Working with area functions Consider the func...

Question

Working with area functions Consider the function ƒ and its graph.

(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.

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Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says consider the graph of F of X, and below this, we're given a graph that has our function F of X plotted on it, and F of X is a sinusoidal curve. That has minimal located at X equal to 0 and X equal to 6. It has maxima located at X equal to 3 and X equal to 9, and it crosses the x axis at X equal to 1.5, 4.5, and 7.5. We're asked to draw the graph of the area function A of X, which is equal to the integral from 0 to X of F of TDT, or 0 is less than or equal to X, which is less than or equal to 9, without a scale on the y axis. And below this, we're given an empty graph on which to plot our function. So the first thing that we're going to do is determine the points where A of X crosses the X axis, and we're going to start by looking at the lower end of our interval, which is going to be X equal to 0. So when X is equal to 0, a of 0 is going to be equal to the interval from 0 to 0 of FFT DT. And recall that anytime that you have the lower limit and upper limit of integration equal to each other, that that integral is always going to be equal to 0. That means that A of 0 is going to be equal to 0. So we're going to plot our first point on our graph at the origin. Now, to determine the other points where A of X crosses the X-axis, we're going to need to use our graph F of X. Now if we look at our function F of X, we noticed that the area underneath the curve between X equal to 0 and X equal to 1.5 is equal to the area underneath the curve between X equal to 1.5 and X equal to 3. However, the area underneath the curve between X equal to 0 and X equal to 1.5 is actually below the x axis, and the area underneath the curve between X equal to 1.5 and X equal to 3 is above the x-axis. And so recall that if the area is above the x-axis, that means that area is going to be positive, and the areas below the x-axis, it's going to be negative. So since these two areas are equal, but one is above the x-axis and the other is below, that means that these two areas will cancel out. That means our area function when X is equal to 3, is going to be equal to 0. So we'll place another point at X equal to 3. On the X axis. Now, if we look at the area on our graph underneath F of X, we see that the area underneath the curve between X equal to 3 and X equal to 4.5 is the same as the area between X equal to 4.5 and X equal to 6. And again here, we have one area that is above the x-axis and another that is below. So the area between X equal to 3 and X equal to 4.5 is above the x-axis, and the area between X equal to 4.5 and X equal to 6 is below the x-axis. So those two areas again are going to cancel out, so we'll have another value of 04 area at X equal to 6. We'll place that point on the X-axis. And finally, if we look at the areas between X equal to 6 and X equal to 7.5, that that area is going to be equal to the area between X equal to 7.5 and X equal to 9. And again, we have one area that's below the X-axis and one that is above, so the area between X equals to 6. To X equal to 7.5, that that area is below the X axis, and the area. Between 7.5 and 9. is above the x axis. So again, those areas will cancel out, so we'll have another 0.4 A of X at X equal to 9. So we'll place that point on the X axis at X equal to 9. And so now that we have the 0 points, we need to look for our extrema. Now, since A of X is equal to the integral from 0 to X of F of TDT, then by the fundamental theorem of calculus that are critical points for AX are going to be equal to the zero points of F of T. So, if we look at our graph of F of X and identify the points where F of X is equal to zero, we see that occurs when X is equal to 1.5, X equal to 4.5, and X equal to 7.5. So, those are going to be the X values for our critical points. So now we just need to determine whether we have a local minima or a local maxima at those points. So, we're going to begin by looking at X equal to 1.5. Now, if we look at X equal to 1.5, and we look at the area to the just to the left of that, we see that that area underneath the curve is below the X axis. So that means that A of X is actually going to be negative when it's reaching 1.5. So that means that A of X is actually going to be a negative value at X equal to 1.5. So, we will know that that will be a minimum. And so we'll just arbitrarily set a. 6 units below the X axis at X equal to 1.5. Since we have no scale on the Y axis, we're free to choose where we place our minima and maxima. So here we've just arbitrarily chosen 6 units. So next, we'll look at our next critical point, which is X is equal to 4.5. Now, recall that our total area. I 0 at X equal to 3. So if we look at the area just to the left of X equal to 4.5, the area underneath F of X on a graph here is actually above the X axis. So that means that A of X is going to be positive when X is equal to 4.5. So that means that we're going to have a local maximum at this point. So, we'll place a. 6 units above the X-axis. Above X equal to 4.5 on our graph. And then the final critical point that we want to look at is when X is equal to 7.5, and recall that the total area for A of X is equal to 0 and X equal to 6. So if you look at the area just to the left of X equals to 5, the area underneath the curve of F of X is below the X-axis, so it's going to be negative. So that means that A of X is negative. That means that we're going to have a minimum value at X equal to 7.5. So we'll place a 0.6 units below the x-axis at X equals to 7.5. And so now that we have all these points identified, we just need to connect them with a sinusoidal curve. And in doing so, we will have the answer for a graph. So we'll make sure that we connect all of our points. And making sure that we have the minima where we have indicated the minima and the maxima where we have indicated the maxima. And once we've drawn off. Of the lines here. The graph that we have shown on the screen here is going to be the answer to this problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

Working with area functions Consider the function ƒ and its graph.
(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.

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Working with area functions Consider the function ƒ and its graph.(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.

Watch next

Working with area functions Consider the function ƒ and its graph.
(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.