Multi-Variable Calculus: Partial Derivatives Using Level Curves

[Dembadon]

Homework Statement

This is a bonus problem on our homework, and I'm having trouble figuring out how to setup what I need.

Homework Equations

Here are my best guesses:
[tex]f_x=\frac{\partial f}{\partial x}[/tex]
[tex]f_y=\frac{\partial f}{\partial y}[/tex]
[tex]f_{xx}=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right)[/tex]
[tex]f_{xy}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)[/tex]
[tex]f_{yy}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right)[/tex]
[tex]f_x=\frac{f(x_0+h,y_0)-f(x,y)}{h}[/tex]
[tex]f_y=\frac{f(x_0,y_0+h)-f(x,y)}{h}[/tex]

The Attempt at a Solution

My professor said that a "sufficiently precise qualitative explanation" (whatever the hell that means) will be good enough. If it's possible, I would rather provide an analytical explanation. Maybe with the limit definition of the derivative? I'm having trouble figuring out what I need to use, and I have a feeling it's embarrassingly simple.

Initial observations:

(a) If [itex]f[/itex] is increasing at [itex]P[/itex] then [itex]\frac{\partial f}{\partial x}[/itex] is positive. If [itex]f[/itex] is decreasing, then [itex]\frac{\partial f}{\partial x}[/itex] is negative, right?

(b) Same line of reasoning from (a), but holding [itex]x[/itex] constant.

(c) For [itex]f_{xx}[/itex] the contours appear to be closer together for [itex]x<x_0[/itex] than for [itex]x>x_0[/itex]. This indicates that [itex]f_{xx}[/itex] is negative, right?

(d) For [itex]f_{xy}[/itex] , I think this means that I'm supposed to observe how [itex]f_x[/itex] changes when [itex]f_y[/itex] changes, right?

(e) For [itex]f_{yy}[/itex] the contours appear to be closer together for [itex]y>y_0[/itex] than for [itex]y<y_0[/itex]. This indicates that [itex]f_{yy}[/itex] is positive, right?

 

[CompuChip]

Intuitively, f_x is something like "the (instantaneous) change in f in the x-direction only" - so what do you expect?

You can confirm this suspicion analytically if you look at the definition of f_{x[/b]. Just take h > 0 and check the signs of the denominator and the numerator (you'll have to assume that the level curves are a reasonable representation, e.g. that f doesn't rise to 12 in between the level curves for 6 and 4).}