MATH 101 Exam 1, Spring 2012

American University of Beirut Mrs. M. Itani Hatab Mathematics Department Spring Semester 2011-2012 Vi } Math 101 Quiz I erSsion 2 Name: ID#: Circle your problem solving section: Section 1: M @8:00 Section2: M @ 9:00 Section3: M @11:00 Answer table for Part I # of correct answers : ----------------------- Grade of Part I # of wrong answers : ------------------------- 30% Grade of Part II Final Grade * > 3. a 5. 10% 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) =V/x+1 —3 isthe graph of f(x) = Vx Shifted : a. | 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 . f 2. Let f(x)=3Vx +1 and g(x) = 3 then domain of | — (x) 7 & a. (1;+00) b. [0;1]U[1;-+00 ) c. [1;+0) d. [0;1)U (1;-+00) aes asa ssa oes seeeeeeeeeoecs - eee 3. The range of f(x) =3+— is x +5 a. [3,4) b, (-00,3) c. (3,400) a. (3,4) . 5x° —2x? 43x47 4.) lim ———————_-= x0 2x" -6x-5 a b. c.0 4 = 5 2 5. The function /(*) = xsinx j,. a. a polynomial b. aneven function c. a cubic function d. an odd function 6. The function f(x) is represented graphically as: J (x) is not continuous at x = 2 because : a lim f(H=f(2) b. lima F(x) doesn’t exist c. it is not defined at 2 d. f(2)=1 Iff(x)=vVx and g(x)= then: (Answer questions 7and 8 ) x- 7. fe g(l0)= 1 1 . ; _— d.— a. 3 b. 9 c. 3 9 8. the domain of the function g° f (x) is: a. [0,1)U(1;+0) b. (0,1 JU 1;4+20) c. [1;+00) d. (1;+00) 1 | = _ 9. If tanx=— , xe] 2, —| then SCCX= ae 2 a2 » 5 5 a 5 2 2 V5 Part II ( 70%) Answer each of the following questions. Explain and show your work. 1. a) Express the given in terms cosx and sinx cos (= —x)+sin(x+ 9 = = Cos hws Sinai sian + Sinn lsh + Wose of oO -| (4%) o ! = —dSinx + Cosx b) Show that: P98 =a 0 sinx 1+cosx — — Sov y+ osx Sine, C4A+¢0x ) A— wsx — Dan @ ~ tose) (14 tox) — Sif (4%) (4- ton) — Sit oe Sinx (4 + Cos) . 2 Sina Bin x = oS Sinn CA + Cos) a) \) c) Show that cos? x = 5a + cos 2x) and then find the value of cos? 2 Cost + tinnx =! (4%) es hel fa = A +Ccosax 1 if x<-2 x+2 2. Sketch the graph of the functions f(x) ={ x+1 if -2<x<0 vx+3 if x20 (7%) (8%) * 3. Assume that for all values of x near 0 the function f(x) satisfies: cot 4xsin 2x < f(x)< sn 2 inl + )) Use the sandwich theorem to find lim f(x) x0 nan A on Qaim Cot tx Sine = Dien cane rl x0 xz 0 . Ss. % Gn , asin YN _— “x90 UNpnt, 2 2. 24 u s\P es %*>0 6 a \ - sie (I S\ = \ : , — el beg eee ae R (x) the gandwich Hou» = Yin 4 Jet by X30 4. For each of the following find the limit (if it exists) lim x +4x—5 oO. ras v+5x 0 Qercr” a = ™ r—2 x? - 5x +6 x32 G2) (3) Ve (5%) peu _ i sind Qian! os ake qe © x@™ x3 &-—20 =e 41) ££ api | sin 2x ee sin 3x on re (5%) Tl y vf I bn 5 (5%) % a (nr 45) a S/S 4-367 (ne +6) _ (65) O_mn = ees | G+6 V2 (5%) ~~ 5-5 cosx V5 COS xX e) lim x—>0 sin? x = bie (e-em) (05 ve os) x © — Yim x-3 0 = dim X30 = lim TL = Qi mn sae — x X—>0 =i sin? Crs 4+V5 Cos x) 5S cos * w —__ 7 Sintac (VEN C08 5 (1—¢oe 2x) an(vs +V5 (t Usx) m 5 See aE) BAe (VES Wy) 2 = (1-6) £3 ns) aca 3-5x if -l1<x<0 a if 0<x<2 5. consider the function f(x) = 3 x = if 2<x<5 (15%) *~ 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, if not find the continuous extension of f atx=c if possible? (Explain) . C1) 2345-8 . - <=> Gen $y) adm 3-3% = % ,. "yy it x -17 _ ~ BA tx) -) = - Ha) = & ok =o g-5xe 3 HL RHE vot ; Qe Tee) Se Yor xc: =O xA0 ey x>0 xe oT =O . QE) fs mot Coot - at xs ; Nine of ° f(s) = unde ad X= 5 2 DA) is Cont: (6) y- Vs continu at c= Oo is not mr a bo. Qs any (x) doesn} exist. Oe x0 k Discont. ab % =O FS Temovable bee Ds on 4 (x) = $ ex\ sts d 2) We Cont. exhension F(x) = x? oS «EF 2) Xe 16x65 Xe