Textbook Question
Sketch the graphs of the rational functions in Exercises 53–60.
𝓍²
y = ------------------
𝓍² ― 4
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 4.7.37
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.
∫7sin(θ/3) dθ
Verified step by step guidance1
Identify the integral to solve: \(\int 7 \sin\left(\frac{\theta}{3}\right) \, d\theta\).
Recognize that the integral involves a sine function with a linear argument \(\frac{\theta}{3}\). Use substitution to simplify the integral. Let \(u = \frac{\theta}{3}\), so that \(\theta = 3u\).
Calculate the differential \(d\theta\) in terms of \(du\): since \(\theta = 3u\), then \(d\theta = 3 \, du\).
Rewrite the integral in terms of \(u\): \(\int 7 \sin(u) \, d\theta = \int 7 \sin(u) \cdot 3 \, du = \int 21 \sin(u) \, du\).
Integrate \(21 \sin(u)\) with respect to \(u\): recall that \(\int \sin(u) \, du = -\cos(u) + C\), so the integral becomes \(-21 \cos(u) + C\). Finally, substitute back \(u = \frac{\theta}{3}\) to express the answer in terms of \(\theta\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integral
An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation and is expressed without limits, showing all possible functions whose derivative matches the integrand.
Recommended video:
05:04
Introduction to Indefinite Integrals
Integration of Trigonometric Functions
Integrating trigonometric functions like sine involves using known antiderivatives, such as ∫sin(ax) dx = -cos(ax)/a + C. Recognizing the inner function and applying the chain rule in reverse is essential for correct integration.
Recommended video:
6:04Introduction to Trigonometric Functions
Substitution Method
The substitution method simplifies integration by changing variables to handle composite functions. For example, setting u = θ/3 transforms the integral into a simpler form, making it easier to integrate and then revert to the original variable.
Recommended video:
07:33Euler's Method
Related Practice
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 + 1) dx
Textbook Question
Finding Extrema from Graphs
In Exercises 11–14, match the table with a graph.
Textbook Question
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = {x² − x, −2 ≤ x ≤−1
2x² − 3x − 3, −1 < x ≤ 0
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Textbook Question
Finding Functions from Derivatives
Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.
Textbook Question
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
g(x) = x√8 − x²