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.
k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0
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 4.7.13a
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
sec²x
Verified step by step guidance1
Recall that an antiderivative of a function is another function whose derivative is the original function. In other words, if \(F'(x) = f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\).
Identify the given function: \(f(x) = \sec^{2}x\).
Recall the derivative of the tangent function: \(\frac{d}{dx}(\tan x) = \sec^{2}x\).
Since the derivative of \(\tan x\) is \(\sec^{2}x\), an antiderivative of \(\sec^{2}x\) is \(\tan x\) plus a constant of integration, \(C\).
Write the general antiderivative as \(\int \sec^{2}x \, dx = \tan x + C\), where \(C\) is an arbitrary constant.
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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
Basic Trigonometric Derivatives
Understanding the derivatives of trigonometric functions like tangent and secant is essential. For example, the derivative of tan(x) is sec²(x), which helps identify that the antiderivative of sec²(x) is tan(x) plus a constant.
Recommended video:
06:35Derivatives of Other Inverse 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 reinforces the connection between differentiation and integration.
Recommended video:
05:53Finding Differentials
Related Practice
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.
−πsin πx
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(x − 1)
Textbook Question
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = sin x − cos x,0 ≤ x ≤ 2π
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(x) = (x + 1)², −∞ < x ≤ 0
Textbook Question
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
f(r) = 3r³ + 16r