MATH 201 Final, Fall 2003

American University of Beirut MATH 201 Calculus and Analytic Geometry III , Fall 2004 Final Exam In? n Exercise 1 a) /2 points]: Find lim n>+co on (hint: this is the same sequence of quiz 1) na +90 b) /2 points/: What can you say about the series : n=l n2 Exercise 2 a) /3 points each] Discuss whether the following series converges or diverges. iii) y nr? sin(1/n) tan(1/n) nal +co Inn b) /8 points] Find the interval of convergence of the power series = +2)” n=l (be sure to check at the end points) Exercise 3 a) /3 points] Find the equation of the tangent plane to the surface z = 2°—2?y?+4 at the point (1, 1,4). b) /5 points} Suppose that the equation 2° — 2?y? + 2yz—8 = 0 defines z as a function of y and 0 z. Find the values of 2 and a at the point (1,1, 4). dy Oz ry4 c) [3 points} Prove or disprove: yw=t ) [8 points} Prove or disp g(x,y) ray 5 can be extended by continuity at (0,0). Justify. Exercise 4 (5 points] Find the absolute minimum and maximum values of the function f(z, y) = x? +y? +22 —y+1on the domain R defined by {(v,y) R22 +y? <4 and y > 0}. 1 t Eo Exercise 5 /10 points] Evaluate I = I | i e” dydrdz. : OQ vz 40 [eee JAMENICAN UNI VEHSIT 4 LIBRARY t CL Le ed ) ee Exercise 6 /12 points] Let V be the volume of the region R bounded laterally by the cylinder x? + (y— 1)? = 1, from above by the cone z = \/z? + y?, and from below by the zry-plane. a) Sketch the region of integration. b) Express V as iterated triple integral in cartesian coordinates in the order dzdrdy (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 u = y/xr,v = zy to find / | en vdA R over the region R in the first quadrant enclosed by the lines y = xz, y = x/2 and the hyperbola y = 1/z and y = 2/z. (sketch both regions of integration) Exercise 8 /15 points] a) Write the complete statement of Green’s Theorem. b) Find é (x? — y)dz + dy, where C is the closed curve positively directed given by the equa- g tions 2° + y? =4, and y > 0. i) By evaluating directly the line integral. ii) By using Green’s Theorem quoted above. eT {amsiic sn ESET wStem RAR Y }