MATH 201 Final, Fall 2012

Math 201 Final Examination ( Time: 2.25 hrs.) (Fall 2012) N. Nahlus and (R. Nassif, F. Mroue, Z. Rawas) Name: ……………………………………….. (VERY CLEARLY) I.D ........................................... Circle your Section number ( - 2 points if incorrect) Sec 8 (12:30 T) — Sec 9 (2:00 T) — Sec 10 (11:00 T) — Sec 11 (5:00 T) --- Sec 12 (9:00 M) — Sec 13 (3:30 R) — Sec 14 (10:00 M) — Sec 15 (5:00 R) --- If you know limits like Simply write the answer ( without proof) 1. (8%) a. Investigate for convergence or divergence. b. Investigate for convergence or divergence. (Hint: = ) 2 . (6 %) a. Find the interval of convergence of Reminder: Check end-points Problem 1 8% Problem 2 6% Problem 3 8% Problem 4 8% Problem 5 8% Problem 6 8% Problem 7 9% Problem 8 9% Problem 9 9% Problem 10 9% Problem 11 10% Problem 12 8% 3. a) (5 %) Let | | | | | | | | b) (3%) Find Hint: Use the (Nahlus) estimates for continous increasing (positive) functions are Continue on the back side. 4. a) (4%) Find the parametric equations for the line L tangent to the curve of intersection of the surfaces at the point P(1,1,1). (Hint: L lies in both tangent planes) b) (4%) Given that and the components of are never zero, PROVE that 5. (8%) Set up (as explained below) the maximum value of the function on the upper disk bounded by . |||||||||| Set up => Find ALL candidate points; THEN STOP! STOP! |||||||||||| (Please Circle each candidate point) 6. a) (4%) Investigate the critical point (0, 2) of the function for local maxima, local minima, saddle points. b) (5%) Let +100) .Assume WITHOUT proof that (0,0) = 7 and (0,0) =8 From definitions, show that f is differentiable at (0, 0). 7. a) (5%) Find the value of b )(4%) Evaluate by changing to polar coordinates. 8. a) (5%) Set up the integrals to find the area of the region that is inside the big loop of the Limacon & outside the circle . 8b) (4%) Set up the integrals in cylindrical coordinates to find the volume in the 1st octant BELOW the paraboloid (Draw very carefully! (Note The needed intersection is at z=1 (not z=0) with the same circle.) 9. a(5%) Let V be the same region in problem 8b in the 1st octant below the paraboloid . Set up the integrals to find the volume of V in spherical coordinates in the order d d d . 9b) (4%) Set up the integral(s) to find the volume of V (above) in the order d d d. ( If you can not solve for cos , put ? & loose only 1 pt.) 10. a) (7%) Evaluate the over the solid ellipsoid Hint: Let x=2u, y=5v, z=10w. Then integrate in the uvw-space. b) (2%) Set up the integrals to find (only) of the centroid of a plane region R in the 1st quadrant bounded by . Note: All curves C below (in problems 10—12) are assumed to be smooth (except for finitely many points) counter clockwise, and traced once. 11. (a)(4%) Use Green’s Theorem to find the circulation of the vector field and C is the circle: . b. (4%) Evaluate for any wild path C lying in the right-half plane (x>0) from A(1,1) to B(2,6) by finding a potential function f(x,y) by inspection. Hint: Try c. (2%) Solve part b above without finding a potential function f(x,y)? 12. a) (4%) Use Green’s theorem (with holes) to find for any simple closed curve C . (To get full credit, check all details) | | | | | | | | | | | | | | | | b. (2%) Use part (a) to compute (Hint: Make a substitution to use part (a) ------------------------------------------------------------------------------------------------- c. (2%) Set up the Line integral to find the area of the “bumped” Cardiod.