MAT 205 SPRING 2000 FINAL EXAM

Name: SSN: THERE ARE TEN PROBLEMS. SHOW YOUR WORK!

1. Let z = x cos y, x = t^{4}, y = e^{2t}. Find \frac{dz}{dt}.

2. Find the equation of the plane containing the points (0, 2, 0), (1, 1, 0), and (−1, 0, 1).

3. Find the derivative of the function f(x, y) = e^{2x+y} in the direction of the unit vector \frac{\sqrt{3}}{2}i + \frac{−1}{2}j at the point (1, −1).

4. The velocity vector of a particle is given by v(t) = 3t^{2}i + e^{2t}j + tk. Find r(1), the position vector of the particle when t = 1, given that r(0) = i + j.

5. Compute

∫ ∫_{D} x cos x^{2}dA

where D is bounded by y = 0, y = π, y = x^{2}, and x ≥ 0.

6. Calculate

∫ ∫_{S} F · dS

where F(x, y, z) = ye^{z^{2}}i + y^{2}j + e^{xy}k and S is the surface of the solid bounded by the cylinder x^{2} + y^{2} = 9 and the planes z = 0 and z = 3 given the outward orientation.

7. Calculate

∫ ∫_{S} curlF · dS

where F(x, y, z) = yzi+xzj+x^{2}y^{2}k and S is the part of the paraboloid z = x^{2} + y^{2}, oriented upward, that lies inside the cylinder x^{2} + y^{2} = 4. You will need to use the identity ∫ cos^{2} tdt = \frac{1}{2}t + \frac{1}{4} sin 2t + C 8. Evaluate

∫_{C} xy dx + y^{5} dy

where C is the triangle with vertices (0, 0), (2, 0), and (2, 1) positively oriented.

9. Let F = (x + y^{2})i + (2xy + y^{2})j.

1. Is F conservative?

2. Find a function f(x, y) such that ∇f = F.

3. Find ∫_{C} F · dr where C is any curve starting at (0, 0) and ending at (1, 1).

10. Evaluate

∫_{C} 3y ds

where C is the curve paramatrized by x = t, y = t^{3}, 0 ≤ t ≤ 1.