Chapter 12.6, Problem 59E

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.

59.     $f\left(x,y,z\right)=1+\sin \left(x+2y-z\right);P\left(\frac{\pi }{\text{6}}\text{,}\frac{\pi }{\text{6}}\text{,}-\frac{\pi }{\text{6}}\right);\langle \frac{1}{3^{\prime }}\frac{2}{3^{\prime }}\frac{2}{3}\rangle$