Reference no: EM13883080

Math 203 - 2008

Part I:

1. (a) Parameterize the line that passes through the point (4, 0, -3) which is parallel to the intersection of the plane 2x - y + z = 2 and the plane x - 3z = 4

(b) Find an equation of the plane perpendicular to the planes 2x - y + z = 2 and x - 3z = 4 that passes through (4, -2, 3).

2. Let f (x, y, z) = x²y + y²z.

(a) Evaluate ∇f .

(b) Find an equation of the tangent plane to the surface x²y + y²z = -1 at the point (2, -1, 3).

(c) Find a unit vector v^→ such that f (x, y, z) has a negative rate of change at (2, -1, 3) in the direction ^→v.

3. Evaluate the integral ∫₀¹∫_y¹ √(1 + x²) dx dy by sketching the region R defined by the limits, changing the order of integration to ∫∫_R√(1 + x²) dy dx and evaluating.

4. 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 = 1}∑^{∞ e-k/√k+1}

(b) _{k = 2}∑^∞(-1)^k 1/x(ln(x))²

(c) _{k = 1}∑^∞(-1)^k 2k + 1/√(4k³+1)

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

_{k = 2}∑^∞ x^k/2^kln(k).

6. Suppose the temperature T is given by T = x³ + y³ - 3xy.

(a) Find and classify all critical points of T.

(b) Find the hottest point on the square plate bounded by x = 0, x = 2 and y = 0, y = 3.

7. Find the volume of the cone with base radius a given by a - z = √(x² + y²), z ≥ 0.

Part II:

8. Let f (x, y) be a differentiable function that is constant on all circles x² + y² = a² for any radius a ≥ 0. If (x, y) ≠ (0, 0), show that

f_y(x, y)/f_x(x, y) = y/x.

Hint: Observe that in polar coordinates, f is a function of r alone.

9. (a) Find a power series expansion for 1/1 + t² and determine its radius of convergence.

(b) Use the fact that

₀∫^1/√3 1/(1+ t²) dt = arctan(t)|₀^1/√3 = π/6

to express π/6 as an infinite series. What is the fewest number of terms you must calculate to guarantee 2 decimal places of accuracy?

10. Let R be the lamina given by a quarter circle bounded by x² + y² = 4, x ≥ 0, y ≥ 0, and suppose the density of the lamina is ρ = √(x² + y²).

(a) Find the mass of the lamina.

(b) Find the the coordinates of the center of mass (x', y') of the lamina.

11. Find the surface area of the part of the sphere of radius 3 given by x² + y² + z² = 9 that lies inside the cylinder x² + y² = 4.

12. (a) Evaluate the limit or show that it does not exist:

_{(x, y)}lim_{→(0, 0)} xy²/(x2 + y2)

(b) Expand f (x) = 1/x as a Taylor series about x = 2. Use the remainder formula to estimate the error in using the first five terms of your series to evaluate 1/e.