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Fall 2010 MT1 Solutions

Fall 2010 MT1 Solutions - Math 3 Midterm 1 Instructor...

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Unformatted text preview: Math 3 Midterm 1 - 10/22/10 Instructor : Nandini Bhattacharya Use of books, notes and graphing utilities is not permitted on this exam. ' Show all your work to get partial credit. Your Name: BOWHW-KQ% ............ Student id# -------------------------------------------------- Your TA -------------------------------------------------- Max Possible Your score - Problem 1: 20 Problem 2: 2O 7 Problem 3: 15 Problem 4: 15 Problem 5: 20 Problem 6: 10 TOTAL: ' I100 1. (20 points) Give a decent sketch of the graph of the following functions. Please include all the information indicated under each part and don’t forget to carefully graph the functions at those points. a)(10 points) f(x) : —3x2 +12x x-intercept(s): @ 3 "*3 X11" [l X 0 K V: O 2:. #3 (20 (wt) ”2-71 "#1)“le __--[;l;t/2 y-intercept: 1/ ~ .._ , ......_—-——-— X20 - . é ”5 .._. ml’rl , 9,2 «3 (of 212(2) :2 Q -- :7: Li 2% ,5 ~63 Coordinates of the vertex By completing the square: ,2 K; 441% 3-": ,3 (Kg/W3 i [.2 ~42 2. ,3 (Kim-w) H1 _’ ._ - ,.- 9‘ V (@2251! 4033242 .._ 3 (a 2) $1.2 'd I Var/xix .2162! [95 Check your answer from above by using the vertex formula: 42 ..._z__2 (-3” ’5 ~31 —3(2) H 6.261): swim .. 2/2232: 2:1, Axis of symmetry VE/F'F’C‘ibl {r‘flc’x ”trawl/t VéI/I‘TLOC; X :1 l P 2. (20 points) Consider the function fix): (x—3)—1 a) Find the domain and range of f(x) {Isazamppmrwe so W+l> o l ~-l RSfl—hwE _ b) Show that the function f(x ) has an inverse. Hint: You can show this either algebraically or graphically. ' 7 Passe/5 'AQFEZO/vtm/ [mg fias‘f‘. c) Find f’1(x) _ ( ~9~ a, m W 201L043 «— y I, w oils» m ,— fig ' :11’y,3--{1C(K> d) Graph both f(x) and f"(x) on the same coordinate system and explain how the graphs are related. may (we, fa‘pzreclliflflfi across fix 9) Find the domain and range of f"1(x) . Hint: You can figure out the domain and range of f"1(x) knowing that of f(x). 3. (15 points) A projectile is thrown upward with an initial velocity of t76tt/sec. After t sec, its height h(t) above the ground is given h(t) = —16t2 +176t. a) Find the projectile’s height above the ground in 2 sec. A (a) r: ”(A (mgr-1% (2.) 2 fiat-4,351: 288‘ b) Sketch the graph modeling the projectile’s height by labeling the x and y intercepts. ~15 {2 +l ea: vléfé} (can X 4 rVMLCfcey'YLS C 0 , H y- ijrarcefis: O c) What is the projectile’s maximum height? What is the value oft at this height? ‘Mé’erfl’lfn/‘VL (”fill {:9 a. '95?!" VE/(‘lLC/ifie ' air-<92 4.!— ' M 41—):-ze(§)i.lzmg):Ll-H d) How many seconds after it is thrown will the projectile strike the ground? ' [l gecmaésl 4. (15 points) a) Use the properties of logarithms to write the following expression - as a sum or difference of simple logarithmic terms. x i ”as w gimp/mam» 1n .. In ‘__._...,2 ...- x+5 ' 4-3 311:1: b) Use the properties of logarithms to write the following expression as a single term: 10g10(x2 _16) _ 310g10(x + 4) + Zloglox 99”th 4?) ~ iagtgt'wo f-{ojiJK} 0) True or False? Justify your answer to receive partial credits: 1 I . 1V=— '- Li“ 7 .._ , )‘Sfi'm' ' . ' . z) n 4 int—gjgm Inc "E!“aj'itg awe: ii) [M3 = 3lnx m iii) In2x3 = 31112;; Fake. 3/}; 1%: ZACH); 31m (Xxx) iv) The domain of g(x)= lnx IS the set of all real numbers Fa! §Cj qufl‘ ring-{:1 in o i; fit as.) mf‘llfé v) logac— - b implies ab — —c (CW/LE; Oiblcé-i? Zggmab‘: Eagflafifl‘) b: [aj~C 4x2+1 xz—l' 5. (20 points) Assume you are given the rational fonction f(x)= Find the following: a) The y—intercept: _ otwl ”" ”l “a b) The x-intercept(s): ‘ Lira-lino 4x1: '~—l (5": c) The equation of the vertical asymptote(s): d) The equation of the horizontal asymptote (if any): ‘l’ef @405 bgmm game: Jay rte. 1%; film e) Does the horizontal asymptote ever crosses the graph? 1‘) Sketch the graph ofi indicating all intercepts and asymptotes as found in a)-e). b) (5 points) f(x) = (x—3)(X+2)(x+1) x- intercept(s): ygo :7 24—— 3 :29 QM x.{»L::O and? ,X H ’20 y-intercept: 7 X” 3 2C 2 “Q” X 2 "if c) (5 points) f(x) = 2*X—2 x-intercept(s s): ~~~~~ > Equation of the asymptote: — ~2< Q {-120 22> at :19 ~x: I:> x34 y-intercept: 3 ‘0 4' y: 2 ~01: 2 *2 f: #2:: ---1 Graph: . I 1 I I F A; QK “-90! 24""249 ”l ‘90 yzi—Q 6. (10 points) in 2002, a newspaper company bought a new printing press for $60, 000. By 2005 the value of the press had depreciated to $42, 000. Find a linear function that model this depreciation and discuss the slope and y-intercept in context. (‘1 051 x 49959) , (163') 771, ’11' ”Loot-a) 53996:: ”[email protected]*é<9aec: "'iXQ-sofi 7,3300 1.991 1166);). g «~— » lit/Fifi. V— é 09639 :2 arm (x niece) >4: ~3Z‘aox 4- 134mm 71m: gigfa. is flog; rah: a g ciefléicéc,‘ 3-95.91}, Mai 3 ,V " (“3“ @163“ 5‘s 3136, (99th year gem; ...
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