Level Curves Graph and Partial Derivatives.

[nikkihendrix]

Delete post.

Delete, please.

 

[micromass]

The level curves of a function z=f(x,y) are shown in the link provided.

f (sub x) (P), f (sub y) (Q), f (sub y) (R), f (sub x) (S), and the NUMBER 0.

Link: http://imageshack.us/photo/my-images/840/64800067.png/

Assume that the scales along the x and y axes are the same. Arrange the following five (5) quantities in ASCENDING order. Give a brief explanation for your reason.

I figured 0 is the least because there are no level curves.

I know that f_y (Q) and f_x (S) are negative. Also, f_x (P) and f_y (R) are positive.
But f_x (S) have more compact level curves therefore it's bigger than f_y (Q). Similarly the reason why f_y (R) is bigger, right?

Does it go: 0, f_y (Q), f_x (S), f_x (P), f_y (R)?

Hi nikkihendrix!

If f_y(Q) and f_x(P) are negative, then 0 is larger than these two values, right?? So 0 can't be the least.

Furthermore, saying that f_x(S) has more compact level curves that f_y(Q) means that it's absolute value is larger. That doesn't mean that f_x(S) actually is larger...

 

[micromass]

How would I determine whether f_x (S) goes before or after f_y (Q)?

Or f_x (P) is greater or less than f_y (R)?

So, it's f(sub x) (P) < 0 < f(sub y) (Q) < f(sub y) (R) < f(sub x) (S)

f(sub x) (P) has negative slope.
f(sub y) (Q) has positive slope. (Flipping x and y can be seen)
f(sub y) (R) less than or equal to f(sub x) (S)?

The partial derivative indicates the rate of change. For example, a partial derivative of -1 means that I'm going downwards with a rate of 1. A partial derivative of -10 means that I'm going downwards with a rate of 10: this is much faster!
So I expect the level lines with a partial derivative of -10 to lie much closer together than with a partial derivative of -1.

The same way, I expect the level lines with a partial derivative of 10 to lie much closer together than with a partial derivative of 1.

If the partial derivative is 0, then you neither increase neither decrease in that direction.

So, in our case: in which case of f_x(S) and f_y(Q) are you going downwards the fastest?

In which case of f_x(P) and f_y(R) are you going upwards the fastest?

 

[micromass]

f_x (S) is going downwards faster than f_y (Q).

What does it mean by ascending order than? f_y(Q) < f_x (S) then.

If you calculate partial derivatives. Which ones is going downward faster: -10 or -1?

 

[micromass]

-10. oh!

okay so...it's actually suppose to be
f_x (S), f_y(Q), 0, f_y(R), f_x(P)!

Which one is increasing fastest? +10 or +1? What does that mean for your last two values?

 

[micromass]

Indeed!

When given a situation like in the picture. Try to give explicit values to the partial derivatives. It makes it easier to see what's going on!

 

[micromass]

Seems good!

 

[SammyS]

Very Interesting !