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 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 4.7.64a
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫tanθ sec²θ dθ = sec³θ / 3 + C
Verified step by step guidance1
Identify the integral to be checked: \(\int \tan\theta \sec^{2}\theta \, d\theta\).
Recall the derivatives of common trigonometric functions: for example, \(\frac{d}{d\theta}(\sec\theta) = \sec\theta \tan\theta\) and \(\frac{d}{d\theta}(\sec^{3}\theta) = 3 \sec^{2}\theta \cdot \sec\theta \tan\theta = 3 \sec^{3}\theta \tan\theta\).
Compare the integrand \(\tan\theta \sec^{2}\theta\) with the derivative of \(\sec^{3}\theta\): since \(\frac{d}{d\theta}(\sec^{3}\theta) = 3 \sec^{3}\theta \tan\theta\), the integrand is \(\frac{1}{3}\) of this derivative.
Therefore, the integral \(\int \tan\theta \sec^{2}\theta \, d\theta\) should be \(\frac{1}{3} \sec^{3}\theta + C\), matching the given formula \(\frac{\sec^{3}\theta}{3} + C\).
Conclude that the formula is correct because the derivative of the proposed antiderivative matches the integrand.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivatives and Indefinite Integrals
An antiderivative of a function is another function whose derivative equals the original function. Indefinite integrals represent the family of all antiderivatives and include a constant of integration, C. Understanding how to find antiderivatives is essential for verifying integral formulas.
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Introduction to Indefinite Integrals
Integration by Substitution
Integration by substitution simplifies integrals by changing variables to make the integral easier to solve. It often involves identifying a part of the integrand as a derivative of another function. Recognizing substitution opportunities helps verify or find antiderivatives involving composite functions.
Recommended video:
04:27Substitution With an Extra Variable
Derivatives of Trigonometric Functions
Knowing the derivatives of trigonometric functions like tan(θ) and sec(θ) is crucial for checking antiderivatives. For example, the derivative of sec(θ) is sec(θ)tan(θ), and the derivative of sec³(θ) involves the chain rule. This knowledge allows one to differentiate the proposed antiderivative to confirm correctness.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 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.
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
20.The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. a.What dimensions will give a box with a square end the largest possible volume?
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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(x) = √(x² − 2x − 3), 3 ≤ x < ∞