Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
a. Does f'(0) 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 4.7.66a
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫√(2x + 1) dx = √(x² + x +C)
Verified step by step guidance1
Identify the integral to be checked: \(\int \sqrt{2x + 1} \, dx\) and the proposed antiderivative \(\sqrt{x^2 + x + C}\).
Recall that to verify if a function \(F(x)\) is an antiderivative of \(f(x)\), we differentiate \(F(x)\) and check if \(F'(x) = f(x)\).
Differentiate the proposed antiderivative \(F(x) = \sqrt{x^2 + x + C}\) using the chain rule: \(F'(x) = \frac{1}{2\sqrt{x^2 + x + C}} \cdot (2x + 1)\).
Compare \(F'(x)\) with the original integrand \(\sqrt{2x + 1}\). Notice that \(F'(x)\) has a denominator involving \(\sqrt{x^2 + x + C}\), which is not present in the original integrand.
Conclude that since \(F'(x) \neq \sqrt{2x + 1}\), the proposed antiderivative is incorrect.
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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 f(x) is another function F(x) whose derivative is f(x). The indefinite integral symbol ∫ represents the family of all antiderivatives, including a constant of integration C, since differentiation of a constant is zero.
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Introduction to Indefinite Integrals
Integration of Composite Functions and Substitution
When integrating functions like √(2x + 1), substitution is often used by setting u = 2x + 1 to simplify the integral. This method helps handle composite functions by changing variables, making the integral easier to evaluate correctly.
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Verification of Antiderivative by Differentiation
To check if a proposed antiderivative is correct, differentiate it and see if the result matches the original integrand. If the derivative does not equal the integrand, the formula is incorrect.
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Related Practice
Textbook Question
52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,
respectively.
a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)
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
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
1
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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
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 < ∞