Textbook Question
Applications
Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).
Find:
∫f(x) dx
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 4.7.9a
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
(2/3)x⁻¹ᐟ³
Verified step by step guidance1
Identify the function to find the antiderivative of: \(\frac{2}{3} x^{-\frac{1}{3}}\).
Recall the power rule for antiderivatives: for \(x^n\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Apply the power rule by adding 1 to the exponent: \(-\frac{1}{3} + 1 = \frac{2}{3}\).
Write the antiderivative as \(\frac{2}{3} \cdot \frac{x^{\frac{2}{3}}}{\frac{2}{3}} + C\).
Simplify the expression by canceling the \(\frac{2}{3}\) terms and include the constant of integration \(C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivatives (Indefinite Integrals)
An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often represented as indefinite integrals. For example, the antiderivative of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration.
Recommended video:
05:04
Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, provided n ≠ -1. This rule is essential for integrating functions with variable exponents, including fractional and negative powers, by increasing the exponent by one and dividing by the new exponent.
Recommended video:
04:04Power Rule for Indefinite Integrals
Verification by Differentiation
After finding an antiderivative, verifying it by differentiation ensures correctness. Differentiating the antiderivative should yield the original function. This step confirms that the integration was performed accurately and helps identify any mistakes in the process.
Recommended video:
05:53Finding Differentials
Related Practice
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) = (x − 1)²(x + 2)
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
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
2x
1
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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.
f(t) = t³ − 3t², −∞ < t ≤ 3
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) = (x − 1)(x + 2)(x − 3)