Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)
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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 7.4.2b
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
2. y' = y²
b. y = -1/(x+3)
Verified step by step guidance1
Start by identifying the given function: \(y = -\frac{1}{x+3}\).
Find the derivative of \(y\) with respect to \(x\). Use the chain rule to differentiate \(y = - (x+3)^{-1}\), which gives \(y' = -(-1)(x+3)^{-2} \cdot 1 = \frac{1}{(x+3)^2}\).
Express \(y^2\) by squaring the original function: \(y^2 = \left(-\frac{1}{x+3}\right)^2 = \frac{1}{(x+3)^2}\).
Compare the derivative \(y'\) and \(y^2\) to check if they are equal: both are \(\frac{1}{(x+3)^2}\).
Since \(y' = y^2\), conclude that the function \(y = -\frac{1}{x+3}\) satisfies the differential equation \(y' = y^2\).
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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 to its derivatives. Solving it means finding a function that satisfies this relationship. In this problem, verifying a solution involves checking if the given function and its derivative satisfy the equation y' = y².
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07:39
Classifying Differential Equations
Derivative of a Function
The derivative represents the rate of change of a function with respect to its variable. To verify the solution, you must compute the derivative y' of the given function y = -1/(x+3) using differentiation rules, such as the chain or quotient rule.
Recommended video:
06:30Derivatives of Other Trig Functions
Substitution and Verification
After finding the derivative y', substitute both y and y' into the differential equation to check if the equality holds. This process confirms whether the given function is indeed a solution to the differential equation.
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04:27Substitution With an Extra Variable
Related Practice
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%.
b. How long will it take to eradicate the disease—that is, reduce the number of cases to less than 1?
Textbook Question
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2
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Textbook Question
2. Express the following logarithms in terms of ln 5 and ln 7.
b. ln 9.8
Textbook Question
Find the inverse of f(x)=x+b (b constant). How is the graph of f^(-1) related to the graph of f?
Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
67. ∫(from 0 to 2√3)dx/√(4+x²)
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