Calculus: Cusp Points and the Derivative

Calculus: Cusp Points and the Derivative

10/07/2009

~ Ferddy

This was hard, I wished that the professor would have shown us how the first derivative was found.

• View all learner reviews »

Curve Sketching
Graphing Using the Derivative
Cusp Points and the Derivative Page [1 of 3]
All right. Well now, I thought it would be fun to actually try one of the exotic examples where we have a fraction in the exponent. These are always interesting and tricky because more often than not there are going to be places where the derivative doesn't exist but the function does. So, always remember a little rule of thumb. If you see an exponent, like x to the two-thirds power, think to yourself, "Uh oh, I better be careful because there may places where the derivative doesn't exist but the function does." Those would give you those unusual critical points. So, be on the lookout.
Okay, so again, the meta question is to graph this function, that's the meta question. The many, many multi-part question is find all critical points, find where the function is increasing or decreasing. Figure out if the critical points are max/min, then study the concavity. Find the points of inflection. Tote that barge, lift that bale and graph the function.
Okay, so let's try this. So the first thing you need to do is take the derivative of this. Okay, so to take the derivative of this to see where the critical points are what you have to use here is the Product Rule. Now, of course, the other possibility is to distribute and then just take the derivative. Anyway, you can try to take the derivative anyway you wish and this is sort of like a cooking show, where I've worked this out in advance.
You might want to try to do this on your own. I think it's a great example. But I think you will see that the derivative is eight x squared minus five x minus twelve, all divided by three x to the one third. Now if you're just watching sort of passively, this is not at all obvious how I got here. You've got to sit down and actually work it out and make sure that you, in fact, can verify that represents the derivative.
Okay, well armed with that derivative, I want to find a critical point. So, in fact, here's an opportunity for you, what I would like for you to do is do the first half of this problem. I want you to find all of the critical points. Then I want you to find out where the function's increasing and where the function's decreasing and then use that chart to figure out if the critical points you find were max or min. Here's an opportunity for you to really try this and then we do it together so you can see and get feedback. So really give this a shot and I'll be back in a second.
Okay, a lot of work. Let's see how we made out. The first thing you have to do is set the derivative equal to zero. So the first thing we do is set the derivative equal to zero and solve. Now, if you remember the only way a fraction can equal zero is if the top equals zero. So, in fact, that scary bottom we can now ignore and just look at when the top equals zero. And so we have to solve that quadratic. You probably tried and hopefully failed in trying to factor that correctly because you can't factor it. You have to use the quadratic formula. Should that bother you? Did you think that maybe there was a mistake somewhere? No, these problems actually get a little more tricky and you have to be willing to use the quadratic formula. Not a big deal.
So if you used the quadratic formula - remember that the quadratic formula says that x equals negative b plus or minus the square root of b squared minus four ac minus all over two a. If you insert in this eight for a, minus five for b, and minus twelve for c, what you see after you simplify a little bit is the following. X equals five plus or minus the square root of four o eight, all divided by sixteen. So we see two answers. We see two places where the derivative equals zero. One of them is five plus the square root of four o eight over sixteen and the other one is five minus the square root of four o eight over sixteen. And what are those two numbers? Well numerically, I actually calculated them for you, so one of them is one point five seven stuff. That's with the plus sign, and with the minus sign, if you compute that number, I think what you will see if minus point nine five one stuff. So that at least gives you a sense where these things are on the number line.
Okay, are there any other candidates we have to look for, for critical points? Yes, we have to ask are there any places where the derivative is undefined but the function, in fact, is defined. And if you look back here, the only time the derivative is undefined is if we have a zero in the denominator. So when do we get a zero in the denominator? Well, only if x were to equal zero. Is x equal zero allowed in the original function? You may say no, but the answer actually is yes, it is allowed. Because this is not so scary. This is just zero to the two third power, which is zero. And here I can plug in zero so, in fact, I see that f of zero is zero. So there's a dot there. The function's defined and yet I see the derivative is undefined. So that's a potential place where we might have to waves coming together and get a cusp formed. So, in fact, there's another place - f prime does not exist at x equals to zero. So I actually see three critical points. I hope that's what you found. Two of them where the derivatives equal zero, and the one where the derivative wasn't defined but the function was defined.
Okay, now armed with that we can look at a little sign chart. Notice how this process is really so multi-step. There's many, many steps to graphing a function. And I insert all these numbers here, just the ones that we need here. So, maybe put one here at minus point nine five one and then I'll place the zero, and then I'll place the one point five seven. You know that I'm not drawing this at all to scale, but I don't care. All that I care about is seeing how it partitions the x-axis so I can see what the sign of the derivative is doing.
Now if you pick a point here to the left, like say minus one, which you'll notice is to the left of minus point nine, and if we insert that back into the derivative, you can figure out exactly what the sign of this is. Is it positive or is it negative? Well, try it and I think you'll see that, in fact, we have negative values there. Between minus point nine five and zero you can pick like minus a half. Take minus a half and evaluate the derivative there and see what kind of sign you get and you'll see positive numbers. Between zero and one point five, you can actually pick one. If you put one in here - let's do that one together. If we put one in here, what's the sign of this? Well, the bottom is easy - the cube root of one is just one. This is three, the bottom is three, but the important thing is that the bottom is positive. And what about the numerator? Well, if I put in a one here that gives me an eight minus five minus twelve. Well, that's putting a negative on top, positive on the bottom, so this whole thing is negative. And you could do that to the other one. You could pick a very large number and plug it in and verify that in this last region, you get positive answers.
So, what do you conclude? Well, let me first of all put in that here we saw the derivative is zero. Here we saw that the derivative, in fact, didn't exist, so undefined. And here we see that the derivative is zero. So what does that mean? Well, that means that the function must be falling and then rising and then falling and then rising. So now, I've found all the regions where the function is increasing and decreasing. It's decreasing from negative infinity all the way up to minus point nine. Then it's decreasing again from zero to one point five seven and the function is increasing in this interval from minus point nine up to zero and then increasing again from one point five seven out to infinity. So now we see where it's increasing and decreasing and for free, we find out that this must be - well, it's going down then up, this must be a min. And this point where the derivative was undefined, a potential one like this, turns out to be just that - there's a little cusp there. It comes up just like that. This is a max and over here, we see another min. So there's two mins and one max. So all of these critical points turn out to be max and mins. Great! Okay. Well, all that work just tells us how the function is rising and falling and max/mins. What about curvature?
Okay. Well, now it gets even more involved because we have a second derivative. So I have to take the derivative of the derivative to study the curvature. So now here's the second challenge for you, I want you to go off now and really attempt to find the curvature issues. So what do you have to do? First of all, you have to very carefully take the derivative this. You could use the Quotient Rule, very carefully simplify and write down what the second derivative is. Then, set that second derivative equal to zero to find out the critical points. The critical points for this, which would be potential points of inflection. Also, don't forget to look at places where the second derivative doesn't exist. Okay, and once you have that, make a little sign chart like this. Put down those points and see if you can find out where this curve is concave up, concave down and see if you have any points of inflection. A lot of work ahead, by the way. This actually takes a long time. It took me a while to figure this all out. Give it a shot though. It's great practice, a great chance to get some feedback. Try it right now.
All right. Well, I'm not going to take the derivative of this for you right now live, but I worked it out in advance and I will try to write it down for you. It is very long. Now, I simplified this a little bit so it might not be exactly what you got, but I got three x to the one third power, all multiplied by sixteen x minus five, minus eight x squared minus five x minus twelve, all times x to the minus two thirds power, all divided by nine x to the two thirds.
Now, in fact, you can see this a little bit, let me just talk through this very fast. I just used the Quotient Rule. So the bottom times the derivative of the top minus the top times the derivative of the bottom, all over the bottom squared. That's a really messy looking thing. That negative exponent is crying out to be put underneath here as a fraction. So you have to combine these things - a lot of algebraic gymnastics and I hope that you tried.
Anyway, when you take this object now and you set it equal to zero and solve, well what do you find? Well, in fact, you find that there are no solutions to this. So if you set this equal and solve it, there are no solutions, no solutions. So this never equals zero, it never equals zero.
Are there any places where this is undefined? Well, it's undefined when the bottom is zero. So it's undefined when x equals zero. A candidate for a point of inflection? Well, potentially, yes. So we need to make a sign chart for the second derivative and put in the candidate zero, where it's undefined, and look to the right and to the left. Plug some points in for here. That alone is a big task, just plugging in a point here. But, I'll let you try it and see you get positive here and positive here.
So, what do you see? Well, you see that, in fact, this thing is concave up before zero and concave up after zero. So, in fact, this is not a point of inflection. The concavity does not change here.
Okay. Well now, how do we put all of this information together to sketch an actual picture? Well we need to know the value of the function at all of the important places, which is going to be f of this number, one point five seven, f of minus point nine five one and then f of zero. Because we have to know where to plot those points, those points are relevant.
So, if we actually do that, what do we see? Well, we see that f evaluated at one point five seven is equal to approximately minus six point eight nine six. Where am I getting that number? I'm plugging back into the original function. There are three functions rolling around here and these all can easily get confused in your mind. There's the original function, what does that do? That spits out the location of the points - how high to go, how low to go, put a point there.
Then you've got the derivative. What does that do? That tells you how the function is moving through that point. Is it increasing, is it decreasing? How fast is it moving through the point? And then you've got the second derivative and what that tells you is what's the curvature at that point? Is it curving up, is it curving down? How is it curving? You have to make sure when you're plugging in to get where the point is located you go back to the point function, which is f of x.
Now if we take a look at f of zero, that's pretty easy and you can see that is just zero. And the other point that we need is f of minus point nine five one dot, dot, dot, dot, dot, dot. And that equals minus four point o o eight and then it goes on. Okay, taking this information and those tables over that, what I want us to do is try to sketch an accurate picture of this function.
Okay. So what do we have going on here? Well, first of all let's just plot the point. At zero we're zero - so that's an easy point just to plot in. At one point five seven - let me mark that down here. At one point five seven, let's put that here - one point five seven something. And this point we'll need here too, I'll put that over here, minus point nine five one. What do I see? Well, at one point five seven, I'm at minus six point eight. So it comes way down here, minus six point eight. And at minus point nine five, I see that my value is minus a four point 0 0 eight, so basically minus four. This is minus six, so that's probably over here, a little bit higher. And those are the points that are going to completely classify what this function does.
Now look at the chart and what do you see? Well, what you see is that the function is first falling. So we're going to go down to here, then we're going to go up to here then we go down to here and up. So we're going to make sort of a w. We're going to go down, up, down, up. That's the activity of the increasing/decreasing issue, but what about curvature? Well, we saw that, in fact, the curvature is always concave up. So we're always like this. So how do you draw that? Well, remember that point is not going to be a nice level smooth point. That's a cusp point. You can really start to see how that cusp point's going to fit in and what we get is something that looks like this. It comes down; it comes up to this cusp point. Now we start to decrease down to here. It levels off and now we're increasing. So we're forming a w, we're decreasing concave up, increasing concave up, max. Now we're decreasing, concave up, and now we're increasing, concave up. Now I've got a real pretty picture for you; you can see exactly what it looks like, a real pretty version of it. You can see it's exactly what we've said -- right here at minus point nine five, we've got that min. We've got that sharp cusp point at the origin and over here at around one point five seven, we've got the other min. Notice it's decreasing, increasing, decreasing, increasing, and it's always concave up. Remember, if you have rational numbers in the exponent you might not be surprised if you have those sharp cusp points. Okay, we'll take a look at another one up next, see you there.