MATH 101 Exam 1 with Solution, Fall 2011

AMERICAN UNIV. OF BEIRUT QUIZ-I-MATH.101 Oct. 29, 2011 (Material Covering Sec. 1.1-----Sec.3.2 of Thomas’ Calculus 12™ Ed.) LECTURERS:Dr. M. BRIGHT Mr. Z. KHACHADOURIAN NAME?.... RGU TOSS eee LDevsssssessessssssessessesssessseesees SECTION: KHACHADOURIAN 5. 6. 7. 8. BRIGHT 9. 10. 11. Question 1 2 3 4 5 6 7 8 TRUE | QUIZ I FALSE | GRADE Max. 6 6 16 |6 16 |6 12 8 24 100 Gr. x if x<2 1. (6 Points) Suppose f(x) = 4 3 if x=2 3x-3 if x>2 Is this function continuous at x = 2? Justify your response. To be carhneess , ust wud Quen Fix) ho vet moa be RO’ eay rl £(2) . We har = Lu flap = fie) = 4 Xo rd fn Ela) = ba (Sx-9) = 3 mt xT $e bala) dew ast eeab) © Cy cok Cohrecs ef 2 2. (3+3=6 Points) You are given the function: f(x) =5x+1+ =, a) _ Is there a vertical asymptote to the graph of this function? If yes, give its equation. Tey oa , ra S) b) Is there an oblique asymptote to the graph? If yes, give its equation. a 42 5x4) ’ 3. (5+3+3+5=16 Points) Given the function : y = f(x) ody #0. x a) Apply the formula: /'(x) = limp LO*9- LO) h to show that the derivative of the function at every point of its domain is -2 fia. x 1 . C'(a] : gen (xk) - ho k = on xt (92s labs lt) (py var | hoe at (x aLyt Common Lite © a ta -2x ah renter Re SAC aby oy ee b) Now find the equation of the tangent to the graph of the given function at x = 2. an Sloge eb x-2 0 -*hs = —Ye , Va farynt has Slepe rs ol dates Treo (2, “hex) Pur Yr-Re* c , __ (2, “tad f(x, 4) ad Salve he he = <'le 2) * ¢ oe al The Mngt a \ A= Vag 4 a\ i c) For what value(s) of x is the slope of ne ‘eaph equal to a) ? Wie nd {, salve CO)= ~8f t. ry, | = 12-27 > y2-3. 4 Ra Slag 4 AA ck 2-3) (od d) Assume now we have another function : g(x) = v¥4-x i) What are the values of (BG), (we AIHZ)s and (ge NH G f)(x)= 9( Cbd » J 4- Ios" gels 9 | - ol) CA) = a - Yay : [4- . se) = [Ao ou = (a = ci)(ta) = I~ Augy = M4 Bhs ii) What is the domain of the function 8° f ? O re) gk ‘a clehued whe wr uh - Ux = oO Pa) 42 xt &> Vu & xt La Ia é \c | G Dar clomar of ak iS (~ 0, -h | ¥ Ch, 2 ) 4. (6 Points) Assume that, for all values of x near 0, the unknown function g(x) satisfies the condition: Apply the sandwich theorem to calculate: lim[g(*)]. Tha Sandwrk Nig ree Seas og £(x) < 9 (x) < La] jcflal= Gobod=l, all © mes ©, od de Ai. (| 3 fy de “ 1 an Ae w(2r se ee 2 LR) a = Ll ad dan (A- se) . Ae ii (=) suet: 3. IG 29 K-90 Se bs Sedocl Mc rting wv oh diducg be g(t) = ™ TO® { Smee 1S SRE SH, we be “ke Sb sh, & 5. (16 Points) Evaluate the following limits: o£ ‘ro (38) : oO be he . 3+2t—cost / ees t etl Nawen. os Sirdeod a) lim sint=1+40" . : Pan CVE - cont Sunday pee — ro ark cok _ f- % 12- al Q | / io Ee fn = 2 = : 2 k-30 Ast - (24k fw wt Wt Ly . vx+2- V2 b) lin -—————_. x30 x me ve — ( fxr “(2 £ (ix el “2 Yo 42 e117 ) . i MAV-D wo a = KN —— r. SST YX nu wd x (daa 1c) won HUET 2) = haw ta = —_ we 47" av WE 2 °) lim? cos (42) Ma . ned eazy ge 2) ( i | _ 2 haw ieee = by eT = ae 5 #2 s oo58 S2t 29° Sat 2x) —x4 d) liim———_; . ; at ) x0 x7? +5x° -L Caw (2x) + Lew (x ) ob L 2x '- oa “4 4 2% +X _ KP ne = PE eh eet ie be) amie te Sa X~Id \ + 5x | + bm (bx ) 6. (6 Points) A function f(x) is assumed to be continuous over the closed interval [-4,4].The following table gives a selection of values of f x I(x) -4 5 -2 -6 0 -8 2 3 4 -10 Explain how you can make use of the intermediate value theorem, in order to locate roots lying between -4 and 4, of the equation: f(x) =0. How many such roots are guaranteed by the mentioned theorem? Give adequate justification. Peas, (la)--b Sma Go baba Sod 6, Ms Ankorudrbe Valve “Thictn sey) fhe Goal cef4-0] sul mt F(c) Oo. 4 L ffoje-& FAI 3 |B Sme Bees ce(Or) wh Mele. § faje3, Cle e-fo, Ge Sle Bern cee] ot Kee. fajeo % C44] All toge tha: , Ye muir be ct lem 3 cost ok 7. (12 Points) Fill in the empty boxes to match the four graphs of the functions on the left to the graphs of their derivatives on the right. Functions Derivatives — = ~e oS 1 ~ 8. (8 Points) a) i) Using the formula sin(4 + B) = sin Acos B+ cos Asin B, show that: sin(6 + 5 =cos0, forall 0. _ - « sw(6t 2) = gE, 4 wer Sm h am| = Cg € 4 |. as€ = as€ fo att & Po \ gia cl, _ ah = =O ii) What does this tell you about how the graphs of y = sin@ and y = cos@ are related? The greyl mt 4° us os fle Sanz grepl od = one S aikte d Gl by. u 2 b) Your friend mistakenly believes that sin(4+ B)=sin A+sin B forall A and B. i) Find a value of A anda value of B for which your friend’s formula does work, Tele ArQ7c Gu(Or1b) = Siler = G Siclshe SACO) - OF6 = O So hy (oomle weler , ii) Show how you would convince your friend that his/her formula is wrong. Gack yodwes f N\ ad fey At dicin't wove : ow Rrecingle , A-w= TW) Sin (A +8) < Se TT) ~ O lat sc Ad seSe SA ¢ SOT) 2 1th 2 Answer the following by circling T (for true) or F (for false). You win 3 points for each correct match. . . ‘ ' [ t() limeos(™* + sin(3x")) =1 Dx limk cos(Z) “Dp vi3 (r) F] Jf tanx= and x [0,51 , then secx = ag eltere xis measured in radians). setae ete = (3) = BAA RG eed Con 9 28 4 &OK Y fos (1) F] The range of the function y = f(x) =—3sin(2x) ;0<x ze sis [-3, 0]. mrs ao) Gach (tT (F} |6.1+[-3.5]]=4 ° Leit (-3 Sl\ = Lo + (-3)| = \3: iS (1) FI kim 5+x-3x° _3 se 2x3 4x? 47 2 . Loc |x? -9 “t x slelth lon tL. 3, A me , [T F] lim =-6 petal = 4-21? Ten [re Al (3-X3*x) _ 3), wo by bak | —— oe 1-3 xed fo i i a eos [ T () y= f(x) = 3x‘ —5 is anodd function. Acwel bo 8 _ (on When the graph of the function y =1+ =" is stretched horizontally bya factor of 5, we get the x graph of y=1 +3. oe are | Suloyhluby % fis x are _ 26 ‘\= —;%