Suppose you are climbing a hill whose slope is given by $$z = 400 – .05 x^2 – .01 y^2$$
where $x,y,z$ is measured in terms of meters. Now you are standing at $(20,20,376)$. The positive $x$-axis points east and the positive $y$-axis points north.
a.) If you walk due southeast, will you start to ascend or descend? At what rate?
b.) In which direction is the slope the largest? What is the rate of ascend in that direction?
Attempted solution a.) – We need to calculate the directional derivative $D_{u}f(x,y)\cdot\textbf{u}$. We have $$D_u f(20,20)\cdot\textbf{u} = <-1.4142,-.2828>$$
Thus we must be descending. The rate is given by
$$\left|\nabla f(x,y)\right| = 2.0396$$
I left some of the details out but I am not sure if this is right or how to proceed with b.). This is a problem I am doing for my client I tutor in calc 3
Best Answer
We have $$\nabla f(20,20) = (-2, -.4)$$
$a)$ The directional derivative going south-east, that is going in the direction of the vector $(1,-1)$, is $$<\nabla f(20,20),(1,-1)>=-1.6.$$ So you'll start to descend at a rate of $1.6$.
$b)$ The direction in which the slope is the largest is the direction of the gradient. So, the direction of the vector $(-2, -.4)$. We have $$<(-2, -.4),(-2, -.4)>=4.16,$$ so if you take this direction you'll start to ascend, at a rate of $4.16$.