MATH 210  EXAM 2  FALL 1992

NAME: (PRINT) SIGNATURE: SSN: Show your work in the space provided.

1. For z = sin(xy^{2}), x = r cos θ, and y = r sin θ, find \frac{∂z}{∂r}.

2. Find the equation of the tangent plane to the surface z = x^{2} − 3y^{2} at the point (2, 1, 1).

MATH 210  EXAM 2  FALL 1992

3. Let f(x, y, z) = \frac{x}{y} − z^{2} and P = (3, −1, 2).

a) In what direction from the point P is f increasing most rapidly?

b) In what direction from P is f decreasing most rapidly?

c) Find the rate of change of f at P in the direction of w⃗ = ⟨3, 0, 4⟩.

MATH 210  EXAM 2  FALL 1992

4. Sketch the region of integration and change the order of integration for ∫_{−1}^{1} ∫_{0}^{1−y^{2}} f(x, y)dxdy 5. Locate and classify all critical points of the function

f(x, y) = x^{2} + y^{2} + x^{2}y + 1.

MATH 210  EXAM 2  FALL 1992

6. Find the points on the surface z^{2} = xy + 1 that are nearest to the origin.

7. Evaluate ∫_{0}^{π/2} ∫_{0}^{1} x sin(xy)dydx.

8. By changing to polar coordinates, find ∫_{0}^{2} ∫_{0}^{√\overline{4−x^{2}}} e^{−x^{2}−y^{2}}dydx