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 5.3.114
Capitolo 5, Problema 5.3.114

Max/min of area functions Suppose ƒ is continuous on [0 ,∞) and A(𝓍) is the net area of the region bounded by the graph of ƒ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of ƒ. Verify this fact with the function ƒ(𝓍) = 𝓍² - 10𝓍.

Guida verificata passo dopo passo
1
Step 1: Understand the problem. The function A(x) represents the net area under the curve of f(t) from t = 0 to t = x. We are tasked with showing that the local maxima and minima of A(x) occur at the zeros of f(x). Additionally, we need to verify this fact using the specific function f(x) = x² - 10x.
Step 2: Recall the relationship between A(x) and f(x). The derivative of A(x) with respect to x is given by A'(x) = f(x). This is because the Fundamental Theorem of Calculus states that the derivative of the integral of a function is the function itself.
Step 3: Identify critical points of A(x). Critical points occur where A'(x) = 0. Since A'(x) = f(x), the critical points of A(x) are the zeros of f(x). Therefore, the local maxima and minima of A(x) occur at the zeros of f(x).
Step 4: Verify this fact using the given function f(x) = x² - 10x. Find the zeros of f(x) by solving f(x) = 0. This involves factoring the quadratic equation: x² - 10x = 0, which can be factored as x(x - 10) = 0. Thus, the zeros are x = 0 and x = 10.
Step 5: Analyze the behavior of A(x) at the zeros of f(x). Since A'(x) = f(x), the sign of f(x) determines whether A(x) is increasing or decreasing. Evaluate the intervals around the zeros (x = 0 and x = 10) to determine whether these points correspond to local maxima or minima. This involves checking the sign of f(x) in the intervals (0, 10) and beyond.

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:
4m

Concetti chiave

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

Net Area Function

The net area function A(x) represents the accumulated area between the graph of a function f and the t-axis from 0 to x. It is defined as the integral of f from 0 to x, capturing both positive and negative areas. Understanding this concept is crucial for analyzing how changes in x affect the total area, particularly in relation to local maxima and minima.
Video consigliato:
05:06
Finding Area When Bounds Are Not Given

Critical Points

Critical points occur where the derivative of a function is zero or undefined. For the net area function A(x), critical points correspond to the values of x where the rate of change of area is zero, indicating potential local maxima or minima. Identifying these points is essential for determining where the area function reaches its highest or lowest values.
Video consigliato:
04:50
Critical Points

Zeros of a Function

The zeros of a function f are the values of x for which f(x) = 0. In the context of the problem, the local maxima and minima of the area function A(x) occur at these zeros because they represent points where the graph of f intersects the t-axis, leading to changes in the accumulation of area. This relationship is fundamental in verifying the behavior of A(x) with respect to f.
Video consigliato:
6:37
Zero and Negative Rules
Pratica correlata
Domanda del libro di testo

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The right Riemann sum for ƒ(𝓍)) = x + 1 on [0, 4] with n = 50.

95
views
Domanda del libro di testo

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      

                                                                                                                                                                                       

 ∫₀⁴ (8―2𝓍) d𝓍

86
views
Domanda del libro di testo

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.



∫₀ᵃ ƒ(𝓍) d𝓍

57
views
Domanda del libro di testo

Explain the statement that a continuous function on an interval [a,b] equals its average value at some point on (a,b).

129
views
Domanda del libro di testo

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 1/(𝓍² + 1) on [―1, 1]

83
views
Domanda del libro di testo

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

                                                                                                                                                                    

 ∫₀^π/⁴ cos² 8θ dθ

57
views