BRIEF ANSWERS TO TEST 1

• • -(b x a)
  • hyperboloid of two sheets
  • (dF/dx) * (dx/dt) + (dF/dy) * (dy/dt) + (dF/dz) * (dz/dt)
• square_root(2546)/2
• Point: (2,-1,3). Angle: arccos(5/7).
• k = 3.
• Cylindric: z = (r*cos(theta))^2 - (r*sin(theta))^2. Spherical: rho*cos(phi) = (rho*sin(phi)*cos(theta))^2 - (rho*sin(phi)*sin(theta))^2
• z = -9(x-1) + 6(y-2) - 3.
• 5.014.
• grad(F) = (2x+2y, 2x+6y). The derivative in the direction of (1,1) is 8*square_root(2). The maximal directional derivative at (2,1) is |grad(F)(2,1)| = square_root(136).

BRIEF ANSWERS TO TEST 2

• The critical points that lie in the indicated region are at (-1,1) and (-1,-1). The first is a saddle point and the second is a local max.
• By reversing the order of integration it can be shown that the value is (1/2)*(1-e).
• The area is (3/2)*Pi.
• The centroid is (0,0,(3/8)*a).
• The surface area is (Pi/6)*(17^(3/2) - 1).
• The area is (1/2)*ln(2).