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
• 5.014.
• (0,0), (+/-square_root(3),0), (1,1), (-1,-1).
• 14*square_root(2).

BRIEF ANSWERS TO TEST 2

• Hypotheses: Assume that F(x,y) assumes an extreme value on the curve defined by G(x,y) = 0 at the point (x_0,y_0). Assume that F and G are differentiable at (x_0,y_0) and that grad(G) is not zero at (x_0,y_0). Conclusion: grad(F) = lambda * grad(G) for some real number lambda.
• D_u(F) = -2/square_root(5).
• (1/2, -1/4). (Use Lagrange multipliers.)
• (0,0) is a saddle, (1,1) is a max, (1.-1) is a max.
• 2*Pi/3.
• (1/2, -1/10).
• 4/3.