 Welcome back to our lecture series Math 12-10, Calculus I for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. Lecture 18, we're concluding our trilogy, which introduces us to the derivative. We've learned previously that the derivative is the limit of the difference quotient, and given a function, when it's differentiable, we can create a new function which we call the derivative. That derivative can measure the slope of a tangent line associated to that function. It also measures the instantaneous rate of change, which has applications to things like velocity, acceleration, and many other problems we haven't even touched yet. In this video and in the remaining part of lecture 18, I want to show you how the shape of the function affects the shape of the derivative vice versa. That is, how are their shapes related to each other? Since the derivative measures the rate of change of F, we can determine the graph of F prime from the graph of G as illustrated in the following examples. We're gonna do one of them in this example right here. So consider the function F, whose graph is given by like so. And the formula that produces this graph, who cares, don't know, I already forgot it. Deliberately giving you a function which would not be easy for the typical calculus student to look and find a formula. We're not gonna do the derivative calculation algebraically, we're gonna do it purely geometrically. So we're gonna take the graph of F and draw the graph of F prime. So we begin by recognizing where the horizontal tangent lines are. These are gonna be very important in our consideration, not just now, but in the future as well. The horizontal tangent lines, these are gonna be the tangent lines to the function which are completely flat. They have a derivative of zero. In which case on this graph, we can see three horizontal tangent lines like so. Some important significance to place upon these horizontal tangent lines. You'll notice that this tangent line corresponds to a minimum value on the graph. This one right here coincides to a maximum and this one coincides with a minimum. In fact, if I were to put a little mark right here, then our function switched from decreasing to increasing, then increasing to decreasing, sorry, excuse me, we went from decreasing to increasing like there. Then you switched from increasing to decreasing there and then you switched from decreasing to increasing here. So these horizontal tangents seem to coincide with the changes in monotonicity of the graph. That's not a coincidence. We'll come and look at that a little bit more in the future in this lecture series. So if we just consider these points right now, there's gonna be a horizontal tangent line somewhere close to one half. Again, this isn't drawn perfectly, but we're just gonna make it an estimate here. There seems to be another horizontal on around one and a half. So I'm gonna put this right here and then our last one, it seems to be on four, you know, a little bit right there, excuse me, on three, X equals three. And so we see that these horizontal tangent lines are gonna coincide with around one half, one and a half and three. Why is this significant? Well, the slope of the tangent line is the derivative, right, the derivative measures that. So when we're at X equals one half, the tangent line slope is zero. Therefore, F prime at negative, or F prime at one half should likewise be zero. That is one half is an X intercept of the derivative. Likewise, if we look at one and a half, because the tangent slope is zero, the derivative should be zero. And likewise, the derivative at three should be zero in the situation. So this is our first observation that I wanna point out to you, that the location of the horizontal tangent lines gives us the X intercepts of the derivative. Let's now focus more on this monotonicity I was mentioning earlier. So our function right here, when we're to the left of one half, we can see that the tangent line has a negative slope. We also mentioned that in this sector up to one half, the function was decreasing, right? It's only when you get past one half before you hit one and a half, then the function, if you look at the tangent line there, the tangent line actually has a positive slope and we see that the function was increasing. Then, coming over here to three, if we look at another tangent line, the tangent lines can have a negative slope again, in which case we see it's decreasing. Then once you get past three, we see that the tangent line's gonna be positive again, the function's actually increasing in that region there. So notice that if we just kind of summarize what we saw right here, our function was decreasing, then it was increasing, then it was decreasing again, and then it became increasing in the finale there. But with respect to the derivative, the derivative, when you look at the slope, the slope of that tangent line was negative. Then when you get past one half, but not quite to one and a half, the derivative was positive because the tangents are positive there. When we get between one and a half and three, the tangent lines are again decreasing. That is the tangents have negative slope. The derivative is the slope of the tangent line, so it's gonna be negative. And then finally, when you get past three, you're over here, the tangent lines are gonna be positive again. That is the slope of the tangent line, so it's gonna be positive. So we see something like this. And so there's this correspondence that whenever the function's decreasing, the derivative's gonna be negative. And whenever the function's increasing, the derivative is gonna be positive. And so we can interpret that on our picture as well. And so let me take some of these things off the screen right there to help us see it a little bit better. So what we see is the following behavior that going from this to the left right here, the function, it starts off with the tangent lines, not very steep, it getting steeper, getting steeper, getting steeper, right? It gets steeper and steeper and steeper, but when I say steeper, I mean the absolute value of the slope is bigger. It's gonna be negative. And so as we are to the left of one-half, we're gonna see a picture that looks something like this. That is it's going off towards negative infinity. Because to the left, we have a negative slope and the farther to the left of one-half we get, the more negative the slope's gonna be. Well, how about between one-half and one-and-a-half? Here, the slopes can be positive. When you're really close to one-half, it should be close to zero. When you're close to one-and-a-half, it should also be really close to zero. But when you're in the middle, that tangent slope's gonna be the steepest, about right there at x equals one. It's where the tangent slope's gonna be about the steepest. So it's giving some type of like maximum value between one-half and one-and-a-half. And so we see our function's gonna go up and come down. Again, I don't have the exact calculation here. I'm just kinda trying to eyeball it a little bit. But we wanna get an idea of what the graph of the derivative looks like just from this synopsis we're doing right now. So then as you go from one-and-a-half to three, well, when you're close to one-and-a-half, the slope is gonna be small. That is, it'll be close to zero. Somewhere in the middle, it's gonna get really steep, your tangent slope. And then it'll get back down to zero when you get close to three. So we're gonna see the same type of behavior. It's gonna be negative, so it's gonna go down. Somewhere, you know, somewhere, it's gonna turn around and come back up. We see like so. And so again, you're gonna see this local minimum somewhere in the middle between one-and-a-half and three. Once you get past three, we're gonna start off with tangent lines which are not so steep, but they're gonna get steeper and steeper and steeper the farther away from three you get. And so we see that our function's gonna be positive, but it's gonna be increasing as well. And so we get a picture that looks something like this and this is the derivative of F when you compare the two side-by-side. My hand-drawn picture's not the best. Let me show you a computer-generated picture. So using the actual formula for F, we graphed it in yellow on the left. And then if you use the formula for the derivative of F, F prime, you graph that to the right, you'll see this is here. And that's actually pretty close to what I had drawn on the screen a little bit ago. And so we can infer the basic shape of the derivative from the shape of the function F. We will see later in this lecture series that this direction is in fact reversible, that the basic shape of the derivative from it, we can infer the basic shape of F. The two are very closely related to each other.