Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫3(2x + 1)² dx = (2x + 1)³ + C
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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 4.7.2b
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x⁷
Verified step by step guidance1
Recall that an antiderivative of a function is another function whose derivative is the original function. For power functions, the antiderivative rule is: if \(f(x) = x^n\), then an antiderivative is \(F(x) = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration and \(n \neq -1\).
Identify the exponent in the given function \(x^7\). Here, \(n = 7\).
Apply the antiderivative formula by increasing the exponent by 1: \(7 + 1 = 8\).
Divide by the new exponent to get the antiderivative: \(\frac{x^8}{8} + C\).
Verify your answer by differentiating \(\frac{x^8}{8} + C\) and checking that the derivative is \(x^7\).
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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 often expressed with an arbitrary constant, C, since differentiation of a constant is zero.
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05:04
Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) + C. This rule is fundamental for finding antiderivatives of polynomial functions like x⁷.
Recommended video:
04:04Power Rule for Indefinite Integrals
Verification by Differentiation
After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps avoid mistakes in integration.
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05:53Finding Differentials
Related Practice
Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = (x − 2)²ᐟ³.
b. Show that the only local extreme value of f occurs at x = 2.
Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
4 sec 3x tan 3x
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Textbook Question
38. What values of a and b make f(x) = x^3 + ax^2 + bx have
b. a local minimum at x = 4 and a point of inflection at x = 1?
Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/2)x⁻³ᐟ²
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Textbook Question
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
b. On what open intervals is f increasing or decreasing?
f′(x) = x(x − 1)