Textbook Question
Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).
b) Verify that A'(𝓍) = ƒ(𝓍).
ƒ(t) = 4t + 2 , a = 0
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8. Definite Integrals
Fundamental Theorem of Calculus
4:41 minutesProblem 5.3.114
Textbook Question
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 guidance1
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.
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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.
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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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