Textbook Question
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = x(x − 1)
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 4.7.64c
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫tanθ sec²θ dθ = (1/2) sec²θ + C
Verified step by step guidance1
Identify the integral to be evaluated: \(\int \tan\theta \sec^{2}\theta \, d\theta\).
Recall that \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) and \(\sec\theta = \frac{1}{\cos\theta}\), so \(\sec^{2}\theta = \left(\frac{1}{\cos\theta}\right)^2 = \frac{1}{\cos^{2}\theta}\).
Consider using substitution: let \(u = \tan\theta\), then \(\frac{du}{d\theta} = \sec^{2}\theta\), which implies \(du = \sec^{2}\theta \, d\theta\).
Rewrite the integral in terms of \(u\): \(\int \tan\theta \sec^{2}\theta \, d\theta = \int u \, du\).
Integrate \(\int u \, du\) to get \(\frac{1}{2} u^{2} + C\), then substitute back \(u = \tan\theta\) to obtain \(\frac{1}{2} \tan^{2}\theta + C\). Compare this with the given formula \(\frac{1}{2} \sec^{2}\theta + C\) to determine if it is correct.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a function whose derivative also appears in the integral, allowing the integral to be rewritten in a simpler form. This technique is essential for verifying antiderivative formulas involving composite functions.
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Substitution With an Extra Variable
Derivatives of Trigonometric Functions
Understanding the derivatives of trigonometric functions like tan(θ) and sec(θ) is crucial. For example, the derivative of tan(θ) is sec²(θ), and the derivative of sec(θ) is sec(θ)tan(θ). Recognizing these relationships helps in identifying correct antiderivatives and verifying integral formulas.
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06:35Derivatives of Other Inverse Trigonometric Functions
Constant of Integration
When finding antiderivatives, it is important to include the constant of integration, denoted by C, because indefinite integrals represent a family of functions differing by a constant. This constant accounts for all possible vertical shifts of the antiderivative graph.
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06:18Integration by Parts for Definite Integrals
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.
-sec²(3x/2)
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Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
b. Does f'(-3) exist?
Textbook Question
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = (x − 1)²(x + 2)
Textbook Question
105. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (c) Acceleration equal to zero?
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 - 5 / x²