Math 241 - Quiz #7

1. Let f(x,y,z) = x y^2 z^3
Determine grad f
Determine D_u f with u = <1/sqrt(3), -1/sqrt(3), 1/sqrt(3)> at P(1,-2,1)

2. Find the maximum rate of change of
f(x,y,z) = x + y/z
at the point (4,3,-1).

In which direction does is occur?

Solutions

1. grad f = < y^2 z^3, 2x y z^3, 3x y^2 z^2 >
at (1,-2,1), grad f = < 4,-4,12>, so
D_u f = 20/sqrt(3)

2. grad f = <1, 1/z, -y/z^2 >
at (4,3,-1), grad f = <1, -1, -3>
so max rate = |grad f | = sqrt(11), and
it occurs in the direction of grad f or grad f/sqrt(11).

ccollins@math.utk.edu
Last Modified: Mar. 1, 1999