Parameterizing implicitly defined functions
At about 152s the narrator says '21 minus -3t to the fourth' when he means 21 minus 3t^4; the next steps use the correct value.
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The video tests four possible parameterizations x(t), y(t) for the curve x^2 + 3y = 7. For each choice, x(t) and y(t) are substituted into the equation, and the choice is kept only if the result is true for every real value of t. Only choice D, x = 2t and y = (7 - 4t^2)/3, works.
You'll be able to
- substitute x(t) and y(t) into an implicit equation to test a parameterization.
- tell the difference between an equation that is true for all t and one that is true only for some values of t.
- pick the correct parameterization of a curve such as x^2 + 3y = 7.
Key words
- Parameterization
- A way to describe a curve by writing x and y as functions of a third variable, t.
- Parameter
- The extra variable (here t) that x(t) and y(t) both depend on.
- Implicitly defined curve
- A curve given by an equation in x and y, like x^2 + 3y = 7, rather than y = some formula in x.
- Identity (true for all t)
- An equation that stays true whatever value of t you put in, such as 7 = 7.
Transcript
- [Instructor] The equation x squared plus 3y equals seven defines a curve. Which of the following, and I can never say this word right, parameterizations, I think I said it right, x of t, y of t satisfies this equation for all real values of t? So pause the video, have a go at it before we do it together.
Okay, so the way I'm going to do it is for each of these choices, I'm going to substitute x of t back in for x and y of t back in for y and see if what I get is going to be true for all values of t. So let's substitute, let's do that substitution. So for x, we're going to substitute x of t in each of these scenarios.
And then for y, we are going to substitute y of t for each of these scenarios. So let's do it and let's see if I can keep my color straight the entire time as well. So here we have t squared, and actually let me just go to a neutral color here because I don't want to have to keep switching after every single little thing.
So I'm going to have t squared for x, x is just t here, and then I'm going to have plus three times all of this business, plus three times 21 minus 3t squared, plus three times 21 minus 3t squared. And I'm going to see if that is going to be equal to seven. So this is going to be the same thing as t squared and then three times 21 is going to be plus 63.
And then I'm going to have -9t squared is equal to seven. I already can tell that this is not going to work out. And let me just write it again. So this is going to be -8t squared plus 63 is equal to seven because as I go down this path, I'm going to solve for only particular ts where this is going to be true.
So it's not going to be true for all real values of t. So I'm going to rule that one out. Let me try the second one here. So for x, I replace it with t squared, but I'm going to square that. That gets us t to the fourth, and I learned my lesson, I use too much space here.
So if I have t to the fourth plus three times all of this business, and actually let me just multiply the three, times all of this business. Three times seven is going to be 21 minus -3t to the fourth. Does that equal seven for all real values of t? So let's see, if we subtract three t to the fourth from t to the fourth, we have -2t to the fourth.
We could subtract 21 from both sides, is equal to -14. So once again, I'm just going to solve for... There are some ts that would satisfy this equation, but not all real values of t, so I don't like that one either. Now, let's go to this one. All right, so if we take t squared and we were to square it, we get t to the fourth again.
And then if we multiply three times all of this business, it just cancels, that three cancels out with this three into the denominator. So we get plus seven minus 3t to the fourth is equal to seven. These cancel out. And so I get -2t to the fourth is equal to zero.
Well, that's not going to be true for all ts. In fact, that's only true for t equals zero. So I don't like that one. Let's hope that choice D actually works. So if I substitute 2t in up here, I get 4t squared. I have a good feeling about this. 4t squared, and then plus three times all of this, that's just going to be three, and then we're dividing by three, so it's just going to be plus seven minus 4t squared is equal to seven.
And this is feeling good because that cancels with that. And no matter what t you have, this is going to be true. Seven is going to be equal to seven. This is going to be true for any real value of t. So I definitely like that choice.
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