# Directional Derivatives and the Gradient Vector

1. Oct 2, 2009

### ktobrien

1. The problem statement, all variables and given/known data

Suppose you are climbing a hill whose shape is given by the equation below, where x, y, and z are measured in meters, and you are standing at a point with coordinates (120, 80, 1064). The positive x-axis points east and the positive y-axis points north.
z = 1200 - 0.005x2 - 0.01y2

a) If you walk due south, will you start to ascend or descend? At what rate?
b) If you walk northwest, will you start to ascend or descend? At what rate?
c) In which direction is the slope largest? What is the rate of ascent in that direction?
At what angle above the horizontal does the path in that direction begin?

2. Relevant equations

Duf(x,y) = gradient f(x.y) * unit vector

3. The attempt at a solution
I have already done a and b and most of c. I am having trouble with the last part of c. I am not sure how to go about finding the angle it makes with the horizontal. I know that it goes in the (-1.2,-1.6) direction and that the rate of ascent is 2. Could someone please tell me how to find the angle? Thanks

2. Oct 2, 2009

Arctan(2)

3. Oct 2, 2009

### ktobrien

Yea thats what I thought. Thanks for confirming that. I just discovered my calculator has been in radians. Thanks.