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Chapter 5: Problem 13

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 \(A^{\prime}(x)=f(x)\) $$f(t)=5, a=0$$

Short Answer

Expert verified
Question: Find the area function A(x) for the given function f(t) = 5, and verify that its derivative is equal to f(x). Answer: The area function A(x) is given by A(x) = 5x, and its derivative A'(x) = 5 is equal to the given function f(x).

Step by step solution

01

Finding the area function A(x)

First, we need to find the integral of f(t) with respect to t from a = 0 to x as follows: $$A(x) = \int_{0}^{x} f(t) dt = \int_{0}^{x} 5 dt$$
02

Compute the integral

Now, integrate the function with respect to t: $$A(x) = 5 \int_{0}^{x} dt = 5 \cdot [t]_0^x = 5(t | _0^x) = 5(x - 0)$$ So, the area function A(x) = 5x.
03

Graph the area function A(x)

To graph the A(x) function, we know already that it is a straight line function (y = mx format) with slope equal to 5. This means that for every increase in x by 1, y increases by 5. The line will pass through the points (0,0), (1,5), (2,10), (3,15), and so on. Draw the line connecting these points with a slope of 5, and this will be the graph of the area function A(x) = 5x.
04

Find the derivative of the area function A(x)

Now, we need to find the derivative A'(x) and confirm if it is equal to the given function f(x). $$A'(x) = \frac{d}{dx}(5x)$$
05

Compute the derivative

Differentiate with respect to x: $$A'(x) = 5$$ Now, we observe that the derivative A'(x) is equal to the constant 5. The given function f(x) = 5 is also a constant value of 5. Therefore, we can verify that A'(x) = f(x).

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