/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Consider the following functions... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 16

Consider the following functions \(f\) and real numbers a (see figure). a. Find and graph the area function \(A(x)=\int_{a}^{x} f(t) d t\) for \(f\) b. Verify that \(A^{\prime}(x)=f(x)\) $$f(t)=2, a=-3$$

Short Answer

Expert verified
The area function is \(A(x) = 2x - 6\) and its derivative, \(A^{\prime}(x) = 2\), which is equal to the original function \(f(x) = 2\).

Step by step solution

01

Determine the Antiderivative of \(f(t)\) and Find \(A(x)\)

The function \(f(t)\) is given as \(f(t)=2\). The antiderivative of a constant function, such as \(2\), is a linear function. In this case, the antiderivative of \(f(t)\) is \(F(t) = 2t + C\) where \(C\) is a constant. Now, we evaluate the area function \(A(x)\) using the given integral: \(A(x) = \int_{a}^{x} f(t) dt = \int_{-3}^{x} 2 dt\). We substitute the antiderivative of \(f(t)\), \(F(t)\), into the integral and evaluate it with the given limits of integration. \(A(x) = \bigg[2t + C\bigg]_{-3}^{x} = (2x + C) - (2(-3) + C) = 2x - 6 + C\). Since the constant \(C\) cancels out, we end up with: \(A(x) = 2x - 6\).
02

Graph the Area Function \(A(x)\)

The area function \(A(x)\) is a linear function in the form of \(A(x) = 2x - 6\). Let's graph it: 1. Create a coordinate plane. 2. Identify the y-intercept, which occurs when \(x=0\). In this case, \(A(0) = 2(0) - 6 = -6\). Plot the point \((0, -6)\) on the y-axis. 3. Determine the slope of the line, which is \(2\). From the y-intercept, move up and down 2 units and left and right 1 unit in each direction, following the slope, and draw the line through the points. The graph represents the area function \(A(x) = 2x - 6\).
03

Calculate \(A^{\prime}(x)\)

To find the derivative of the area function, we differentiate \(A(x) = 2x - 6\) with respect to \(x\). \(A^{\prime}(x) = \frac{d}{dx}(2x - 6) = 2\)
04

Verify that \(A^{\prime}(x) = f(x)\)

The calculated derivative of the area function is \(A^{\prime}(x) = 2\). The original function \(f(t) = f(x)\) is also given as \(f(x) = 2\). Since \(A^{\prime}(x) = f(x)\), we have verified that the result is indeed correct. In summary, the area function \(A(x) = 2x - 6\) and its graph is a linear function. Additionally, the derivative of the area function \(A^{\prime}(x) = f(x) = 2\).

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