t), 0<t<1
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Textbook Question
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
8. cosh(3x) - sinh(3x)
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Textbook Question
82. Find a curve through the point (1, 0) whose length from x=1 to x=2 is
L = ∫(from 1 to 2)√(1 + 1/x²)dx.
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