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 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 4.7.66a
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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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 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.
Recommended video:
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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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05:50Antiderivatives
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
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.
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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 < ∞