Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = (x − 2)²ᐟ³.
a. Does f′(2) exist?
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- 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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Video duration:
2mKey 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.
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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.
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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.
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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.
−3x⁻⁴
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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) = x / 2 − 2sin (x/2), 0 ≤ x ≤ 2π
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.
(3/2)√x
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) = 12t − t³, −3 ≤ t < ∞
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) = csc²x − 2cot x, 0 < x < π
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