(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant 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]

3. 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] isnegative, 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] ispositive, right?

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# Homework Help: Multi-Variable Calculus: Partial Derivatives Using Level Curves

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