MATH 101 Exam 1, Spring 2012

Qoluton American University of Beirut Mrs. M. Itani Hatab. Mathematics Department Spring Semester 2011-2012 Ve rs . 0" Math 101 Quiz I ee Name: ID#: o Circle your problem solving section: Section 1: M @8:00 Section2: M@9:00 Section3: M @ 11:00 o Answer table for Part I or Q b d Cc lb fa ce d iy ee pel Grade of Part I # of correct answers ; ----------------------- ~ 30% # of wrong answers : ------------------------- Grade of Part II ri 1. 2. 3. 4, 5. Final Grade 70% Part I (30%) 10 multiple choices with 3 points for each correct answer and -0.5 penalty on each wrong answer. Circle the correct answer then, copy _your answers as a, b,c or d on the table provided on page 1: 1. The graph of the function is the graph of F (x)= Vx-1 +3 is the graph of f(x) =Vx Shifted : a. 1 unit to the right and 3 units down b. 1 unit to the left and 3 units down c. 1 unit to the right and 3 units up d. 1 unit to the left and 3 units up x-1 . tf 2. Let f(x)= 3yx +1 and g(x) = 3 then domain of | + (x) = & a. (1;-+00) b.[0s1)U(1;+00 ) c. [1;-+00) d. [0;1]U[1;+00 ) eee - wovrroonsasasesesccconsasernnnnanaaancccoaaaesecconananeneneeninananecnans 3. The range of f(x) =3+ 7 is xo +5 a. (3,400) b. (00,3) c. [3,4) d. (3,4) 7x? —2x? 43x45 4 1 7 = x30 5x” -6x-2 a. = b. c.0 d as 5 2 5. The function /() = xcosx ;, a. a polynomial b. aneven function c. a cubic function d. an odd function 6. The function f(x) is represented graphically as: a F(x) is not continuous at x = 2 because : a. lim F(x) doesn’t exist b. lim f()=f(2) c. it is not defined at 2 d. f(Q)= —s If f(x) = Vx and g(x)= i then: (Answer questions 7 and 8 ) 7. f°g(10)= x- a 3 b. 9 a a4 9 3 8. the domain of the function go f(x) is: a. (0,1]U[1;+0) b.[0,1)U(1;-400 ) c. [1;+<0 ) d. (1;-+00) 9. If tanx=1 vxe[n, | then S€CxX= a. = b. x c M5 d. — 5 2 2 5 Part II (70%) Answer each of the following questions. Explain and show your work. 1. a) ra the given in terms cosx and sinx sin —x)+cos(x + = = we Cosme — yee 4+ (OS% ve Sinn Sin = — Wom — Sins (4%) l-cosx | sinx sinx 1+cosx re 42 jJ-twosn _ Sine — 4_— Wsx 14 c05%) Sine & Sinw 1+ losx Sinx Cig tosx ) . (4. wos) — din x o j= Sinx Cl + esx) rs) _ mo o\ nx _ a) @ _—_——_ eran _ Binx CV 405%) b) Show that: =0 (4%) iV it} c) Show that cos? x = 5a + cos 2x) and then find the value of cos? ~ Cos. + Bin = | Qe ax. (© Cokx _— daw = COS (G cose 7 os cos x. = 4 - COS2XL. G -i (C+ coo 2x ) * & (4%) 6 2 Cos & 2 W.il 44s a) * Cos 3 C a Ky - 4 (A + cos IL) =+t 5 Ong) = a (7%) 2. Sketch the graph of the functions f(x) = 1 xt+l x+1 vx+2 if x<-l if-1<x<0 if x20 3. Assume that for all values of x near 0 the function f(x) satisfies: (8%) cot 4xsin2x < f(x)< sal 2 inf» + =)) Use the sandwich theorem to find lim f(x) x0 P Sinan Dn cote site = Dime x — Lan Ww “zw 30 oe as 7 bem UREN u s\ r e5 l DO ws) fos ae ie Y/Y / Sx HN \ a= Qin Cot Hes SHV = ies Sin (= sin (+t)) = = 2 a = Qien Lx) = = by the Randwick Lirica Xo 4. For each of the following find the limit (if it exists) Y -—4x-5 fact ‘* o£ ( \ Sin HGS) _ Dim Coe) (4%) - m Cir X25 «x \ ve 3 eo (4 aide e—-0 A. As =\1 . sin 3x = -u c) lim - x— 0 sin 2x \ (5%) X30 nm (A (5%) Gui - (+5) G | (4%) & ) — bm VS VS oe \UENG ate (re) Aw _Badetxn CO ~ x0 Sint ne Ces +V3 wo) elim v3- v3 cosx x—30 sin? x (5%) Re _»S im Y3 C4- cosx ) a) A_ cog x =m 3G-e8) 0 xo Sine Ces +N cos) - ty, apt te x— 0 gh (VE +s ex ) Oru ein HY | Pac = | _ Boe _ =f x20 ae r mur 2. 5. consider the function f(x) = 20 (15%) x° -2x if -l1<x<0 if O<x<2 if 2<x<5 if x=2 a) For what values of x is f(x) continuous ?(write the answer in interval form) b) If f(x) is not continuous at an interior point c determine whether discontinuity at c is removable, ifnot find the continuous extension of f atx=c if possible? (Explain) («) cont: at ~ =-) 2-3-5 . Qin (3 -2x) =5 x->-17 ~2(o)=0 Qin 3-2%= > Lee FRY XI = «an at =0 x->o A Qiw 2.) clues} exist we HG) ts tt ato at nae (A Pay wm wot WV xe Ee) 0G) 20Gs) Qpadireatn abr X=O is not rmovable 4 bec Bex) look} exile X50 v discon ab x= TS removable bee. Dion Plale 4 a Aexey @ onel She cont. extension Foe 3 meets uy “ye pa exc