Textbook Question
101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.
Ch. 4 - Applications of Derivatives
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 4, Problem 4.7.79
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dr/dθ = −π sin (πθ), r(0) = 0
Verified step by step guidance1
Identify the given differential equation and initial condition: \( \frac{dr}{d\theta} = -\pi \sin(\pi \theta) \) with \( r(0) = 0 \).
Recognize that this is a first-order ordinary differential equation that can be solved by integrating both sides with respect to \( \theta \).
Set up the integral: \( r(\theta) = \int -\pi \sin(\pi \theta) \, d\theta + C \), where \( C \) is the constant of integration.
Perform the integration by using the substitution method or recalling the integral of \( \sin(kx) \), which is \( -\frac{1}{k} \cos(kx) \).
Apply the initial condition \( r(0) = 0 \) to solve for the constant \( C \), then write the final expression for \( r(\theta) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Separable Differential Equations
A separable differential equation can be written as a product of a function of one variable and a function of another, allowing variables to be separated on opposite sides of the equation. This technique simplifies solving by integrating each side independently.
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Solving Separable Differential Equations
Initial Value Problems (IVP)
An initial value problem involves solving a differential equation with a given initial condition, which specifies the value of the unknown function at a particular point. This condition helps determine the unique solution among many possible ones.
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Integration of Trigonometric Functions
Solving differential equations involving trigonometric functions requires knowledge of integrating these functions, such as sine and cosine. Recognizing standard integrals and applying them correctly is essential to find the explicit solution.
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Related Practice
Textbook Question
99. In Exercises 99 and 100, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f.
Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.
g(θ) = θ³ᐟ⁵, −32 ≤ θ ≤ 1
Textbook Question
Solve the initial value problems in Exercises 71–90.
y⁽⁴⁾ = −sin t + cos t;
y′′′(0) =7, y′′(0) = y′(0) = −1, y(0) = 0
Textbook Question
Business and Economics
60. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫x⁻¹ᐟ³ dx