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.R.105b
Capitolo 5, Problema 5.R.105b

Area functions and the Fundamental Theorem Consider the function
ƒ(t) = { t      if  ―2 ≤ t < 0
t²/2    if    0 ≤ t ≤ 2
and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.
fig
(b) Use the Fundamental Theorem to find an expression for F '(𝓍) for ―2 ≤ 𝓍 < 0.

Guida verificata passo dopo passo
1
The problem asks us to find F'(𝓍) for the interval -2 ≤ 𝓍 < 0 using the Fundamental Theorem of Calculus. Recall that the Fundamental Theorem states that if F(𝓍) = ∫ₐˣ ƒ(t) dt, then F'(𝓍) = ƒ(𝓍), provided ƒ is continuous at 𝓍.
The given function ƒ(t) is piecewise defined as ƒ(t) = t for -2 ≤ t < 0 and ƒ(t) = t²/2 for 0 ≤ t ≤ 2. For the interval -2 ≤ 𝓍 < 0, we are concerned only with the first piece of the function, ƒ(t) = t.
Using the Fundamental Theorem, F'(𝓍) = ƒ(𝓍). Since ƒ(t) = t in this interval, we substitute 𝓍 for t, giving F'(𝓍) = 𝓍 for -2 ≤ 𝓍 < 0.
Verify that the function ƒ(t) = t is continuous on the interval -2 ≤ t < 0. This is true because the function t is a simple linear function, which is continuous everywhere.
Thus, the expression for F'(𝓍) in the interval -2 ≤ 𝓍 < 0 is F'(𝓍) = 𝓍.

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:
3m

Concetti chiave

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

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if F is an antiderivative of a function f on an interval [a, b], then the integral of f from a to b can be computed as F(b) - F(a). This theorem also implies that the derivative of the integral function F(x) is equal to the original function f evaluated at x, which is crucial for finding expressions like F'(x).
Video consigliato:
Percorso guidato
06:11
Fundamental Theorem of Calculus Part 1

Definite Integral

A definite integral represents the signed area under the curve of a function f(t) from a lower limit a to an upper limit b. It is denoted as ∫_a^b f(t) dt and provides a numerical value that corresponds to the accumulation of quantities, such as area, over the specified interval. In this context, F(x) and G(x) are defined as definite integrals of the function f(t) over different intervals.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral

Piecewise Function

A piecewise function is defined by different expressions based on the input value. In this case, the function f(t) is defined differently for the intervals t < 0 and 0 ≤ t ≤ 2. Understanding how to evaluate piecewise functions is essential for correctly applying the Fundamental Theorem of Calculus, especially when determining derivatives or integrals over specific intervals.
Video consigliato:
Percorso guidato
05:36
Piecewise Functions
Pratica correlata
Domanda del libro di testo

Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. 


∫₀² (𝓍²―4) d𝓍

155
views
Domanda del libro di testo

Use geometry and properties of integrals to evaluate the following definite integrals.

∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍 . (Hint: Write the integral as sum of two integrals.)

114
views
Domanda del libro di testo

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.


(c) Evaluate the definite integral by taking the limit as n →∞ of the Riemann sum in part (b).

77
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(f) ∫ₐᵇ (2 ƒ(𝓍) ―3g (𝓍)) d𝓍 = 2 ∫ₐᵇ ƒ(𝓍) d𝓍 + 3 ∫₆ᵃ g(𝓍) d𝓍 .

37
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(b) Given an area function A(𝓍) = ∫ₐˣ ƒ(t) dt and an antiderivative F of ƒ, it follows that A'(𝓍) = F(𝓍) .

46
views
Domanda del libro di testo

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

                                                                                                                                                                    

 ∫ (cos 7ω) /(16 + sin² 7ω) dω 

71
views