# Jsee x dx = In Isec x + tanxl + C Jcsc x dx = - In I cscx + cotxl + C

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1 MAC 2312 Final Exam Review Instructions: The Final Exam will consist of 15 questions plus a bonus problem. All questions will be multiple choice, which will be graded partly on whether or not you circle the correct answer and partly on your work. So be sure to show your work for all the questions. The point total and breakdown is listed after each question. The will be no bonus problem on this exam. A scientific calculator may he used but no graphing calculators or calculators on any device (cell phone, ipod, etc.) which can be used for any ot her purpose. You may print the unit eircle and trigonometric identities that I have posted on my website, hut they cannot have any writing on them at all. Additionally, you wi ll be giveu the following formulas on the test: Jsee x dx = In Isec x + tanxl + C Jcsc x dx = - In I cscx + cotxl + C J (U) du = -tan 1 - I - + C u 2 + a 2 a a n! lim - = 0 n_oo nn lim ;;In = 1 n -->oo n L ] - r l- r i =O 1= 1 n i a (I - rn+l ) L i- 1 a (I - rn) ar = or at = ----c-----' Ix - CI"+1 Taylor's Inequality: IRnI ::; (n + 1)! [max lj<n+ I)(z)ll The exam will be similar to t his review, although the nnmbers and fu nctions may he different so the steps and details (and hence the answers) may work out different. But the ideas and concepts will be the same.

2 (1) Integrate the function. (10 points - 5 points for the answer and 5 points for the steps) Jxe- " dec (a) xe- ' - e- x + _ xe- x - e- x + C (c) xe- x + e- x + C (d) - xc' + c- x + C (e) None of the above - )( -x f' ::.. - l(~ - (. tv

3 (2) Evaluate the integral. (10 points - 5 points for the answer and 5 points for the steps) l' sins x COS 2 x dx (b) 1 ~5 (e) ':;;5 (d) 0 - f T'5ln 'X. S',W'I'( )(. COS7.-X ctx o ~ ('3\~1.xl = (l-to?l<y -:: )011" 'f>it\'w:: ( l-ur,l.xyc.do:. 1 XcMt \A" tq~~ ~ t{u ~ - ~lr\xj.o,c ~ - ~ (HA"1.\'\..'l. au :: -) [1-2u 1 ~\.I.'I )lj.'1 ~v. =... r(il - 2\.1 '( +UIo)J""

4 (3) Integrate the function. (10 points ~ 5 points for the answer and 5 points for the steps) -:--=== dx J 1 X 2 V X 2 +4 (a) ~~+C (b) 4. + C ~ ( c ) - "'.'H + C (d) ix'h + C >/:c'2+4 7x~ 4x 4:t: V ~ (7 :: ~ -:::?;> X":: 2~e ~l\."l. ::. ~ ~Z.e -::?; ou< :: L<;tc,'f1-Je '5tc e ~ Jx:t"l ~ ~ It:t~ :::.;tst(.9 M ~ r I. 7 StL~JJj :: ~ S~ ( e- d.. It J ~-htti"t.e ~ *'~ IJ. ~si" \1 ~u -:otp; \} 49 I ( e - - 'ts'ivl.e - '1 '.: \ ~ -"t{ X

5 (4) Evaluate the integral. (10 points - 5 points for the answer and 5 points for the steps) 4 ~x_- _9 _ dx 3 X Z + 3x In 1~18 (b) In ~ (e) In (d) III ~ (e) None of the x- 'f 5~ ~ == J." x.- 9 k & X... -\-~'x -\0 ;3 (l(~s)(x - ~) 'A (It- l) f,&x-ts) x-9 A ~ - - ~+S" X-d.- (I\-t-S)( ~-J-) t - ('(i's)(\-c).) A('(-~) -\- D('(1-5)::- ),- 4?H.k X::: J: 15::- 1 ~~ =- -1 Plcl x ~- s : <~ A -:.- l'" ~A- ~ J. above '\ x- ~ r" \ 1 J.A-I'I I ~ ( )( d., M: =J J 'K +" ~ 3 )\- J. "&,(+':) ~- "3 -:. ;).\'" \ ~+\$ \ - \VI \ x-~\ \: ~ \V\ \ l:~~\r ~ \_ 11~~n- \:\(~~\ ~ l.\ ; \-1-\ ':\,,\-\~ \_ \. \~~\ -= \.( ~~),IJ J

6 (5) Evaluate the improper integral or state that it is divergent. (10 points - 5 points for the answer and 5 points for the steps) 1= _(X-22-: -2"")2 dx (a) Divergent (b) ~ (d) 0 (e) None of the above --

7 (6) Using the integral test, determine whether the series converges or diverges. (10 points - 4 points for the answer and 6 points for the steps) BConvergent (b) Divergent I.A.: '"')< ~ 'f..::"<' u d",-= *~ ~ h, ::.- ~:: J-t. ~~ em ":' Je~ u cau ::: Ju e-"1 Jv.

8 (7) Using the comparison test or limit comparison test, determine whether the series converges or diverges. (5 points each - 2 points for the answer and 3 points for the st eps) (i) G1 Convergent (b) Divergent _I.,... - \;"" 00 3n.L 2 n - 3 n= l (a) Convergent e OiVergent V\ "3 &L. 'L... -~ s1, ~(...wr ~~IW {\vtil'tp -0'" DO -;\ _ "» no;., )

9 (8) Determine whether the series is absolutely convergent, condit ionally convergent, or divergent. (10 points - 4 points for the answer and 6 points for the steps) 00 1 " -e _<. ri~ L-::: 1>0 L -\ L VI~ e t\~ '(i<>l.. f e \ vi" (a) Divergent Absolutely convergent( c) Conditionally convergent VI""'l ~ r=-1. V-. ~w- l~::3'71) i~ 5~ ~ Gl. CHI~ SU\t'i. So ',+ ~ JSiJ

10 (9) Using the rat io test, determine whether the series converges or diverges. (10 points - 4 points for the answer and 6 points for the steph) 00 L e- nn! n=l (a) Convergent 90ivergent -" \ J I'. e f\. "1" li~ n... ".,... \1'"" V\~00 e"' ~~0 ~ (\+' \ e ",. t' (~>.1\ 1 }7e -/ ~;rt\ \'\-'100 ti- \ e _CO ~1)~

11 (10) Find the interval of convergence. (10 points ~ 5 points for the answer and 5 p oints for the steps) f (- 1)n(2x + 3)n n='l, n In n (a) [~5,1 1 (b) [-2,-1) (c) (-2,-1) (d) (-5,11 ~ ' o n e of t he U bove (z.m3)"'" (...,) ('" (...,) ~(2~+3) ~ l~~ ~("'~) t"(i\~\ \ - '(\ \,,.., -' ::- \l:x.t:\ )\-,,00 I'\~' \..v- 1 \i,.., \ ~ ~ \ \iv"i \ 2~~)\ - \ 11lC.nl --!.-. h-"" '" I'" ~ ~ L\~i+J') ~(.1. _ L. k.n,,-"0! C/{OifitJ's fj.;1( I \ I."", Ih~31 v \ - _'-\ L 1: (.- "2.. - \. "Z L')( L - \

12 (ll) Using Taylor's Formula, find the Maclaurin series for the function. Express your answer in summation notation. (10 points - 5 points for the answer and 5 points for the steps) f(x ) = (1 + xt" 00 (-l)n(n + 2)(n + l )x" (b) ~ 2 n =: l (e) None of the above ":> ;I. I. ~ lo) :: {l-kjy ::. I -:> 2 {"(Xl:: - 3( I~Y'l r" (K) '; - 3'1{') (I-nY'" s\ r \ (Ol -=- 3 ' ~ S(I+Or"' :. -3 I.{ S ~- ~ ttl<' (Xl -:. "~S ' I..( IrIC) +.f(~\,"::, 3 ' ~ ' )'(',( I-H)Y\ s 'H <,,-=

13 (12) Find the power series representation for the function. (10 points - 5 points for the answer and 5 points for the steps) x 3 f(x) =,,2 _ 5 00 (_1 )n+j. x 2n+3 00 x2n+3 (b) L 5 n +1 (e) L 5 n + 1 n = Q n = O (e) None of the above ~> -- -s ()o -'L

14 (13) Solve the differential equation. (10 points - 5 points for the answer and 5 points for the steps) dy dx = y2 sin ct (c) y = 1_ (a) y = - cosx + C (dl y = co~x + C (e) None of the above ~ tkj Q.I<. V( ) r-""' ~, :: J,&,"'1( ~ J t ()'" 'X. -+- c.. \ ~~ (If> k. +C

15 (14) Solve the dilierential equation. (10 points - 5 points for t.he answer am I 5 points for t he steps) 1 dy x y = 6.1: xdx. y = 2e 2x3 + Ce x3 X (a) (b) y = 2 + Ce x3 (rj U= 2+ ce- ' (d) '!I = 2e- 2,,3 + Ce- x 3 (e) None of the above t X J ~"X) ~ ~ +3).1.\j -= (0,,1. c}j<. '""""J?{J.) J'P()(loVK =j3l~ = ",1 '("!J fl~'r= e ~ } "!, e 7t. -t.. Au 3'l.'4 /1.-X ~ e j:: \Dx. e (e... 1 'j)' ~ Co \"- l':?' r(~"~1)rjn.:: Jbli~ok u:= X:~ ~ ala =?>\~ ~ 2Jv:::lDlk -:::2.J/'d.u -= 2i":C =: 2e" -+C

16 (15) Find the length of the curve on the given interval. (1 0 points ~ 5 points for the answer and 5 points for the steps) x 5 1 Y =6 +lox 3, 1.<Sx.<S2 (b) ;;~ (e) ;~! (el) ~~ (e) None of the above L~r (-fk~+ (O~r t4, fl(s" '1 3 ~I. ~ J, ~X + loll" )CJV'K. I )"2. to):.!. \ l~e-o -- ~,-\O ~-ac. ~ (ii.,,\o

17 Scratch Paper lj.::' \" l< d.m -;-); \ ol«j r)..x ":- S-& ~::: \" 1...1:: 1",0" )() K.\.. " ~~:1 t - { ', ~ ::: I,~ I~ (I... ' l/{ ~ ~ ':.~(ldh'(i.l) "l1>~ 1 X"X -\.-., L~

18 Seratch Paper

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