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Derivative at a Point - Example 2

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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Okay, let's bring in some devil's advocates to kind of shake things up. We're talking about these derivatives finding the slope of the tangent line. Everything's great, but what about if a function is not differential at the point? So here you have this function GFT t to the two thirds. We want to determine whether the function is differential at the point. And so remember that differential. What does that mean? It just means that f prime in this case of zero exists. Yeah, so we need to figure out whether or not f prime of zero exists as a limit. Okay, I guess I should say g here, right? Because it's actually G, not F and you'll see kind of a We go along, we'll start mixing up. The notation will use different letters for the variables. Different letters for independent variables, dependent variables, because in different contexts you use different variables. So here we're using GFT. So maybe t represents time or something like that. So for GFT, we want to know whether or not g prime of zero exists. So let's just write down what g prime of zero. It's so g prime of zero is going to be H to the two thirds. It's a really zero plus h to the two thirds minus zero all over h. And so just a reminder. This is g of a plus itch. This is G A and in this case is equal to zero. So let you verify that. And of course, very easy to forget or taking the limit is h goes to zero, and it really is important. As you're taking these limits toe leave the limit. That's just good notation. It's good practices, good mathematical practice. It will also help you get full credit on problems or partial credit on problems that maybe you don't get exactly right. It will make your greater, very happy if you use good notation. So you lead this limit until you actually evaluate the limit. So this is just the limit is h goes to zero of H to the two thirds divided by H or limit is h goes to zero. So H to the two thirds divided by H to the first is the same thing is one over age to the one third. But this is something that we've seen before. When we talked about infinite limits. So we know that the limit is h goes to zero from the right of one over age to the one third is going to be infinity because we're dividing by a small positive number. But the limit his h purchase zero from the left. Now we're going to be dividing by a small negative number, so it's going to be approaching minus infinity. So what does this mean? G is not differential At T equals zero. And why is that? Because this limit does not exist. So again, we see this connection between does Geep Ryan of Zero exists and she not being differential AT T equals zero. So both of these are, of course, equivalent ways to say the same thing. But the point is, dysfunction is not differential. And actually, if you remember our discussion on cusp sis dysfunction G has a cusp. At T equals zero. So let's see a picture. Okay, so here's a plot of the function. G F T equals T to the two thirds and we see we have a problem. This function is not smooth at this point. It has a nice sharp corner, but it's actually stronger than a corner is a cusp and it's a cuss. We actually showed that it was a custom because the limit from the left we approach zero goes to minus infinity Limit from the right goes to infinity, so it's really, really sharp. It's like if you put the ends of a piece of paper together, Andi kind of bend it inwards. It kind of goes like If here's the piece of paper and I put the two ends of the piece of paper like that, they're meeting at a really, really sharp corner. That's kind of like a cusp. If you want to kind of think about where these occur naturally and we see that if we try to find a tangent line through this point, we're gonna be out of luck, right? So we see, Really. Actually, we can almost argue that any line passing through this point could almost be a tangent line. And in fact, if you go on and learn more math, it's actually the case. It's not that a tangent line doesn't exist here. It's actually that they're infinitely many tangent lines at this point, so the tangent space really is the whole space. Instead of being kind of a one dimensional line. It's actually just the whole plane, so I mean, this is probably more than you signed up for. But the point is, this function has what's called the cusp. You have infinite limits from the left and the right, not equally. And so it's not smooth. The derivative doesn't exist. The tangent line doesn't exist or is not well defined anyway. It's not differential, etcetera, instantaneous rate of change. We don't know what's happening, right? So we don't actually know how the function is changing. At that point, it really could be doing a lot of different things. It could be just changing direction. Or it could be. We really don't know. That's the point. We don't know how the function is changing at that point at that corner, her cusp. Okay, so let's look at one more example. This time we have h of our is equal to our to the one third again R equals zero. So let's take the limit now. This is gonna be a little bit confusing, but not so confusing. His h goes to zero. Of course, this H is not the same as this H but we're not going to explicitly use h here, but we're going to plug in H to the one third minus zero over H. And again, this is a church of well, here it is going to get confusing zero plus h. And this is a job zero. And this h is actually the limit. H And in this age is, of course, the function H. But the point is, the limit is just equal to that. And this is h prime of zero. So this is really the limit is h goes to zero of h two, the one third over H and very similarly to the last example we can rewrite. This is one over H to the two thirds in this case. Okay, so what does this do? Well, what this does is because this age and the denominator is being raised to an even power as well as being cute bruited. We know that the limit is h purchase zero from the right of one over H to the two thirds is going to infinity. This is one over a small, positive number, but actually the same thing happens. Is agent purchase zero from the left because even though we're plugging in negative, small, negative numbers, they're becoming positive because we're raising them to the second power. So this is again one over a small positive number. So the limit is infinity. So, actually, in this case, now we're getting that. The slope of the tangent line is undefined, but there's a consistency from the left and the right, So the limit from the left is infinity. The limit from the right is infinity. And what we'll see when we look at the picture is that this is a vertical tangent. Okay, so here's the graph of the function. H bar is our to the one third, and here we see this vertical tangent. And so it's a little bit interesting because on one hand you can see that this function is smooth. I mean, if I trace this with my finger, I don't feel any sharp points. It is smooth, but the problem is coming from the fact that to be smooth, my slope actually has to become undefined. So my slope at X equals well are zero Sorry at Article zero is undefined. So even though the function appears smooth, the tangent line is vertical, so the slope is undefined, so the function is not differential. So the answer are original. Question. Its function is not differential at zero, so it's a little tricky. But that's that's how it goes. Undefined slope functions, not differential, even though in some sense the function is smooth. But what we'll see is what that actually means is that the derivative function is not defined, so that's why it's not differential.