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 4.7.13c
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)
Verified step by step guidance1
Recognize that the function given is \(-\sec^{2}\left(\frac{3x}{2}\right)\), which resembles the derivative of the tangent function, since \(\frac{d}{dx}[\tan(u)] = \sec^{2}(u) \cdot \frac{du}{dx}\).
Identify the inner function \(u = \frac{3x}{2}\) and compute its derivative: \(\frac{du}{dx} = \frac{3}{2}\).
Set up the antiderivative integral: \(\int -\sec^{2}\left(\frac{3x}{2}\right) dx\).
Use substitution: let \(u = \frac{3x}{2}\), so \(dx = \frac{2}{3} du\). Rewrite the integral in terms of \(u\): \(\int -\sec^{2}(u) \cdot \frac{2}{3} du = -\frac{2}{3} \int \sec^{2}(u) du\).
Recall that \(\int \sec^{2}(u) du = \tan(u) + C\), so the antiderivative is \(-\frac{2}{3} \tan\left(\frac{3x}{2}\right) + C\).
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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 is also called the indefinite integral and includes a constant of integration since differentiation loses constant terms.
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05:04
Introduction to Indefinite Integrals
Derivative of Trigonometric Functions
Knowing the derivatives of basic trig functions like tan(x) and sec(x) is essential. For example, the derivative of tan(x) is sec²(x), which helps in recognizing antiderivatives involving sec² terms.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Chain Rule and Substitution
When functions involve compositions like sec²(3x/2), the chain rule applies. To find antiderivatives, substitution reverses the chain rule by adjusting for the inner function's derivative.
Recommended video:
05:02Intro to the Chain Rule
Related Practice
Textbook Question
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
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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
Finding displacement from an antiderivative of velocity
a. Suppose that the velocity of a body moving along the s-axis is
ds/dt = v = 9.8t − 3.
iii. Now find the body’s displacement from t = 1 to t = 3 given that s = s₀ when t = 0.
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?