Gradient and Directional Derivatives

dmalwcc89

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

Suppose, in the previous exercise, that a particle located at the point P = (2, 2, 8) travels towards the xy-plane in the direction normal to the surface.

a) Through which point Q on the xy-plane will the particle pass?

b) Suppose the axes are calibrated in centimeters. Determine the path c(t) of the particle if it travels at a constant speed 8 cm/s. How long will it take the particle to reach Q?

2. Relevant equations

Gradient of F: <dF/dx, dF/dy, dF/dz>

3. The attempt at a solution

I completed the "previous exercise:" I found the gradient of f after given the equation z^2 - 2x^4 - y^4 = 16 and asked to find vector n normal to this surface at P = (2, 2, 8) that points in the direction of the xy-plane. After normalizing a vector and finding the gradient, I was left with 1/(sqrt. 21)<-4, -2, 1>. The option was either + or - this value, and since (2, 2, 8) lies above the xy-plane, I needed the negative value. My answer was finally -1/(sqrt. 21)<-4, -2, 1>.