MATH 201 Final, Fall 2003

UIRED, “a PARTIAL CREDI Finok® fall L008 - o 5 Gedy (c> Gt Bo) y S$... ANSWERS, NO JUSTIFICATIO 1 (2 pis/part, total 10 pts). Which of the following series converge and which diverge? Circle your answer. ont? la) ) Converges Diverges 1b) Converges Diverges 1c) Converges Diverges Converges Diverges Converges Diverges aT CAS : fees " lin +l 2 (2pts). Fill in the blank: the radius of convergence of the power series 3 aot 27): n n=O is R= 3 (1pt/part, total 4pts. Note 1 blank = 1 part.). Fill in the blanks for the Arst few coeficients in the Maclaurin series for the following integral: 4 (2pts each for parts a-d, 4pts each for parts e-f, total l€pts). The following picture shows level curves for a function f(z, y), and the value of the gradient Vf at some points. —!DAT DATA VS(h,)2 52447 =, ° i V5 (Pel= 32527 VS (Pe)=- a+4 4a) Draw a vector on the figure at P, pointing in the direction of maximum increase of f. g ’, Circle the correct answer for 4b, 4c, 4d: of : was . . . 4b) Sn is positive zero negative aoes not exist. IL i Py . of | . . toe : _ 4c) Ba | 1S positive zero negative does not exist. wc lP, 4d) lim f(x,y) is positive zero negative does not exist. (x,y) (0,0) : 4e) Fill in the blank: The equation of the tangent line to the level curve passing through P,(7, 3) is e\ 17: : . . . < 3 . 1 4f) Fill in the blank: We start from the point 5(5, 0), and move a distance of ds = 706 units in the direction of the vector 7 =7+ 2). Then the change in the value of f(z, y) is approximately “(0.5 St/part, total 10.5pts. Note 1 blank = 1 part.). Consider the shaded region R elow (Ris common to the two circles shown and lies ahave the r-avis ) Fill in the blanke for the following integrals in rectangular and polar coordinates: A'S a! same] dr dé. 6 (0.5 pts/part, total 9.5 pts. Note 1 blank = 1 part.). In each of the pictures below, we give a o-cimensional picoure and a cross-section of a region D (which is always part of the hali-ball 2? +y?+2z? <1,z> 0). Ineach case, fill in the blanks for integration in spherical coordinates: u My plane dp dy dé. LY DIONE < pn i L his. ee ne are ) , ~~ / 6b) If/ 1dV =| | | [same] dp dy dé. = ee p= D HOLLOWED ouT HALF - BALL (SUNDER oF RADIUS A REMovED) z «i [for [same] dp dy dé. sp oP 7 {2 pus/part, total 8 pts. Note 1] blank = 1 part.). The following picture shows two curves C; and C2 in the plane. D; is the region inside C;. Dz is the region that is inside Cy AND outside Cy. We go around each of C; and C> counterclockwise, and the norma} vector points outwards of C, and C> in each case, as drawn or the figure. , & (ipt/part, 4 pts total). i t Stokes’ Theorem says: = | Far, VCO where the curve C is the boundary of the surface § S, and we go around C in the direction compatible with fi, according to the ens hand rule. Below is a picture where S is the part of the sphere of radius 2 (x? + y? = 4) with 2 < V3; the normal vector points away from the origin. C is the circle which is the boundary of 3. pace neem em JAMESGAN UNTCEESIT LIBRARY Da BREE 8a) On the figure, indicate the direction that one has to go around C for Stokes’ Theorem. 8b) Parametrize C in the orientation that you have described. You need to enter a constant number in each of the blanks below: ‘PART Il. FULL SOLUTIONS REQUIRED, PARTIAL CREDIT AVAILABLE. 9 (19 pts). Qa) Find the second degree Taylor approximation, centered at a = 2, to the function f(z) = Inz. (Your answer should have the form P2(x) = co +¢1(x@~2)+cp(x—2)? for appropriate cg, C1, C2.) 9b) Use the Remainder Theorem -for Taylor series to show that: | Useful information so you can avoid doing long calculations by hand : 18 < (1.9)? < 4, 6 < (1.9)3 <7, 4< (2.1)? <5, 9 < (2.1)3 < 10. ~l 10 (10 pts). Evaluate the following limit: AMeb eS PN The 7 tiharany | eee) ce ce et Url =3i44j, 7 = 314 2j, ofl = -f4j. /| ° : flew , J» Poa) rd > B i B lo) ct Bb @ ot R iH & o Il ” + v and y = y(s,t) = st? ~ 4. Find the value of 9 \ 2 \ “12 ()# pts!. We are given the function f(z,y,z) = 2° -2z+-zy, and we restrict (2,y,2) to lie in the solid cylinder D: 27+? < 2 (and z arbitrary). At what point(s) of D does f attain its minimum value? : (Hints: (i) remember to check the interior and the boundary of D, (ii) remember that we are in three dimensions, (iii) do not try to do the second derivative test — we didn’t cover it in class for three dimensions!)