Screencast 1.4.3: Sketching a derivative graph


[MUSIC] Hello and welcome to a screen cast about sketching a derivative graph. The question that we’re looking at today is, given the graph of a function, how do we sketch the graph of its derivative? Okay, so sketch an approximate graph of the derivative given the following function, and the key word here is approximate. We’re not gonna model anything that’s gonna be exact, but we’re gonna have a good idea of what’s happening. Okay, so as we read this graph from left to right, you can tell that the graph is increasing, until we get to about this point up here. And if we were to draw a tangent line in, that would be horizontal. So we know that on the derivative that’s going to be then be a zero on our derivative graph. Okay now, our function is gonna be decreasing until we get about to I’d say about right there, and again that’s going to represent a zero on the derivative. And then we’re increasing until we get to about right here. And again, we’d have another horizontal tangent, so that’s gonna represent then a zero on this graph. Okay, so let’s go back then and look at the pieces in between. So this piece right here to the left of our first zero is, our first horizontal tangent, is going to be increasing. So we know that increasing means it’s got a positive slope. So that means that our derivative is gonna have to be something that’s positive, so that means above the axis. Now again you notice that right in here the slope is really steep, where over here it’s not quite so steep. So we know it’s going to be something that’s then gonna have to look something like this. Because here the derivative is bigger, here the derivative is smaller. Okay, now between these two points, so between these two horizontal tangents, we know that the graph is decreasing. And again, you can kind of go through and eyeball the rates that it’s decreasing at. But we know that we’re gonna have to have something that’s negative on our derivative, and we’re gonna have to connect it up with this point over here. So we could say the graph’s gonna do something kind of like that, and pretend like this actually fairly smooth in here. This got a little bit out of control, there we go that’s better. Okay, then now from this horizontal tangent to this horizontal tangent we have something that’s increasing. And again if you were to look at the the rates of increase you know that they’re gonna be changing, they’re not gonna be constant. So we’re gonna have to then connect this horizontal tangent to this horizontal tangent. So that’s gonna have to look something like this, that was a little bit more smooth. And then after this last horizontal tangent we know the graph is decreasing, so it’s gonna have to be something that’s negative. Now, how do we know where these peaks or valleys are? Honestly, at this point, we don’t. So until we know more information, this is a good approximate sketch of the derivative. Okay, looking at a graph that’s similar, but yet different enough. So again we want an approximate graph of the derivative, but you notice this graph is not so curvy, this graph is more choppy. So here, we have a graph that is increasing right away, until we hit this point up here at the top. But you notice that it’s increasing at the exact same slope. Now I don’t have any grid lines on here, but we do know that it’s gonna be flat, it’s gonna be a constant slope. So we know we’re gonna have to have a flat or constant derivative. So I’m gonna go ahead and put that, let’s just say right in here. So where this is at, like I said we don’t know what the grid lines are, so we don’t know exactly what we’re doing as far as the scale goes. So as long as you make it flat and consistent, then you know you’re correct. And if we eyeball over here to this other part, we know that it’s gonna have the exact same slope and again it’s gonna be constant. So I’m gonna fill that in here. Okay, then I could say that then the pieces on the downward parts of these slants are the same slope. But it’s exactly the negative of that slope, because these are definitely decreasing. So I’m gonna skip down here. And then we’ve got that here, and then we’ve got that here. Now what do we do with these endpoints? That’s gonna be definitely the kicker. So what is happening up here at this peak, this valley, and this peak? Now the end points we really can’t do anything about, cuz we don’t know what’s happening after those. So those we can kind of leave alone. But, at these two peaks and this valley down here, what would you say is happening with the graph? Well, the fact that it’s at a sharp corner, you guys have learned, or will be learning soon, that that means our function is undefined, or our derivative is undefined there. So I’m gonna wanna put an open circle here, and an open circle here. Open circle here, and an open circle here, open circle here, and an open circle here. Okay now, if you did not put in those open circles you would not be indicating that that derivative is undefined, and that’s a very important point. Because, how can the derivative jump from an immediate point where it’s positive to negative without having some sort of jump in the graph. So that’s why you wanna make sure you show those points where the derivative is not defined by putting in those open circles. So again, our derivative is gonna be this kind of step piece wise function here we’ve drawn in red. Thank you for watching. [MUSIC]

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