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.
(2/3)x⁻¹ᐟ³
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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 4.7.109a
Applications
Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).
Find:
∫f(x) dx
Verified step by step guidance1
First, identify the functions f(x) and g(x) given in the problem. We have f(x) = \(\frac{d}{dx}\) (1 - \(\sqrt{x}\)) and g(x) = \(\frac{d}{dx}\) (x + 2).
Next, compute the derivative f(x) by differentiating the function inside the derivative: 1 - \(\sqrt{x}\). Recall that \(\sqrt{x}\) = x^{1/2}, so use the power rule for differentiation.
After finding f(x), set up the integral \(\int\) f(x) \, dx. Since f(x) is the derivative of (1 - \(\sqrt{x}\)), integrating f(x) with respect to x will give you the original function plus a constant of integration.
Write the integral result as (1 - \(\sqrt{x}\)) + C, where C is the constant of integration, because integration is the inverse operation of differentiation.
Finally, verify your result by differentiating your integral expression to ensure it matches f(x). This confirms the correctness of your integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative and Differentiation
The derivative of a function represents the rate at which the function's value changes with respect to its variable. Differentiation rules, such as the power rule, help find derivatives of expressions like √x or x + 2. Understanding how to compute derivatives is essential to identify f(x) and g(x) as given.
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Finding Differentials
Antiderivative and Indefinite Integral
The antiderivative or indefinite integral of a function reverses differentiation, finding a function whose derivative is the given function. The integral symbol ∫f(x) dx represents all functions whose derivative is f(x), plus a constant of integration. This concept is key to solving ∫f(x) dx.
Recommended video:
05:04Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is used to integrate functions involving powers of x, such as those derived from derivatives of √x or polynomials. Applying this rule simplifies finding the antiderivative of f(x).
Recommended video:
04:04Power Rule for Indefinite Integrals
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) = 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
views
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
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
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)