MATH 201 Final, Fall 2003

bch SON IWER: LIBKARY OF BEIRUT Math. 201 Final Exam. (Time: 140 minutes) N. Nahlus & H. Yamani Jan 28 , 2004 Name cevesegeeeeeea nner e nee e eee e eee Circle your section number (Sec 9 at 1W) (sec 10 at 12:30T) (sec 11 at 2T) (sec 12 at 3:30T) Part 1 (50%) 10 multiple choice problems (No Penalty ) 2 (_4)" 1. Consider the series > ( ay . Then the series is nai NO A) Conditionally convergent B) Absolutely convergent C) Divergent 3" +5 ; )" . Then the series is ni 2. Consider the series > (-1)" ( n=] A) Conditionally convergent B) Absolutely convergent C) Divergent 3. Consider the series > (-1)" mn and y(n" n n=2 n=2 ni (n+5)! (2n)! . Then A) 1® series diverges & 2™ series diverges B) 1° series converges & 2” series diverges C) 1" series diverges & 2" series converges st . nd . D) 1" series converges & 2™ series converges = x-1)" 4, The domain of convergence of x (-1)" a a2 3°ninn A) -2sx<4 B) -2<x<4 C) -2<x<4 D) -2sx<4 E) x=1 &x=4 F) None of the above 1— cos??? 5. The Maclaurin series of f(x) = f a ae i ca (-1)"" x" A ) 2 2n! 3n ES -1)"*! onl 3n —1 om (-1)""! xn = «2n! 3n+] = (1) x6" D ) » 2n! 6n E) None of the above n=l Do not forget to integrate ! x=) oa 6. Suppose z = f( where fis a differentiable function. Then x— = ky — 3y A) k=-3 B) k =1/3 C)k=3 Dyk =-1 E) k=1 F) k=-1/3 G) None of the above 7. Let f(x,y) = e*5 Then its critical point is A) a local max. B) a local min. C) a saddle point. 8. Consider the paraboloid x? + y? —4z =1 and the sphere x’? + y? + 27 = 3. Then the tangent planes to both surfaces at the intersection point (1, 1, 1) are A) perpendicular B) parallel C) neither perpendicular nor parallel. 9. Given that F(x, y,z) =8. If the components of VF are never zero, then AZAD . @& aoe =F A)t1& e resp. B) +1& -& resp. a a a a C) ~1 & — resp. D) -1 & ~— resp. E) None of the above yy Yy 1 2 [ave" dxdy is ya 10. The value of the double integral f 0 A) 7/2 (e-1) B) 9/2 (e-1) C) 2(e+1) D) 8(e-1) E) 25/2 (e-1) Part (50 %) (Subjective) 11. (5%) Find the area of the surface cut from the bottom of the paraboloid z = 2x7 +237 by the plane z= 8. (Grading: 4pts for setting it up & changing it to polar) 12. (7 %) Use Green’s Theorem to find f (2xy? + x)dx + 4x7 y7dy é where C (traversed counterclock wise) is the boundary of the “triangular region in the 1* quadrant enclosed by the x-axis, x=] and y= x? 13. (5%) Set up (but do not evaluate) the double integral(s) in polar coordinates to find the area of the “triangular” region in the first quadrant bounded by y=3x" ,x=0 & xty = 4. Hint: The point (1, 3) is a corner point of the region. 14. (8 %) (i) Show that F = (x? ~ y)i — (x + y?)j is a conservative vector field (ii) Find a potential function for F (ii) Evaluate qe — y)dx -(x+y7)dy where C is the line segment from (0, 1) to (2,0). ¢ 15. (5%) Set up (but do not evaluate) the triple integral(s) in Spherical coordinates to find the volume and in the first octant of the surface inside the cylinder x + y = 4 and inside the sphere x? +3? +77 =29 16. (5%) Set up (but do not evaluate) the triple integral(s) in Cylinderical coordinates to find the volume in the 1“ octant common to the cylinders x° + y? = 4 and xi eer at, 7 / 17. (5 %) Consider the transformation u = x ~ wy &v=xy (sox=ut+v & y= (1) Show that the Jacobian J = A ¥) -! Ouy) uty” (1) Use the above transformation to find ff R xdydx where R is the region bounded by the curves X¥-xy=1, x-xy=5, xy=2, xy =3 18. (5%) Use Lagrange multipliers to find the points closest to the origin on the hyperbolic cylinder x* — y? =1 19. (5%) Find by inspection potential functions for the following conservative fields () F= ae its vt j for(x,y) ¥ (0,0) Gi) F2—* i+ <j for(xy) # (0,0) x+y xi +y 7 x . y . i) Fe : fe ; 0,0 (iii) wap Gay! or (x, y) # (0,0) Box your answers