Gradient Partial Derivative Problem

[cmajor47]

Homework Statement

The elevation of a mountain above sea level at (x,y) is 3000e^$$\frac{-x^2-2y^2}{100}$$ meters. The positive x-axis points east and the positive y-axis points north. A climber is directly above (10,10). If the climber moves northwest, will she ascend or descend and at what slope.

Homework Equations

$$\frac{d}{dx}$$e^u=e^u$$\frac{du}{dx}$$

The Attempt at a Solution

$$\nabla$$f(10,10)=-600e^-3i-1200e^-3j

I know that the climber will descend but I don't know how to figure out the slope that she will descend at. Can anyone help?

 

[Dick]

To find the slope you want to take the dot product of your gradient with a unit vector pointed in the northwest direction. Can you find one?

 

[cmajor47]

Would the unit vector be 600e^-3i-1200e^-3j since west is the opposite of east?

 

[Dick]

No. A UNIT vector pointing west would be -i. A unit vector pointing north would be j. Do you see why? Northwest is at a 45 degree angle to both of those. Finding a unit vector pointing NW has nothing to do with your gradient vector. It's a whole different problem.