MAT 113 Final with Solution, Fall 2007

NO" Vie IVERSTTY DATE: FRIDAY FEBRUARY 1/ 2008 1-(8 points) Find the following limits: o-Ve—s _ 2-N 9-5 ae Spt a) lim - r9-3 x43 - =3+% = ar VE es Vb Wits) = be a 2, a> Qa Sees Xs -3 6x43) 24Uxis ) ong “ss ? 2 =X eK @-0G yp ~*~ Cx4 3M 24S x ES ) XK 3-3 CPN 24 SES) : (1) = a SX es 3 -¢€-3) =a 6 [=2 6 ~~” if sf = = 2Nxrts a+Vo-s 22 4 ZB wie <% ae — | ' W) fim SEIS 545-15 0 cD a gH ap Cx +5) OB) + a - BS C45) K-33 (2 -x) exe X45 CA (30) x-5 2-(8 points) Let f(x) = (8 points) Let f) 335-10 a) At which points does f (x) fail to be continuous? b) At which points, if any, is the discontinuity remov extend the function to become continuous there. - = able? Give reasons, and then soly a) om . - ; KS Cx-5) (X+2) Soy ‘ C Gh < . a So eB t end K=5 [2 4 ieee Seer cafrrwel " be Vid a de fan oo) | SS GERTA es be ache cl (fp —— . oO” be become Cott x =-2- bin Cre = o 2 = \ = A —— —_—_- -—e———- XS ° M55 Xte F (se 1 Gre Gan oxtard on te Caw cod WAY. = 5, and the actincs felon Fos Coen X45 \ . = 9 K=$ Hes accn hip REE Vy ap 7 C2 xt 4x-l0 XAG ~ - LL 3 fe ) cok at ? 78 X=5 3-(13 points) Verify that the point A (-1, 3) is on the curve: xy” =9 and then find a) The equation of the line that is tangent to the curve at A. rl eeers “eo EIt(s)ts9 ee Core Oy" r ot, wre we Be spl ot ilfuatision W\b a: = =o a, 29 4 "(x = a 7 =) Me a 2xg* > ne, at = a AS rey nan { -B-3 - @ JRE Slane « at 4-(16 points) Find “ for each of the following functions: a) y =+/tan(x’ +3) ay \ _ $60°C x*x3) (2x) - ¥ sec. CX 43) “ : banvt42) (yt dx 2 4 Ae _% E Xx ) 2 BF c(h) hale) = — Se cx ate “AX ae harry 5-(15 points) Consider the function f(x) = _x? —2x +3 on the interval [- 4 40]. a) Find the absolute extremes of f(x) on [- 4 50]. Cen vA Cok-an clored « Whine A be, oJ =7 fs extremes at ~cncticalloucls ov enel rt D re = 2x-2 20 axe) C1, Pene 4 ) of or ‘ Sc Crrticed H-Uu): “1 x ! 4 (- an ae kes) Of (© AS i ) = ( osm) b) Sketch f(x) on [4,0], then find the curve of f(x) and the X-axis over the interval [- 4,0 i wo 1-5 1° Zyl ig (2) i 3 Oo (2 IG Az fata (* 2 a xxs)AK ° mrt j 6-(20 points) Evaluate each of the following integrals: = TL 44 ws +sinx 4 q af cos? x -\ A as Sax Ax ° o On a CO Gig lh beng pl = Voara seer . ts : e Drea Ate" -d) x aPRela lekucyt : acy =) du s2xdx | letuzx a5du aa x Av ce? Ux au: 03k : i AN 20° a> Vee ~*~ z L $X K xX ot -2 GPP \' 2 *Ax | 3 ip f 2 3x 5x La is ree >” Ze \e Ax > * SX gx BREN 2x Es 4 C 9 <a 2 ; 3X 2x wa oie eye +C -« 2 27 7-(5 points) Find fQ) 7 j f(t)at= csc(22x). ( S. Carat) (ac QT )) = =) Cir = -Cre (27K) coh(enx) (20) ee a 2-2 gngel27rXx).: Cok rx). we l<x<0 x ,0<x<l 8-(8 points) Given f(x)= a) Is f(x) continuous at x=0.(explain) b) Is f(x) differentiable at x=0.(explain) a) Q —_ ( hetfee - - Cro se) ot Co) = —(oc) sane te - how or Coot) eae (o> = ot C Cet) s_(e)t = 4 oo Roo cot of x20 =- 0 =0 Wy ParS2et exh -2tX Vox —\. Cee 2ia§Ge on ae a ‘ Po = Cory = F Cot) C CS) Q2o*) = 07 Cra) excwk 9-(7 points) Find the positions of the local extreme values of the function f(x) = j 9-1? dt. (Without finding f(x) ) Fnce Con B + dafpreoh On an open be lined Eas 2) = focck tabiine ort wt entree a Le 9.~xX* ten5 (3x) (B4 x) 2 0 =) v- x". % SASS \ FFF: » me - Hane u Pc HK @em_¥. a gee! Max at K = 3