Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.3.114

Chapter 5, Problem 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.

Verified step by step guidance

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.

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Key Concepts

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

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.

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.

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.

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Related Practice

Textbook Question

Areas of regions Find the area of the following regions.

The region bounded by the graph of ƒ(𝓍) = x /√(𝓍² ―9) and the 𝓍-axis between and 𝓍 = 4 and 𝓍= 5

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Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.

∫ 2𝓍(𝓍² ― 1)⁹⁹ d𝓍

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Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.

∫ [ 1/(10𝓍―3) d𝓍

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Textbook Question

Suppose an object moves along a line at 15 m/s, for 0 ≤ t < 2 and at 25 m/s, for 2 ≤ t ≤ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≤ t ≤ 5.

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Textbook Question

Explain why ∫ₐᵇ ƒ ′(𝓍) d𝓍 = ƒ(b) ― ƒ(a)

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Textbook Question

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.

∫ₐᶜ ƒ(𝓍) d𝓍

102

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