교과서 질문
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = (x + b) / (x − 2), b > −2 and constant
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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 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
7장, 문제 7.2.87
Solve the initial value problems in Exercises 87 and 88.
87. dy/dx = 1 + 1/x, y(1) = 3
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
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.
추천 영상:
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.
추천 영상:
6:04Intro to Rational Functions
관련 실천
교과서 질문
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))
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교과서 질문
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
73. y = log₄ x + log₄ x²
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교과서 질문
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
14. y = ln(2θ+2)
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교과서 질문
Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).
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교과서 질문
Evaluate the integrals in Exercises 33–54.
53. ∫ (e^r / (1 + e^r)) dr
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