The question:

A skier is on a mountain described by the equation $\displaystyle h(x, y) = 2000 - x^4/10^8 - y^2/10^2$ at the point (100, 1). He skis down the mountain, always moving in the direction of the steepest descent.

a) In what direction does he start moving?

b) Describe the curve along which he skis. [You will need to solve a separable first order ODE.] [HINT: Let the skier's position at time t be (x(t), y(t), z(t)) where z(t) = h(x(t), y(t)). What does the condition say about the skier's velocity in the xy-plane, ($\displaystyle \dot{x}, \dot{y}$)?]

My attempt:

I can almost solve a), I ended up getting $\displaystyle \frac{10^{-2}}{2}(-2i - j)$ which is in the opposite direction. I'm not sure why it's the wrong direction, my working is:

P = (100, 1)

$\displaystyle \frac{dh}{dx} = \frac{-4x^3}{10^8}$

$\displaystyle \frac{dh}{dy} = \frac{-2y}{10^2}$

$\displaystyle \nabla f(x, y) = \frac{-4x^3}{10^8}i + \frac{-2y}{10^2}j$

When I subbed in the point, I got my answer. Where am I getting the signs wrong?

For b), I have no idea how to attempt it. I think I'm missing some intuition here. Any ideas?

Thanks.