# 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*