MATH 201 Final, Fall 2009

Math 201 Final Exam. (Jan 200A) (Time: 140 minutes) N. Nahlus & H. Yamani Name ................................... ……….. I.D ................... Section number: Circle your Recitation section (section 13: 11:00 Th) (section 14: 3:30 Th) (section 15: 2:00 Th) (section 16: 12:30 Th) 1) (5 %) Investigate the series for Abs. /cond. Convergence or Divergence Problem 1 Problem 6 Problem 2 Problem 7 Problem 3 Problem 8 Problem 4 Problem 9 Problem 5 Problem 10 Total Total 2) (4 %) Investigate the series for convergence or divergence. 3) (12 %) INSERT True-False. WITH Penality : for each wrong answer . Note: The extra spaces below are for your convenience a) is Convergent: ………………… (INSERT TRUE/FALSE) Problem 11 Problem 12 a, b Problem 13 Total 1 Problem 14 a,b Total 2 Problem 15 a,b Total 3 Bonus TOTAL Total b) a) is Convergent: ………………… (INSERT TRUE/FALSE) c) (for all n) implies is Divergent: ………… (INSERT TRUE/FALSE) d) is Divergent: ………… (INSERT TRUE/FALSE) 4) (6%) Find the domain of convergence of | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 5. (5 %) (i) Find . (ii) Deduce the value of . Hint: From tabular integration, | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 6. (6%) Find the points on the surface where the tangent plane is parallel to the xy-plane. (Hint: Two planes are parallel if their normal vectors are parallel) | 7. (6 %) The function at a point p increases most rapidly in the direction of the vector v=(3,4,5) with directional derivative . (i) Find (ii) Find the directional derivative of at p in the direction of the vector w=(4, 0, 3) 8. (7 %) Investigate the critical points of for local maxima, local minima, saddle points. 9. (7 %) Find the value of (Hint: Reverse the order of integration) | | | | | | | | | | | | | | | | | | | | 10. (4 %) Evaluate (Hint: Change to polar. Then use substitution) 11. (6 %) Set up (but do not evaluate) the double integral(s) in polar coordinates that represent the area of the region that is inside the lemniscate & inside the circle . Hint: 12a. (4 %) Set up (but do not evaluate) the triple integral(s) in Cylinderical coordinates that represent the volume of the region bounded by the cylinders and 12b. (5 %) Set up (but do not evaluate) the triple integral(s) in Cartesian coordinates that represent the volume of the region in the first octant bounded by the coordinate planes BELOW the surface (Hint: The surface hits z=0 at ………….) 13. (6 %) Set up (but do not evaluate) the triple integral(s) in spherical coordinates that represent the volume of the region inside the sphere , inside the cylinder , and above the xy-plane. Note: All curves below are assumed to be (smooth, counter clockwise & traced once) 14a. (5 %) Use Green’s Theorem to find where C is the circle: 14b. (4 %) Find where C is a complicated curve (not passing through the origin) from A(1,1) to B(3,2). (Hint: By inspection, By inspection, find your potential function f(x,y)) 15a. (3 %) Use Green’s Theorem (in a clever way) to find where C is the circle . 15b. (5 %) Use Green’s theorem (with holes) to find where C is any complicated simple closed curve around the origin. Hint: Assume without proof that except at the origin Bonus 4 % Use Gauss Theorem (or the divergence Theorem) to find the outward flux of the vector field (i) across the sphere (ii) across the sphere Reminder: JUST use Gauss Theorem. (Do not, Do not, calculate )