MATH 201 Final, Fall 2010

Math 201 - Final (Fall 10) T. Tlas • Please answer questions 1, 2, 4, 5 and 6 on the same sheet of paper on which it is written (after the line following the question). Any part of your answer written on the wrong page will not be graded. Question 3 has an extra sheet of paper on which you can write your answer. • When finished leave your work on your desk for it to be collected by the proctors. • There are 6 problems in total. Some questions have several parts to them. Make sure that you attempt them all. ================================================ Name : ID # : Section Number : ================================================ Q1 Q2 Q3 Q4 Q5 Q6 TOTAL Problem 1 (10 points each) Which of the following series converge and which diverge? For those which converge, do they converge conditionally or absolutely? When possible, find the sum of the series. You should justify your answers. i- ∞ � n=1 (−1)n2 n ii- ∞ � n=1 3n πnen iii- ∞ � n=1 e ln(n) n iv- ∞ � n=1 32n n! ================================================ Problem 2 (20 points) Find the minimum distance from the surface x2 − y2 − z2 = 1 to the origin. ================================================ Problem 3 (40 points) Integrate the function f(x, y, z) = 1 � (x2 + y2)(x2 + y2 + z2) over the region R which is the set of all points outside the sphere x2 + y2 + (z − 1)2 = 1 but inside the sphere x2 + y2 + z2 = 4. ================================================ ADDITIONAL SHEET FOR PROBLEM 3 ANSWER Problem 4 i- (10 points) Write the Taylor series for sin(r2). You do not have to derive it. ii- (20 points) Evaluate the integral ��� R sin(x2 + y2) 2π � x2 + y2 dV with an error of no more than 0.001. The region R is the set of all points (x, y, z) in 3 dimensions which satisfy 0 ≤ z ≤ 1 and 0 ≤ x2 + y2 ≤ 1. ================================================ Problem 5 Consider the square in the plane whose vertices are the four points (1, 0), (0, 1), (−1, 0), (0, −1). Assume you’re given the following three vector fields −→ F1 = ( −y x2 + y2 , x x2 + y2 ) −→ F2 = (x2 , y) −→ F3 = −→ F1 + −→ F2 i- (25 points) Find the circulation counterclockwise of −→ F3 around the square above. ii- (15 points) Find a function f(x, y) such that −→ F2 = −→ ∇f. ================================================ Problem 6 (30 points) Consider the vector field −→ F = � (y + 1)ey2sin(y3) , 0 � and the curve consisting of the three line segments A, B and C. Where A goes between (1, −1) and (1, 1). B goes between (1, 1) and (−1, 1) and C goes between (−1, 1) and (−1, −1). Find the flux of −→ F accross this curve (i.e. find � −→ F · ⃗n ds). The direction of the normal is away from the origin (i.e. on A it is to the right, on B it is up and on C it is to the left). ================================================