Consider the function z=x / y^2 a. Compute zx and zy b....

Key Concepts

Directional Movement Along Coordinate Lines

Analyzing the function’s behavior when moving along horizontal or vertical lines (i.e., when one of the variables is held constant) is a practical application of partial derivatives. This examination reinforces the interpretation of partial derivatives, as moving along a line where one variable is constant will change the function's output according to the partial derivative with respect to the other variable.

Level Curves

Level curves are the sets of points in the domain at which the function takes on a constant value. In the context of multivariable functions, these curves help visualize the function's behavior by showing where the output remains unchanged, providing insight into its topography and distribution.

Partial Derivatives

Partial derivatives measure the rate at which a multivariable function changes with respect to one of its variables, while keeping the other variables fixed. They are computed by differentiating with respect to one variable at a time, offering a window into the local behavior of the function along the coordinate directions.

Transcript

00:01 Okay, so we have this function, this function z, or this curve z, which is equal to x over y squared.

00:11 First thing we want to do, we want to find the partial derivatives with respect to y and with respect to x.

00:20 Excuse me, we're going to start with the derivative with respect to x.

00:24 So see, x, which is the same thing as the partial derivative respect of x of x over y square.

00:39 And y square, 1 over y square is simply a constant in that case.

00:44 So we have d or d.

00:47 So this term here has nothing to do with x.

00:54 It's simply a multiplies x.

00:57 So we can write 1 over y square times the derivative of x with respect to x, which is simply 1, and we are left with 1 over y square.

01:12 And we can do the same thing with the derivative of z with respect to y.

01:18 So we're going to get the derivative with respect to y of x over y square.

01:25 Now we can do the same thing as we did before, since x is just a scalar, it will simply multiply 1 over y square.

01:37 We can take it out and we simply have to compute the derivative with respect to y of 1 over y square, which is something hopefully you're familiar with.

01:50 And now we simply get x times minus 2 over y cube so if you don't know how to do this derivative i encourage you to go back or check in a manual and we can simply write the final result minus 2x divided by y cube so here we added the derivative with respect to x and respect to y now we're going to and do something that's always been a bit annoying for me, which is to compute, which is to draw level curve.

02:29 So i, if you, if you have to do it long hand or do it by hand, good for you, but i encourage you to use a software that could draw it for you to make your, to make it as pretty and as readable if you have to do this for a homework, for example, if you have to do this in an exam, of course, just do your best.

02:51 So if we draw the level curve level curves so that's e level curves for z is equal to one that simply means we have x over y square is equal to one which leaves us with this parabola x equals y square so let's say that this is one and this is one so this is one and this is minus 1 minus 1 that means that our level curve we're going to draw it in red so i'm going to draw the best parable that i can this so this is not too bad so this is not too bad so this is the level curve for z equals 1 if we do it for z equal 2 we're left with x over y square equals 2 which is x equals 2 y square so another parable but this one will be a bit more narrow so we're going to draw it let's say in blue and we know that if y is equal to 1 that means y square is 1 so 2 times y square will be 2 so if y equals 1 that means x will be 2 and we can draw another parable this one a little bit more ugly than the previous one of z equals 2 now we go on we continue so if for z equals 3 we have x over y square equals 3 which means we have x x equals equals 3 which means we have x equals 3 y square i'm sorry 3 y square we're going to draw this in green so if y is equal to 1 that means that x is equal to 3 so 1 2 and 3 and 3 because it's symmetrical it's a parable so if y is equal to minus 1 we have minus 1 square is 1 3 3 2 2 2 3 2 times 1 is 3.

05:42 So we're going to try to do this a little better.

05:46 Yeah, that's not too bad.

05:52 And we have z equals 3.

06:00 And for z equals 4, we have x over y squared, which is equal to 4, which is the same thing as x equals 4 y squared and what color can we take uh i don't know maybe we can take orange so we can draw an orange and once again if y equals one we will have that x is equal to four so i'm going to try to draw it once again i'm i really encourage you if you have to draw like these curves in a homework not to do it by hand and to use a software drawing software because you will get much better results and results that you can hopefully move around with depending on the software you use but this this should give you an idea basically the idea is as the bigger z gets the narrower the level curves will be they will all tend to look increasingly more we try to do that.

07:13 It will be increasingly more narrow like this.

07:17 So that's what you're supposed to get by drawing those, in my case, those ugly level curves.

07:25 Oh, let me scroll down a bit.

07:27 Here we go.

07:29 So for part, for part c, let me check my note to make sure away.

07:36 So we, let me read the question.

07:41 So we want to see and what happens.

07:46 We want to move along the horizontal line y equals 1 in the xy plane and describe out the corresponding z value change and explain how this observation is consistent with the partial derivative of z.

08:02 So we remember that z of x, let's just write this here.

08:08 Here is equal to 1 over y squared.

08:16 That's the partial derivative.

08:20 So i'm going to write it in words first.

08:23 So the further, the farther, we move along.

08:36 Y equals 1.

08:39 So i .e.

08:41 X goes to infinity...

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