11. For f(x,y,z)= x28\": , nd a) The directional derivative of f at point P(1, 1, 1) in the direction from P to point Q (2, -1, 3) (6 points) b) The minimum directional derivative of f (x, y,z) at point P0, 1, l). (2 points) 12. The surface KAY): x3 1214:)! + 83/3 has two critical points. Using the discriminant method (involving 2nd order partial derivatives), classify these two critical points as maxima, minima or saddle points of the function. Show your work. [8 points) 13. At t = 0, a particle is moving in three dimensions with a position vector given by . The particle has a velocity vector 170) = 6r? + 9'}: sin(t) . Calculate the particle's position vector as a mction of time. (4 points) 14. 15. The linesLl: x=5+2t,y=8+3t,z=3 andL2zx= -s, y=2,z=7+4s intersect at a point P. Find the following: a) The coordinates of point P (2 points) b) The distance from point P to line L3: = -1 + 3v, = -1 + 4v, 2 = 2v (4 points) Three forces V1, V2 and V3 are located in the xy plane as given in Figure 1. Their magnitudes and angles with respect to the x axis are given in Figure 1. a) Write each force in component form (6 points) b) Determine the resultant force acting on an object located at the origin in Figure 1 above. (2 points) 16. Use the Divergence theorem to calculate the value of the surface integral given as ffs F - 11:15 Le calculate the ux of F across S where F is given by the vector F = and S is the surface of the box bounded by the coordinate planes and the planes x = 3, y = 2, and z = 1. (The triple integral in the Divergence theorem will be much shorter to do). (5 points)