Question

The Fellowship of the Ring traveling through the mountains of Moria (which are described by the functionf(x,y)=12x2y2e(-x2-y2)) and they find themselves at the point A=(1.5, -1, 1.05). They need to get to the top of the mountain they are climbing so they decide to travel along the plane y=-1. How steep is their climb when they start at point A? A diagram: https://www.geogebra.org/3d/jbkkyugx

1. What if they decided to go in a different direction, say they started walking towards the point (0,0,0) on a plane that is “vertical”. How steep would be the mountain going in that direction?

2. What is the general equation for planes that are vertical, like the one that you used to travel from A to (0,0,0)?

3. What would change in your calculations if we change where point A is? Can you generalize how to find the steepness in different directions from a given point A?

Answer 1

Part 1

So to find the steepness of the mountain along a certain direction, you need to calculate the partial derivative of the function along that direction, that is basically

So it is a saddle point, there is absolutely no steepness at that point.

Part 2

The equation satisfying the plane on these two points lie can be determined in the following way.

Take the general equation of the plane to be . Given the plane passes through , we get that . Now, if we put the other point, we get that .

We put that in the preceding equation and get the equation of the plane to be

Part 3

If we change the point A that would change the equation of the plane altogether.

The steepness of different directions from a given point A can be measured by calculating the partial derivative of the function with respect to that direction.