Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
d. Determine all extrema 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 4.7.3c
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x⁻⁴ + 2x + 3
Verified step by step guidance1
Identify the function to find the antiderivative of: \(x^{-4} + 2x + 3\).
Recall the power rule for antiderivatives: for \(x^n\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).
Apply the power rule to each term separately: for \(x^{-4}\), the antiderivative is \(\frac{x^{-4+1}}{-4+1} = \frac{x^{-3}}{-3}\).
For the term \$2x\(, rewrite it as \)2x^1$ and apply the power rule: \(2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2} = x^2\).
For the constant term \(3\), recall that the antiderivative of a constant \(a\) is \(ax\), so the antiderivative is \$3x$.
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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 is also called the indefinite integral and includes a constant of integration since differentiation removes constants. Finding antiderivatives involves reversing differentiation rules.
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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) plus a constant. This rule is essential for integrating polynomial terms like x⁻⁴ and 2x in the given function.
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04:04Power Rule for Indefinite Integrals
Linearity of Integration
Integration is linear, meaning the antiderivative of a sum of functions equals the sum of their antiderivatives. This allows us to find the antiderivative of each term separately and then combine the results, simplifying the process for functions like x⁻⁴ + 2x + 3.
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Related Practice
Textbook Question
Applications
Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).
Find:
∫[f(x) + g(x)] dx
1
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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:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = x(x − 1)
Textbook Question
105. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (c) Acceleration equal to zero?
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.
2 - 5 / x²
Textbook Question
106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (d) When is the acceleration positive? Negative?