 Hello! Welcome to the Grand Valley State University Calculus Screencasts. Today we're going to talk about how to sketch the graph of an anti-derivative. We've already made connections between a function and its derivative, particularly if a function is increasing on an interval where its derivative is positive, decreasing when the derivative is negative, functions concave up on the intervals where the derivative is increasing, in other words where the second derivative is positive, and concave down where f prime is decreasing. We have critical numbers of a function where the derivative is zero or doesn't exist. These are important numbers because these are the inputs at which our function might have extreme values. And we talked about inflection points as being points where the graph changes concavity, in other words where the second derivative changes sign. This kind of information gives us a general picture of what the graph of an anti-derivative will look like. Now in this video we're going to talk more about how we can get some specific information, how we can refine those graphs of anti-derivatives, in particular using the fundamental theorem of calculus. We're going to work through an example to try to illustrate. Here we have the graph little f of the derivative of a function big f, so big f is an anti-derivative of little f. And our goal is to draw as accurate a graph as we can of an anti-derivative. In particular we want to draw the graph of the anti-derivative big f that passes through the point zero one. Now before we get into specifics about the graph of this anti-derivative, let's think about some overall properties of this anti-derivative. For example, based on the derivative that we know, where is the graph of this anti-derivative continuous and why? Where is the graph of our anti-derivative smooth and why? Pause the video for a moment, think about these two questions, and then resume the video when you're ready. Now notice that big f has a derivative. Its derivative is this function little f, and the derivative is defined on this entire interval. So our function big f, our anti-derivative, is a differentiable function. And remember a differentiable function is continuous, so our big f has to be continuous on this interval. Differentiability is a stronger condition than continuity. In fact, differentiability means continuous and smooth. So our function, our anti-derivative big f, will have to be a smooth curve on this interval. We want to keep those pieces of information in mind, that global behavior, as we draw the graph. Now we're going to determine some more specific information about this anti-derivative. In particular, let's think about the intervals on which our anti-derivative is increasing and decreasing. So again, pause the video for a moment, look at the graph of the derivative, and think about how would you identify the intervals where the graph of f, this anti-derivative, is increasing and where it's decreasing. Then resume the video when you're ready. Recall that a function is increasing on the intervals where its derivative is positive. In this example, the derivative, this graph of little f, is positive on the intervals from 0 to 1 and from 3 to 5. And notice we don't include 1 and 3 because that's where the derivative is 0. That function is not increasing at those points. And note that f is decreasing big f on the intervals where the derivative is negative and that would be on the interval from 1 to 3. Next, let's think about the concavity of our anti-derivative big f. Again, pause the video for a moment and think about the intervals on which f, big f is concave up and why is big f concave up on those intervals? So recall that big f is concave up when the first derivative, in this case, the graph of little f is increasing and this occurs on the interval from 2 to 4. So big f is concave down when the derivative of little f is decreasing and in this example, that's on the intervals from 0 to 2 and from 4 to 5. So knowing where the graph of big f is increasing and decreasing, concave up, concave down, helps us see the general shape of the graph of this anti-derive. However, it doesn't help us find any specific points that lie on the graph of f. For that, use the fundamental theorem of calculus and this is where that original assumption we made about big f passing to the point 0, 1 will come into play. For example, if we want to find big f of 1, we know big f of 0 and we could relate big f of 1 and big f of 0 using the fundamental theorem. The definite integral of little f of x dx from 0 to 1 is big f of 1 minus big f of 0 and this works because big f is an anti-derivative of little f. So we can solve this equation for big f of 1 and see that big f of 1 is big f of 0 plus the integral from 0 to 1 of little f of x dx. Now we can evaluate this definite integral from 0 to 1 of little f of x dx by interpreting this integral as a net signed area and notice that on the interval from 0 to 1, the graph of little f is above the x-axis so this integral from 0 to 1 of little f of x dx is going to give us an area. In fact, the area of this triangle, this green shaded triangle with height 1 and base 1. So the integral integral from 0 to 1 of little f of x dx is a half. That means that big f of 1 is big f of 0 which we're assuming to be 1 plus this integral which we found to be a half so big f of 1 is one and a half. Now take a minute and determine the value of big f of 2. Use the same idea we just talked about and then resume the video when you're ready. To find big f of 2, we use exactly the same idea. Big f of 2 minus big f of 1 and remember we just calculated big f of 1. Big f of 2 minus big f of 1 is the definite integral of little f of x dx from 1 to 2. Solving for big f of 2, we get big f of 2 is big f of 1 plus this definite integral from 1 to 2 of little f of x dx. And again we can use the net signed area to determine the value of this definite integral. And in this case, the graph of f is below the x-axis and so the definite integral from 1 to 2 of little f of x dx is going to be negative. And it's going to be the opposite of the area of this yellow shaded triangle of height 1 and base 1. So the integral from 1 to 2 of little f of x dx is negative one half. And we add that to the value of big f of 1 that we found previously and so big f of 2 is one and a half minus a half or one. Now continue to calculate the values of big f at 3, 4 and 5. Work on these on your own, pause the video and then resume when you're ready. So we continue to calculate these values the same way we did before using the force to find big f of 3. We use the fact that big f of 3 minus big f of 2 is the definite integral from 2 to 3 of little f of x dx. And then again we interpret this integral as a net signed area and in this case again the graph of f little f is below the x axis so the integral from 2 to 3 of little f of x dx is the opposite of the area of that yellow triangle which is a half. So big f of 3 is big f of 2 that we calculated earlier minus a half so we get 1 minus a half or a half. Similarly big f of 4 can be found by integrating little f of x from 3 to 4 and then that's equal to big f of 4 minus big f of 3 and we use the net signed area interpretation to calculate the integral from 3 to 4 of little f of x dx. It's the area of that green triangle which is a half. So big f of 4 is big f of 3 which we found to be a half plus this integral which is a half so we get big f of 4 to be 1. Do the same thing to find big f of 5. Big f of 5 minus big f of 4 is the integral of little f of x dx from 4 to 5. The graph of little f bounds this region above the x axis this green shaded region. Now we're assuming that this curved arc that's part of the graph of little f is circular so the area, this green shaded region is going to be one quarter of the area of a circle with radius of 1. So the integral from 4 to 5 little f of x dx is going to be pi over 4. Add that to big f of 4 which was 1 and we get big f of 5 is 1 plus pi over 4. So now we have these different points that we can plot and we can see the graph of this antiderivative big f taking shape and all we have to do now is fill in regions in between these points to get a really accurate graph of big f. So for example use the information we know about big f, remember earlier we determined where big f is increasing, decreasing concave up and concave down to fill in the graph of big f from 0 to 1. Pause the video to do that and then resume when you're ready. On the integral from 0 to 1 we determined that the graph of big f is increasing and concave down so it's going to look like this little green arc that we've drawn on the graph and connecting the two points between 1 and 2. Now why don't you pause the video and then draw in the rest of the graph of big f as best you can using f increasing, f decreasing, concave up, concave down to fill in the video when you're ready. On the interval from 1 to 2 we determined that the graph of big f is decreasing and concave down so it looks like that green arc that we've drawn here connecting the points from 1 to 2. And note that little f of 1 is 0 which means that the point 1 and 1.5 is a critical point of big f and the fact that big f is increasing to the left of 1 and then decreasing to the right makes f of 1 or 1.5 a relative maximum value Now in the interval from 2 to 3 we showed that the graph of big f is decreasing and concave up so it looks like this green arc that we've drawn and notice that big f changes from concave down to concave up when x is 2 so that means there's an inflection point for big f at x equals 2. On the interval from 3 to 4 big f is concave up and increasing so it looks like the green arc that we've drawn here and again we see that little f of 3 is 0 so big f has a critical point at 3 and because big f is decreasing and increasing at this point then we have a relative minimum value of big f at x equals 3. Finally, on the interval from 4 to 5 we have that the graph of big f is increasing and concave down as shown in the green arc here and then we also have an inflection point at x equals 4. So we've drawn based on the global information we were determined about big f and the fundamental theorem of calculus a pretty accurate graph of our anti-derivative big f that passes through the point 0, 1. Now this graph that we drew of big f was determined by that initial condition that we set that big f of 0 should be 1. Suppose we had a different condition that we set. Suppose we wanted an anti-derivative of little f that went through the point 0, negative 1. We'll call that anti-derivative big g. How was that from big f? So pause the video for a moment and describe as best you can what this anti-derivative big g has to look like and how it's related to big f and then resume the video when you're ready. Now big f and big g both have the same derivative of little f which means that they have the same slopes at the same points. In other words, the graphs of big f and big g have to look the same. They have to have the exact same shape. So all big g is is a vertical translation of big f and we're just translating it down to units so that we go through the point or anti-derivative goes to the point 0, negative 1. And in fact, any other anti-derivative of little f is going to be a vertical translation of one of these graphs. So that can be a good question cast. In the future, we'll talk a little bit more about how we can even do a better job of creating anti-derivatives of our functions.