MTH 201 Final, Spring 2009

1. (8%) Determine if the following improper integrals converge or diverge: co a a Ora [fs mM Veg, : p-serz, p >/ (b) [a dx a (x de poy on I ne) ~~ 20 con (ede la co ege, 2 2. (6%) Determine whether the following sequences{an} converge or diverge: sinn - id (@) an = In(n + 1)! ~ bad —z? Oo. oO / ts ro) - 2 ( > (b) an = ncos(1/n) ~n wl, —> .@O 3. (16%) Determine whether the followi determine the type of convergence (a) yey Qn = nA —_-_7 / — ng series converge or diverge: In case of a converging alternating serics, BS xo _ | < Ala-3)< | => Zz x > (heh end yt an ; ayy! a Sten 71 Z sme “arty Aan “= (anv. fn 5h S% <7l2 5. (7%) Find the McLaurin serles Tor f(z) ae): Cees Sree LE. 14€ é 2* ( 2 ty x ~ =H p.. LQ, (+ at) x =—- t+ = " En( (447) ~ 2? a a? 7 ~ Ff =f ; t 6. (7% + 3) _ 3 ) Evaluate, using power series the limit below: lim sin(2)~= sn€z t—t7! tLe -€%) t-. ‘ o\ 4 ts a a OS Po ag ath ~ 7 Alt q yy --. we (9) _ve ~ “/gl t xe] - e/,/ r 3 3 6 enter — 37 t2. 4 9. (6%) Find the following limit, if it exists: > (a) lim ey : Tr x J = Kw (2 2,0)=(0,0) 208 + yf ~ J 1 b. Z a. Ao _ A+ ke “pe om & 3 bait does wh excot 10. (16%) Evaluate the integrals: (a) / / ye® dxdy. (Hint: reverse the order of integration.) i) y/2 n= H/o og Y <2 Lp Vize ay (b) / / 7 dydz.(Hint: use polar coordinates) -1J 0 (x? + y?) seme wy . recent ® sin® r drdé- ct 0 O a [((u?6 me drdvO Lm... Le dn. oO oO 11. (6%) Discuss in a short paragraph, usin & your own words, the relation between the following: fe(1,2) , fy(1,2) , the surface (S):z= f(z,y) and the planes Y= 2,and2=1. by & fa a a one B= (Cu Y) , hoy we cab (5) ley Le flrxe Jo a Jnr longat p He wae GO) Dy? Q Caley Some fa eb ly yu phe wer fy Cr lye Qo