Chapter 12.6, Problem 57E

Gradients in three dimensions Consider the following functions f, points P, and unit vectors u.

1. a. Compute the gradient of f and evaluate it at P
2. b. Find the unit vector in the direction of maximum increase of f at P.
3. c. Find the rate of change of the function in the direction of maximum increase at P.
4. d. Find the directional derivative at P in the direction of the given vector.

56.     $f\left(x,y,z\right)=4-x^2+3y^2+\frac{z^2}{z};P\left(0,2,-1\right);\langle 0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\rangle$