Skip to main content
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 5.3.107d
Capitolo 5, Problema 5.3.107d

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.                                                                          
                                                                                                                                                                                     (d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for ƒ, then                                                                                                                                   
     B(𝓍) = 3𝓍² ― 𝓍 is also an area function for ƒ.

Guida verificata passo dopo passo
1
Recall that an area function for a function \( f \) is defined as \( A(x) = \int_a^x f(t) \, dt \) for some fixed lower limit \( a \). The key property is that the derivative of the area function equals the original function, i.e., \( A'(x) = f(x) \).
Given \( A(x) = 3x^2 - x - 3 \), find its derivative to identify \( f(x) \). Using the power rule, \( A'(x) = \frac{d}{dx}(3x^2) - \frac{d}{dx}(x) - \frac{d}{dx}(3) = 6x - 1 - 0 = 6x - 1 \). So, \( f(x) = 6x - 1 \).
Next, check the function \( B(x) = 3x^2 - x \). Find its derivative: \( B'(x) = \frac{d}{dx}(3x^2) - \frac{d}{dx}(x) = 6x - 1 \). Notice that \( B'(x) = f(x) \) as well.
Since both \( A(x) \) and \( B(x) \) have the same derivative \( f(x) = 6x - 1 \), they differ by a constant. To confirm if \( B(x) \) is also an area function for \( f \), check the difference \( A(x) - B(x) = (3x^2 - x - 3) - (3x^2 - x) = -3 \), which is a constant.
Therefore, \( B(x) \) can also serve as an area function for \( f \) because area functions differ by a constant. This aligns with the Fundamental Theorem of Calculus, which states that any two antiderivatives of the same function differ by a constant.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Area Function and the Fundamental Theorem of Calculus

An area function A(x) for a function f is defined as the integral of f from a fixed point to x. According to the Fundamental Theorem of Calculus, the derivative of this area function A'(x) equals the original function f(x). This relationship is key to verifying if a given function is an area function for f.
Video consigliato:
Percorso guidato
05:22
Fundamental Theorem of Calculus Part 2

Derivative of a Polynomial Function

To check if a function is an area function for f, differentiate it and compare the result to f. Differentiation of polynomials involves applying power rules term-by-term. For example, the derivative of 3x² - x - 3 is 6x - 1, which must match f(x) for the function to be an area function.
Video consigliato:
07:00
Taylor Polynomials

Effect of Constant Terms on Area Functions

Adding or subtracting a constant to an area function does not change its derivative, so it remains an area function for the same f. This means if A(x) is an area function for f, then A(x) + C, where C is any constant, is also an area function for f.
Video consigliato:
05:06
Finding Area When Bounds Are Not Given
Pratica correlata
Domanda del libro di testo

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.

59
views
Domanda del libro di testo

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

{Use of Tech} ƒ(𝓍) = e ˣ/₂ on [1,4]; n = 6

(d) Calculate the left and right Riemann sums. 

119
views
Domanda del libro di testo

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

(d) If ∫ₐᵇ ƒ(𝓍) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍, then ƒ is a constant function. 

63
views
Domanda del libro di testo

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

(a) ∫₀³ 5ƒ(𝓍) d𝓍

47
views
Domanda del libro di testo

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(d) 1 + 1/2 + 1/3 + 1/4

80
views
Domanda del libro di testo

Sigma notation Evaluate the following expressions.

(d)     5                                                                                                                                                                              

       ∑ (1 + n²)                                                                                                                                                                          

       n=1                         

75
views