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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.3.5
Chapter 5, Problem 5.3.5

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

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Step 1: Understand the problem. The linear function Ζ’(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 Ζ’(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 Ζ’(𝓍) with respect to 𝓍. Mathematically, the area function A(𝓍) is defined as: Ax=∫0f(x)dx. The derivative of the area function, A'(𝓍), is equal to Ζ’(𝓍).
Step 3: Analyze the behavior of Ζ’(𝓍) on the interval [0, 3]. The function Ζ’(𝓍) = 3 - 𝓍 is linear with a negative slope (-1), meaning it decreases as 𝓍 increases. Specifically, Ζ’(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) = Ζ’(𝓍), the area function A(𝓍) increases wherever Ζ’(𝓍) is positive. On the interval [0, 3], Ζ’(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 Ζ’(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].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 (Ζ’(x1) > Ζ’(x2)). In this case, the linear function Ζ’(𝓍) = 3 - 𝓍 decreases as x increases, indicating that as we move from 0 to 3, the output values of the function get smaller.
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Area Function

The area function A(x) associated with a function Ζ’(𝓍) represents the accumulated area under the curve of Ζ’ from a starting point (in this case, 0) to a variable endpoint x. Mathematically, it is defined as A(x) = ∫[0,x] Ζ’(t) dt. The behavior of the area function depends on the values of the original function; if Ζ’ is decreasing, the area function will reflect this change.
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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.
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Related Practice
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