Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 5 - Integration

Problema 5.3.5

Capitolo 5, Problema 5.3.5

The linear function ƒ(𝓍) = 3 ― 𝓍 is decreasing on the interval [0, 3]. Is its area function for ƒ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain.

Guida verificata passo dopo passo

Step 1: Understand the problem. The linear function ƒ(𝓍) = 3 - 𝓍 is given, and we are tasked to determine whether its area function (integral) is increasing or decreasing on the interval [0, 3]. The area function represents the accumulated area under the curve of ƒ(𝓍) starting from the left endpoint 0.

Step 2: Recall the relationship between a function and its area function. The area function is the integral of ƒ(𝓍) with respect to 𝓍. Mathematically, the area function A(𝓍) is defined as:

A_{x} = \int_{0} f (x) d x

. The derivative of the area function, A'(𝓍), is equal to ƒ(𝓍).

Step 3: Analyze the behavior of ƒ(𝓍) on the interval [0, 3]. The function ƒ(𝓍) = 3 - 𝓍 is linear with a negative slope (-1), meaning it decreases as 𝓍 increases. Specifically, ƒ(𝓍) starts at 3 when 𝓍 = 0 and decreases to 0 when 𝓍 = 3.

Step 4: Determine the behavior of the area function A(𝓍). Since A'(𝓍) = ƒ(𝓍), the area function A(𝓍) increases wherever ƒ(𝓍) is positive. On the interval [0, 3], ƒ(𝓍) is positive (above the x-axis), so the area function A(𝓍) is increasing throughout this interval.

Step 5: Visualize the problem. Draw the graph of ƒ(𝓍) = 3 - 𝓍, which is a straight line decreasing from (0, 3) to (3, 0). Shade the area under the curve from 𝓍 = 0 to a variable endpoint 𝓍. As 𝓍 moves from 0 to 3, the shaded area grows, confirming that the area function A(𝓍) is increasing on [0, 3].

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:

Concetti chiave

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

Decreasing Functions

A function is considered decreasing on an interval if, for any two points x1 and x2 within that interval, where x1 < x2, the function value at x1 is greater than the function value at x2 (ƒ(x1) > ƒ(x2)). In this case, the linear function ƒ(𝓍) = 3 - 𝓍 decreases as x increases, indicating that as we move from 0 to 3, the output values of the function get smaller.

Video consigliato:

07:32

Determining Where a Function is Increasing & Decreasing

Area Function

The area function A(x) associated with a function ƒ(𝓍) represents the accumulated area under the curve of ƒ from a starting point (in this case, 0) to a variable endpoint x. Mathematically, it is defined as A(x) = ∫[0,x] ƒ(t) dt. The behavior of the area function depends on the values of the original function; if ƒ is decreasing, the area function will reflect this change.

Video consigliato:

05:06

Finding Area When Bounds Are Not Given

Relationship Between Function and Area Function

The relationship between a function and its area function is governed by the Fundamental Theorem of Calculus. If the original function is decreasing, the area function will increase at a decreasing rate. This means that while the area function A(x) is increasing as x moves from 0 to 3, the rate of increase diminishes because the heights of the rectangles (representing area) are getting smaller as the function value decreases.

Video consigliato:

05:23

Finding Area Between Curves on a Given Interval

Pratica correlata

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

views

Domanda del libro di testo

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁⁸ 8𝓍¹/³ d𝓍

views

Domanda del libro di testo

Average distance on a parabola What is the average distance between the parabola y = 30𝓍 (20 ― 𝓍 ) and the 𝓍-axis on the interval [0, 20] ?

123

views

Domanda del libro di testo

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Explain why your result is consistent with the figure.

∫₀¹ (𝓍² ― 2𝓍 + 3) d𝓍

132

views

Domanda del libro di testo

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₋₂⁻¹ 𝓍⁻³ d𝓍

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.

(d) If ƒ is continuous on [a,b] and ∫ₐᵇ |ƒ(𝓍)| d𝓍 = 0 , then ƒ(𝓍) = 0 on [a,b] .

views