Consider F and C below. F(x, y, z) = yz i + xz j + (xy + 2z)k C is the line segment from (1, 0, -1) to (6, 5, 1) (a) Find a function f such that F = Vf. (b) Use part (a) to evaluate fc Vf. dr along the given curve C. ×

No. 6 (a, b, c)

Find that F = 12x²yi + (4x³ + 8yz?)j+ 8y² zk is conservative, and find a function f such that F = Vf, and use the result to evaluate , F •dr, where C is any curve from A = (0, 0,0) to B = (-3,2, 3).

Define f(x, y) = 9x? + y?. (1) Sketch the level curve of this function for z=36. Draw it on an xy-plane. (2) Find the gradient Vƒ at the point (2,0) and the point (0, 6). (3) Sketch the graph (surface) of this function using f(x, y) as the z-coordinate. Show all your work on your own paper. You may leave the box below blank.