Sketching level curves

[TsAmE]

Sketch the level curve for the function \(\displaystyle k = (x^2 + y^2)e^{-(x^2 + y^2)}\). What does the graph of this function look like?

Attempt:

Let z = k:

\(\displaystyle k = (x^2 + y^2)e^{-(x^2 + y^2)}\)

Now I am not sure how I can draw this graph

 

[tkhunny]

What would happen if you shifted to cyllindrical coordinates? It look slike a bunch of circles, to me. Have you seen a nice, symmetrical ant hill, lately?

 

[TsAmE]

This is what I got:

\(\displaystyle z = r^2e^{-r^{2}}\)

\(\displaystyle k = r^{2}e^{-r^{2}}\)

Im not sure how to draw this though :?

 

[tkhunny]

There is a reason you studied circles in analytic geometry.

x^2 + y^2 = r^2 is what on a cartesian coordinate system?

 

[TsAmE]

There is a reason you studied circles in analytic geometry.

x^2 + y^2 = r^2 is what on a cartesian coordinate system?

Its a circle, but I dont see how \(\displaystyle k = r^{2}e^{-r^{2}}\) is a circle :?

 

[tkhunny]

Let's ignore the 'e' for a moment.

Given a value of k, what have you? k = x^2 + y^2

Isn't that a cirle of radius sqrt(k) centered at the origin?

Generally, the x^2 + y^2 should suggest to you that whatever you find will have lovely symmetries across either the x-axis or the y-axis and radial symmetry.

Now, let's ponder just the 'e'- piece. k = e^-(x^2 + y^2)

x and y still appear ONLY in this same form, creating radial symmetries. By this time, we should conclude that we will go only circles, no matter what we do. How close are they? Good question?

Let's look at a simpler example, f(x) = x/e^x

Finding a first derivative. f'(x) = (1-x)/e^x - There is a nice Global Max at x = 1 after increasing sharply from zero and then falling off nicely. This should give you a rough shape.

If you add the x^2, it's a little sharper up to x = 1 and then sharper down.

I'm starting to get the idea of a nice, round volcano. You?

 

[TsAmE]

I can see the volcano, but still dont see why \(\displaystyle e^{-(x^{2} + y^{2})\) doesnt change the shape in any way

 

[tkhunny]

I can see the volcano, but still dont see why \(\displaystyle e^{-(x^{2} + y^{2})\) doesnt change the shape in any way

It does. Without that it would just continue to grow. This was the point of the x/e^x demonstration and the other version with the x^2. It doesn't change it from circles as projections on the x-y plane.