Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.3.107c

Chapter 5, Problem 5.3.107c

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(c) The functions p(𝓍) = sin 3𝓍 and q(𝓍) = 4 sin 3𝓍 are antiderivatives of the same function.

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Recall the definition of antiderivatives: Two functions are antiderivatives of the same function if their derivatives are equal.

Find the derivative of the first function \(p(\mathcal{x}) = \sin 3\mathcal{x}\) using the chain rule: \(p'(\mathcal{x}) = 3 \cos 3\mathcal{x}\).

Find the derivative of the second function \(q(\mathcal{x}) = 4 \sin 3\mathcal{x}\) similarly: \(q'(\mathcal{x}) = 4 \cdot 3 \cos 3\mathcal{x} = 12 \cos 3\mathcal{x}\).

Compare the derivatives \(p'(\mathcal{x}) = 3 \cos 3\mathcal{x}\) and \(q'(\mathcal{x}) = 12 \cos 3\mathcal{x}\). Since they are not equal, \(p\) and \(q\) are not antiderivatives of the same function.

Conclude that the statement is false because the derivatives differ by a constant factor, so \(p\) and \(q\) cannot both be antiderivatives of the same function.

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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). It represents the family of all functions differing by a constant, since differentiation eliminates constants. For example, if F'(x) = f(x), then F(x) + C is the general antiderivative.

Linearity of Differentiation

Differentiation is a linear operation, meaning the derivative of a constant multiple of a function is the constant times the derivative of the function. For instance, if g(x) = k·f(x), then g'(x) = k·f'(x). This property helps compare functions to determine if they share the same derivative.

Checking if Two Functions are Antiderivatives of the Same Function

Two functions are antiderivatives of the same function if their derivatives are identical. If their derivatives differ, they cannot be antiderivatives of the same function. For example, p(x) = sin(3x) and q(x) = 4 sin(3x) have different derivatives, so they are not antiderivatives of the same function.

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Related Practice

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.

∫₁⁷ 1/𝓍 d𝓍 ; n = 6

Textbook Question

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(d) ∫₀⁸ 3𝓍(4 ― 𝓍) d(𝓍)

Textbook Question

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(d) ∫₆³ (ƒ(𝓍) + 2g(𝓍)) d𝓍

Textbook Question

Use Table 5.6 to evaluate the following definite integrals.

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Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.

∫₁⁷ 1/𝓍 d𝓍 ; n = 6

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. (c) The functions p(𝓍) = sin 3𝓍 and q(𝓍) = 4 sin 3𝓍 are antiderivatives of the same function.

Verified video answer for a similar problem:

Key Concepts

Antiderivative (Indefinite Integral)

Linearity of Differentiation

Checking if Two Functions are Antiderivatives of the Same Function

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(c) The functions p(𝓍) = sin 3𝓍 and q(𝓍) = 4 sin 3𝓍 are antiderivatives of the same function.