Need help with Calc 3- differentiation in several variables

[kolodz13]

Hi my friend and I are struggling in understanding the concept of traces. If someone could explain how to do/solve #24 in detail (we already looked at slader and don't understand it) we would greatly appreciate it.

 

[tkhunny]

Trace:

1) Pick a value for f(x,y) = z
2) Use your best two-dimensional skills to draw what you get.
3) Repeat as necessary.

12. f(x,y) = 12 - 3x - 4y
1) f(x,y) = 12
2) 12 = 12-3x-4y ==> -3x - 4y = 0 ==> Look like a line through the Origin?

1) f(x,y) = 0
2) 0 = 12 - 3x - 4y ==> 3x + 4y = 12 ==> x/4 + y/3 = 1 ==> Look like a line with x-intercept at 4 and y-intercept at 3?

You may need several values before you get a vision for the whole surface.

Ever read a topographical map? What does it mean when the lines are really close together?

 

[tkhunny]

Careful with 22). You'll be tempted to expand the Domain.

With 24), try f(x,y) = -4.

Why do you think 24) is difficult? No 'x'? Why does that matter?

Oh, and why is this entitled "...differentiation..."?