Math 203 - 2007

Question 1

a) Find an equation of the plane that contains both the line

(x-1)/2 = (y - 2)/3 = (z - 3)/4 and the point (1, 2, 0).

b) Find parametric equations describing the line through the point (1, 2, 3) which is perpendicular to the plane z = x + 2y - 4.

Question 2

a) Given the curve r^→(t) = < √t, 4/t , t²/2>, find a parameterization of the tangent line at (2, 1, 8).

b) Find an equation of the tangent plane to the surface x³/z² + 4(y-1)² = 5 at the point (1, 2, -1).

Question 3

Consider the function f (x, y) = x/(2x+3y).

a) Find the directional derivative of f (x, y) at the point P (2, -1) in the direction toward the point Q (-1, 1).

b) Find the directional derivative in the direction of maximum increase of f (x, y) at the point P .

Question 4

a) Let R be the region bounded by y = x, y = 2x and y = 6. Write ∫∫∫_Rf(x,y) dA an iterated integral.

(b) Reverse the order of integration in the integral ₀∫²_x∫^2x f(x, y)dydx

Note for both parts: You CAN NOT evaluate the integrals, since no function f (x, y) has been specified.

Question 5

Let the region R in space be above the plane z = 0, inside x² + y² + z² = 4 and outside x² + y² = 1.

a) Sketch the region

b) Use a triple integral to find its volume.

Question 6

State, for each series, whether it converges absolutely, converges conditionally or diverges. Justify each answer.

(a) ∑_n=1^∞(-1)ⁿ(n/n+1)/n     (b) ∑_n=2^∞ (-1)n/n(ln(n))  (c) ∑_n=1^∞ (-1)n cos(n²)/n²+1

Question 7

Find the set of all points x for which the following series converges:

∑_n=0^∞ (x+1)ⁿ/(n²)/2ⁿ

Remember to check convergence at the endpoints, if applicable.

PART II : Answer all parts of any three of Questions.

Question 8 Find all critical points of the following function and classify each critical point (as a local maximum, local minimum, saddle point, etc.)
f (x, y) = xy² - 2xy + x².

Question 9 Let E be the region bounded the cylinders x² + y² = 1 and x² + y² = 4 and the planes z = x and z = 4. Evaluate ∫∫∫_E zdv.

Question 10 Recall that ₀∫^x 1/1+t² dt = arctan(x).

a) Find a formula for the nth term of the power series centered at x = 0 for the function arctan(x)

b) What is the minimum number of terms that you need in the series for arctan(x) to compute arctan(1/10) with an error of less than 1/10,000? Justify your answer.

Question 11 a) Show that the following limit does not exist:

_{(x, y)} lim_{→ (0, 0)} x² + xy²

b) Graph the quadric surface x² - 2x + y² - z² - 2z + 1 = 0, labelling the coordinates of all vertices, if there are any. Show the trace of the graph in the coordinate planes.

Question 12 Let S be that part of the surface z = 4 - x² - y² that lies above z = 1. Find the area of S.