MTH 201 Final, Fall 2012

Lebanese American University — Byblos Calculus Ill - Best? Final Exan Name: Fall 2012 Date: 25/01/2013 Duration: 2h 15 1. (5 points) Evaluate the integral J = fxe" ax ‘8 ew Ax 2. (20 points) Study the convergence of these improper integrals without calculating them, then find their values when they converge: 7-2 , 1 x°(x? +4) Linn j Xe CHEY LC 4>6 a.) by GOT az“ ~ Ah tf \ i n Meh Saas 2S deme both onerge ord }! eersya_ & tate yea) ez4¥o! tf Can eS 5 %* Dover ge => SHR a , Let ux? atPeuy alata A i =a@t ute { = [ _ = UA ata+y) --_ 3 ~ “4 k™ 4 Cara 4) , 2 if \ ax _ ) aa ' tea) ten Ca Re YG) bb i _ A = Bey Simm Yee \ b = W "ay ae tf WAG bare 4e ~F Ce sy J = lm -t— _~ tins . b 2+) 4b gt z)+¢ <5 bon - 8G) te pee ain 400 1 we +e* +] 3 1 3. (6 points) Determine if the series > Nand CONVErges or not. mi (1+ n°) tan” (n) ae z wat Ciset) toa! a lok £&K) new of Sy becuse > 18") toa (x) nay > £CA) Se) £vx cont VA posi he fe AZO (> ec x2) 2 iaregal ‘test. “=f CY) Ee a art 2 EMO) both onveye Or \ ne Hherge 4. (20 points) Determine if each of the following series are absolutely convergent, conditionally convergent or divergent. 2, EOD m 3” n! f) )&* e | YUrome Q< n (nt < Aye Mion Me ° ° nt@ - vary < ¥ Cat € ~ unce b. 3 Cl ee or / A Un Anti o Lekbiez = werram =) cmH vegemt v, j nazi Aten) he Sin Wate VACnAY Vien a 4 _ a4 Pp oooo—< rt “ _ n , 4. n—2 +f Alab yD) D van haconan € sesias alrvesgen\ : ow, wy Litgnck f? nt € ote - ~ ace Vlas) oat, Ssecies i> nek& abso lute l+ Cankigen’& we (-1)"vn ° x In(n) oo VA 7 Bien (Oni 2 Airs = + Db sO. Nm ef Ap Bn A , EO 32 Bn U5 2 SEV 4 a ak = diew On zo =) S(-1) OVA ; 2 a7 A—3— A “4a™ Lian LkOrgg, 5. (10 points) Find the interval of convergence and the radius of convergence of the power series) “(x —3)" n=l lian | gnt CA 3)" ; ot n>+P \Cnt1)* 2rCn-3y = kim 2.07 Cd) | NBFP | Caniye = We a not Ux%e7\ Cana ye = A\ke3\ <} +2 ¥ ~ te oe FB az Cy loy LE (bare thes ern sovies — Cenvesfe® _ for ha te Ss. — e-series nite G nee 0 Pnages \¥=4 5 Boe Be Mie 1 6. (9 points) Use Maclaurin series to estimate f 1~cosx dx with an error of magnitude 0 x? less than107°. & A ou a x x6 AP C “A - / cosAzA = tL— + qi” 67 € &) r An) i * 4 72 _ ~ 7~e _ xt x as fee Ba eT & 2 6 \owsx _ 4 47 44% x8 ye x= — Zz a! f Xl fem __intescahan: ‘ ; 3 A tna ne &! Ad at _ xf , x . 4 COs ) x pee Fae a OSs A? ‘ : ; £3) g | 3 5 at A -— x + _ = 2 Bla * Bed RF Sxtol b, \ } \ _ \ + \ - = 3 Bx¥) "Sy + Bl 3 vlot “ : -¢ game B54xlo S Feo” “<le | \ \ | —Cos x _- ~ (2) So a > 4 agr taper 7 Sra e . Al- etcor Sy \ = 3 eSrlo ® ae lo ¢ . % «ld 4 2,2 y~x*Y x+y! 7. (10 points) Let f(x, y) = Determine if the limit of fas (x, y) — (0,0) exists or not. Lek (42 %4> or yet Liew FAM) = him v= Cuyy'y" Caylee) FRAT. Cayyt+ 9 4 BMY 2. YW 4 = Wm yiak Ye ey ay4 7 bin YC -«*) - a 8. (10 points) Find the volume under the parabolic cylinder z = x? whose base is the region enclosed by the parabola y = 6 —x” and the line y=xinthe xy-plane. a] =