MATH 203 - 2009

PART I:

1. Let P be the plane that contains the points (1, -2, 2), (-2, -1, 4) and (-1, -3, 2).

(a) Give an equation for the plane P.

(b) Parameterize the line of intersection between P and the plane z = 1.

2. Given that f (x, y, z) = x³y² + ye^2z.

(a) Evaluate ∇f .

(b) Find the rate of change of f at (2, -1, 0) in the direction of (-2, -2, -1).

(c) Find an equation of the tangent plane to the surface x3y2 + ye2z = 7 at the point (2, -1, 0).

3. Find all local maxima and minima and all saddle points of the function

f (x, y) = 9x² + 2y² + 2xy².

4. A laminar region R in the first quadrant is bounded by y = √x, y = 0 and y = x-2.

It has density given by ρ(x, y) = 3y. Sketch the region and compute its mass.

5. Find the volume of the solid in the first octant, which is bounded by the coordinate planes, the cylinder x² + y² = 9, and the plane x + z = 6.

6. For each of the following series, state whether the series is absolutely convergent, conditionally convergent, or divergent and show why your answer is correct. (No credit for any part unless your reasons are given.)

(a) ∑_k=2^∞(-1)^k ln(k)/k³

(b) ∑_k=1^∞(-1)^k 1/kln(k)

(c) ∑_k=1^∞(-1)^k k+4 / √k⁶ + 2k² + 2

7. Find the interval of convergence including possible endpoints for the power series

∑_k=1^∞ (x - 5)^k / 2^{k 3}√k

Part II:

8. (a) Write the first three nonzero terms of the Maclaurin series of sin(x).

(b) Write the first three nonzero terms of the Maclaurin series of sin(2x²).

(c) Estimate the value of the definite integral ₀∫^1/10 sin(2x²)dx, expressing the estimate as a sum of two fractions. Give an upper bound for the error of your estimate with an explanation of how the error bound was obtained.

9. (a) Sketch the region of integration and then change the order of integration for:

₀∫¹ _√y∫¹ √x³ + 1 dx dy

(b) Evaluate the double integral you obtain in (a).

(c) Evaluate the limit or show that it does not exist:

_{(x, y)}lim_{→(0, 0)} x⁴ - x²y² + y⁴/ x⁴ + y⁴

10. Find the surface area of that part of the hyperbolic paraboloid z = 2x² - 2y² which lies inside the circular cylinder x² + y² = 9.

11. Use spherical coordinates to find the z-coordinate of the center of mass of the uniform density solid hemisphere given by x² + y² + z² ≤ a², z ≥ 0.

12. For the function f (x, y, z) = z√x + 2y, use differentials to approximate f (1.98, 1.01, 1.02).