Calculus: Domain-Restricted Function & Derivative

Calculus: Domain-Restricted Function & Deri...

10/07/2009

~ Ferddy

thanks.

• View all learner reviews »

Curve Sketching
Graphing Using the Derivative
Domain-Restricted Functions and the Derivative Page [1 of 3]
All right, so we saw in that last example that when you have those fractional exponents, things can get pretty scary. You can have these sort of cusp forms when things come together. What I wanted to point out really fast to you is that in fact sometimes you have to be really, really careful with fractional exponents when the denominator on that exponent is even. Because remember that means root. So, for example, if there were two down in the denominator; that means you're taking the square root, and you've got to remember you can't take square roots of negative numbers.
So, I wanted to illustrate this with one final graph example before moving on, and that's x times six minus x all to the one-half power--exactly the issue I was talking about. This is actually the square root of six minus x.
So, before we even attempt to graph this function, let's just think about it and make sure we understand a couple of basic rules. This cannot be negative, because I can't take the square root of a negative number. So, what does it mean for this thing to be negative? Well, that means that I have to subtract a lot away from six. In fact, anything greater than six, and I'm in negative territory. If this were seven, for example, six minus seven would be negative one. I can't take the square root of negative one.
So, this tells me right away that the graph of this function, in some sense, is going to be truncated. It's not going to exist off on the right-hand side of our graph sheet. It's going to start at six and then move this way.
So, that's going to be really important when we try to make these sign charts. We have to remember we have to ignore everything to the right of six, because it just doesn't exist. I can't plug it into the function.
Okay, well now armed with that, we can just try to use the technique we've developed in these past couple discussions. So, first off, let me actually have you attempt to do the first half of this problem. Now, taking the derivatives, again, is going to be a little bit involved. You have no choice in the matter. You must use the Product Rule.
So, you want to use the Product Rule here, and then find all the critical points, remember, where the derivative equals zero--where the derivative's undefined--but the function is defined there. And then make up a little sign table for the derivative, see how the signs go, see if it's increasing, decreasing, find out if you have a max, a min, da da da da da da. And always keep in mind the fact that you can't have x go beyond six.
Give it a shot right now--a lot of stuff--and we'll see if we can do it together.
Okay. Well, this one is a pill. Now, if you take the derivative and simplify it a bit, I think where we end up is twelve minus three x on top, and a denominator of two times six minus x to the one-half power. You might wonder where that denominator came from, in case you really aren't trying this, and that's because when you take the derivative of a square root, I get blop to the minus one-half. That minus one-half actually goes underneath.
Also, please remember that you have to use the Chain Rule here. There's a blop to the one-half you need to take care of, and then also, the derivative of the inside turns out to be a negative one. So, don't forget that negative one when using the Chain Rule.
Okay, well that seems to be the derivative, and so I want to find out when that thing equals zero, so I set it equal to zero. And the only way a fraction can be zero is if the top were to equal zero. So, I can ignore all that scary bottom, look at the top, and see that would force x to equal four. So, in fact, x equals four is a critical point.
Any other critical points? Well, let's ask are there any places where the derivative is undefined, but the function is defined? Well, the derivative is undefined whenever the bottom is zero, and the bottom is zero when x equals six. Is the function defined there? If you go back and plug in six, the answer is yes. Six minus six is zero, and I can take the square root of zero. So, in fact, I have zero times six, which is zero. So, in fact, x equals zero is another candidate. It's a place where the derivative doesn't exist.
Okay, so let's mark all this in. Oh, I'm sorry. That should be x equals six, because the derivative doesn't exist when the bottom equals zero, and that happens when x equals six. So, that's actually a six there. Caught that one myself. ________.
So, those are the critical points. How do we find out if we have max/mins? Well, let's also figure out where this function is increasing and decreasing by looking at a sign chart. Okay, well what happens to the left of four? You could pick, for example, zero. Do I plug back into here? No. This gives me where the point would be located. I want to know how the function is moving through that point, so I go to the derivative.
If I go to the derivative and plug in zero, what do I see? Well, on the bottom, I see the square root of six times two. That's positive. On the top, I see twelve minus three times six. Well, that's eighteen. Twelve minus eighteen is negative. So, I see negative and positive. This net gain is negative.
Okay. Let's see. So, I put in a zero in here. Let's make sure that's okay. I think I said something wrong. Let's try it again. Putting a zero into here. If I put a zero in here, this becomes six, which is positive, but I don't put a six in there. I am actually putting a zero in there. If I put a zero in here, I see twelve minus zero, not twelve minus eighteen, because I'm plugging in the point zero.
So, then I see twelve over a positive number, so, in fact these should be positive. So, again, we have to be really careful. It's easy to make mistakes as I'm demonstrating. Got to be really, really careful. In fact, it would probably be better if you were to have a lot of paper, as I don't. You might want to actually write these things down. You might want to write down f prime of zero and carefully work through that and see that, in fact, it's positive.
Between four and six, maybe you want to pick five. And you can try that. Plug in five in here. What do you see? On the bottom, we see six minus five, which is one. So, in fact, that's just a positive number on the bottom. And I put a five in here, I see fifteen, and now twelve minus fifteen is definitely negative. Negative over a positive is, indeed, negative.
What happens out here? Well, stop. Remember, nothing happens out here. This function doesn't exist past six. So, you might just want to block that out, because you cannot go in there. The function is not defined.
Okay, well what do we see? Well, we see that, in fact, here we must have a max because the function is increasing--positive slope--and then decreasing, and then it stops. In fact, what happens at six? If you have something decreasing, and then it stops, is it a max or a min? If it's decreasing and stops, that's a low point, isn't it? It's just hanging down there all by itself. It's low.
So, in fact, here we have a min--an end point min, in fact, if you want to be fancy. This is an end point min, and this is a max. It's called an end point min, because it's at the end of the point of the graph. Okay, great. So, now we see where it's increasing, decreasing, and we see that we have a max and then this end point min at six.
All right. What about curvature? Well, that's a whole `nother story. So, we need to take the second derivative and analyze that. And, in fact, if you want to practice doing this, here's an excellent opportunity.
Take the second derivative of this. That will require a Quotient Rule, and within the Quotient Rule, be careful of that little Chain Rule in there. See if you can take the second derivative and simplify it. Once you've got that, find out the candidates for points of inflection, and then see if you can find out the curvature on this region. Try it right now, we'll do it together in a bit.
Okay. How'd you make out? Well, let's see. I took the derivative of the derivative, giving me the second derivative, and I got three x minus twenty-four, all divided by four over six minus x. Okay, well where does the second derivative equal zero? Well, if you set the top equal to zero, we see that that--we set equal to zero, we see that x equals eight. So, that's a candidate for a point of inflection, until you remember that point is not allowed. You are not allowed to stray past six. Okay, you've got to be back by six. And so, therefore, this actually is not even a point to consider, because the function's not even defined there. The function stops right here.
Okay. Well, now what about places where the second derivative is undefined. Well, we see it's undefined when x equals six, and we've already talked about that point. Nothing happens afterwards, so there can't be a change in concavity. What is the concavity? Well, so pick a point in this region here. Let's look at what the sign of the second derivative is.
Pick, for example, zero, and what do I see? I see, well, a negative number, because the bottom is positive, and the top becomes negative. So, in fact, this is negative, and so I see that this is concave down. So, it is a sad kind of thing.
So, now we're ready to sketch the picture--actually, not too complicated. What do we see? We're going to be increasing up to four, leveling off, decreasing and stopping at six. And if you think about it, it's always concave down, so it's a very nice curve of this sort.
Thing to notice is that since the derivative is undefined at six, what's the slope of the tangent there? The slope is undefined, so it's horizontal--I mean, so it's vertical. So, in fact, we're going to have a vertical slope there. So we have to come down--we have to come down--I'm going to do this for you right now. When you come up here, come down--when we come down we have to land just like that--vertically, vertically, because the derivative is undefined.
So, you put this together. It's not too hard to see the picture. The picture actually looks like this. Notice what's going on. We're increasing up to four. This is ten, so you can see this is five and that's four. So, we're increasing up to four, and then we're decreasing, and we stop at six. And notice how the tangent line there is, indeed, if we were somehow to keep going--it's undefined. It's a very sharp drop. It's like a cliff right there.
And what do we also see? Well, we now see about the concavity, it's always concave down. There's no change in concavity, just looks like there's a very gentle thing, which all of the sudden stops and becomes very, very steep and becomes sort of arbitrarily steep right at that point.
So, that's the curve of the function--the graph of the function. You might, in fact--I guess, one thing you will need to do is define the value of the function at four to see how high you go. You can try that on your own. It's four times the square root of two. Just plug back into here and you can easily see that four times the square root of two, which is roughly five point six something, and that's why we have this go up to five point six something.
So, the moral of the story here is that when you have exponents that have rational numbers in there, you've got to be really careful, especially when the denominator is even, because parts might just be left out, and also, you might have these cusp forms.
Okay, well those are now a lot of examples of how to use this process--a very long, elaborate process, for graphing very accurate sketches of function. What I want to do for you before we leave this topic is take up the issue of--well, what if we add rational denominators in there. So, I want to take a look at rational denominators. First off, though, I'm going to take a moment to make a little comment about something called the second derivative test--something you might have heard about in class or in the book.
I'll mention that really fast, then we'll go on to looking at what happens when we put denominators down there. Gets really, really fun. See you there. Bye.