Textbook Question
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²
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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 4.7.12a
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
πcos πx
Verified step by step guidance1
Identify the function you need to find the antiderivative of, which is \(\pi \cos \pi x\).
Recall that the antiderivative (indefinite integral) of \(\cos(ax)\) with respect to \(x\) is \(\frac{1}{a} \sin(ax) + C\), where \(a\) is a constant and \(C\) is the constant of integration.
Apply this rule to the function \(\pi \cos \pi x\). Since the function has a constant multiplier \(\pi\), factor it out and integrate \(\cos \pi x\).
Integrate \(\cos \pi x\) to get \(\frac{1}{\pi} \sin \pi x + C\). Then multiply by the constant \(\pi\) outside the integral.
Combine the results to write the antiderivative as \(\pi \times \frac{1}{\pi} \sin \pi x + C\), which simplifies to \(\sin \pi x + C\). Remember to verify your answer by differentiating it to check if you get back the original function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivative (Indefinite Integral)
An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is expressed with an arbitrary constant since differentiation of a constant is zero.
Recommended video:
05:04
Introduction to Indefinite Integrals
Integration of Trigonometric Functions
Integrating trigonometric functions like cosine involves recognizing standard integral forms. For example, the integral of cos(kx) dx is (1/k) sin(kx) + C, where k is a constant multiplier inside the function's argument.
Recommended video:
6:04Introduction to Trigonometric Functions
Verification by Differentiation
After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps identify any missing constants or errors in integration.
Recommended video:
05:53Finding Differentials
Related Practice
Textbook Question
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
g(x) = (x − 2) / (x²−1), 0 ≤ x < 1
Textbook Question
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
a. What are the critical points of f?
f′(x) = 1− 4/x², x ≠ 0
Textbook Question
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
i. y = x² − 4
Textbook Question
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.
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Textbook Question
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
g(x) = x² − 4x + 4, 1 ≤ x < ∞