Final_Fall-2003-04 - American University of Beirut MATH 201...

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Unformatted text preview: American University of Beirut MATH 201 Calculus and Analytic Geometry HI ‘ Fa132004 Final Exam . , , _ . inn-n Exerc1se l a) [2 paints]: Find hm Rad-00 n (hint: this is the same sequence of quiz 1) +00 1 n , , in n .p b) [3 points]: What can you say'about the series E n2 ? Justlry. 71:1 ExerciSe 2 a) [3 points each] Discuss whether the following series converges or diverges. iii) if n2 sin(l/n) tan(l/n) n:1 +00 inn 1)) [3 points] Find the interval of convergence of the power series 2 7(3: + ‘2)?1 n:l {he sure to check at the end points) Exercise 3' a) [Spot-rim] Find the equation of the tangent plane to the surface 2 = I5 —$2y2+4 at the point (l,l,4). b) [5 points] Suppose that the equation $5 — :6ng 4* 2yz — 8 2: 0 defines x as a function of y and 6 8 2. Find the values of j and j at the point (1,1,4). ‘ fig 32' $6y4 2 + 2 can be extended by continuity at (0,0). 1' y c) [3 pointsf Prove or disprove: g($,y) : Justify. Exercise 4 [15 points] Find the absolute minimum and maximum values ofthe function flz, y) = $2 + y2 + 21: — y +1 on the domain R defined by {tiny} E R2322 + y2 S 4 and y 2 0}. 1 l x 7 Exercise 5 [10 points] Evaluate I = / f / er“ dyd'ccdz‘ A 0 z D I [M lAMrililflax I‘NIVEZHMT' LIISH.AI{Y “I m;|l!’|"1 '4 Exercise 6 [12 points] Let V be the volume of the region R bounded laterally by the cylinder :52 + (y —- l)‘2 = 1, from above by the cone z = #332 + yg, and from below by the mgr—plane. a) Sketch the region of integration. b) Express V as iterated triple integral in cartesian coordinates in the order dzdxdy (do not evaluate the resulting integral). c) Express V as iterated triple integral in cylindrical coordinates, then evaluate the resulting integral. Exercise 7 [15 points] Use the transformation a = 9/22, u 2 ry to find [/erydél . - R over the region R in the first quadrant enclosed by the lines y = 3:,y :- :r/2 and the hyperbola y = l/ct and y = 2/55. (sketch both regions of integration) Exercise 8 [15 points] a) Write the complete statement of Green’s Theorem. b) Find % (a:2 —— yjdz + scaly, where C is the closed curve positively directed given by the equa- C tions $2 + y2 = 4, and y 2 0. i) By evaluating directly the line integral. ii) By using Green’s Theorem quoted above. ; r ...
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