Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,y,z)=(3x+6y)/z, (6,3,1). You have to find the maximum rate of change and the direction vector.

Question

tkhunny · Answer

Does that sound like a Gradient to you?

anonymous · Answer

yes it does! That's just about all I know

tkhunny · Answer

Well, how do you do that?  There are some partial derivatives in your future.

anonymous · Answer

Would (3/z, 6/z, (-3x-6y)/z) be on the right track?

tkhunny · Answer

z^2 on the last one?

anonymous · Answer

whoops ya

anonymous · Answer

and then I get stuck on what to do

tkhunny · Answer

You've a point, (6,3,1) - Substitute away!

anonymous · Answer

and then do you have to make it a unit vector?

tkhunny · Answer

??  Why would you do that?  If we wanted a Directional Derivative in some other direction, that is what we might do.  Here, we are tasked with the maximum rate of change.  This is always int eh direction of the gradient.  What is the Gradient at (6,3,1)?

anonymous · Answer

there is a negative in front of the 1 that I missed. so then I got (-6,-3,-36)

tkhunny · Answer

(3/z, 6/z, (-3x-6y)/z^2)

(6,3,-1)

3/(-1) = -3 -- How did you get -6?

What am I missing?

anonymous · Answer

I keep messing it up when I type it... (-3,-6,-36)

tkhunny · Answer

Why is that different from (1,2,12)?

anonymous · Answer

It's not?