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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.3.107c
Capitolo 5, Problema 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. 

Guida verificata passo dopo passo
1
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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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.
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Introduction to Indefinite Integrals

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.
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Linearization

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

(c) The average value of a linear function on an interval [a, b] is the function value at the midpoint of [a, b] .

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

(c) ∫₃⁶ (3ƒ(𝓍) ― g(𝓍)) d𝓍

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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

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Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    

 (c) ∫₃√₂^⁶ d𝓍/(𝓍² ―9)

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Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

{Use of Tech} ƒ(𝓍) = cos 𝓍 on [0. π/2]; n = 4

(d) Calculate the left and right Riemann sums.

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

(c) Calculate the left and right Riemann sums for the given value of n.

∫₁⁷ 1/𝓍 d𝓍 ; n = 6

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