Soln. Mid 1. Version2, 2-9-09

# Soln. Mid 1. Version2, 2-9-09 - Solidirn Math 3 Midterm...

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Unformatted text preview: Solidirn Math 3 Midterm 1 (version 2) 9—! Cl 1 09 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: Student id# ------------------------------------------------ -- Your TA and section time: ------------------------------------------------ -- Max Possible Your score Problem 1: 10 Problem 2: 10 Problem 3: 20 Problem 4: 20 +5 (EC) Problem 5: 10 Problem 6: 10 Problem 7: 10 TOTAL: 100 1. (10 points) Give a decent sketch of the graph of the following functions. Please include all the information indicated and don’t forget to carefully graph the functions at those points. f0”) 2 x2 — 225w 3 (i X 3W are x-interceptis): :21] mi y-intercept: X- 3'33 Coordinates of the vertex H h 5:" t" gm 3 u, M. x V.) i 5 W} By completing the square: a “' .i nvwWQx+1i> “W t g x: ijtiw- zi ‘ gs! cod 3in i» Vertex is at} 0) Mi Graph: Axis of symmetry: .x I: i 2. (10 points) Find the domain of the following functions. Write your answer using the interval notation. 8) f(x) = sz + 3.x —— 4 76+ 9% ii '2: O ( W i) J2 ‘ WW g J C36) 1%?!” t, .w‘. J on «it"s iw-Vuxt’”? clack-5:20 K w ’0 , Ail M PM ‘ b) g(x) = 111(4 — x2) 3- (20 poms) Assume you are given therational function_ _ , f(x)=w X ( w “+4; 30mm n g All reel __..«— 8x2 + 8x W”‘““””””’T"TW WWW (31% J X“ palm 7W “"3 Q“ hole @353 “'l ” ' ; Find the following: a) The y~intercept:( (kl “)5:- o) .‘l 01 t -.:- jl “Lu/tote l‘i heel b) The x—intercept(s): [all L3: 0) O Knit?“ .. t \ ~ m -: 2 5% :37. o U:ka it» ~ we) l c) The equation of the verticalmawsxrpptote(s): B} A ww.,,_w_....m.w..t,4“ d) The equation of the horizontal asymptote (if any): . I: X t . 71,1253” 0 ,w. i 3., :5" 1w] ‘ , Wm“ M Wig“? -»' f x, ' ta : X t X I? i If,“ /*> . .m Mg.p"“"‘ , 9) Does the horizontal ﬁsymptote ever crosses the graph? it ~ h“ ‘23 tut/w.” we - e ‘ > { l ﬁx ‘ e: s; "X - SE’X' “ O .. 1‘) Sketch the graph of f indicating all intercepts and asymptotes as ’” found in a)—e). 4. (20 points) Find the coordinates of the point P on the line 5,: 2x+1i closest to (4,0). a) (4 points) Draw a picture and label all quantities. - -7:thl [N b) (6 points) Express the distanced between P(x,y_) and (4,0) as a function of x alone. . _ if, . 1‘ “£252 ﬁt: ‘]'(Xr'xb)l+ {grim/)1» (X1 ) m: ( X I ‘ j )flv) r( f) > at: \l tx~4)‘+ We)" I ‘ yeast! W at: We)" r J x584x+2b+4xl+4m : W7 ‘ c) (6 points) Now find the value for x for which the distance is minimal and give the coordinates of P. Hint: You will be better off working with the Square d2 of the distance. ‘ ‘ 0L2: 5%é-4XH? eundx‘etﬁc ﬁmctian. . 46: I VET‘W Pom-t I W mmgmal ﬂzaxf‘: Z(%)+[a %.ti 7 i — ”¢ 4 4 :5 t . 5 9:5 2 , .4-H&.hj§. '_ * .aw k: Méﬁl-Mgﬂlwa—g 5 5+ 5 ,‘ 5 cl) (8 points) How do you know that this value of x gives the minimal distance and not, say, the maximal distance? SIM,» oL~ 5% 4x“? is o: quadrant fbk . _ the lamina elven“; «fwecotsq The: vertex Point gives us ‘WMInimaI’ but we have no wag ‘tze final 33M r maternal r e). (Extra Credit 5 points) If 0 s x s 6, give'the coordinates of the point O that is farthest from (4,0). For this Pmbtxzm, we can find ﬂu ram xlmaf [far-WAZJ become we have 0. restn‘cti‘oo on me. domain .~ - I ' as 76 5'6 . _ Sui Maximal?» might ‘98 ﬁmw'ja-t' el‘mr x50 or 0656 '~ ' (that): ,1: J7? ' . 1616 t a: W:W}:ﬁ 7J3, SD, [96 5. (10 points) Find the function that is finally graphed after the following transformations are applied to the graph of y = 1, x>0 : x a) Reflected about the x-axis. 7: v :2- b) Shift up 2 units. 6- (10 points) Consider the function f(x)= em a) Show that the function f(x) has an inverse. Hint: You can show . K h I“; this either aigebraicaliy or graphically. _4 Hi 1 i i - , fl - a as t ' Cit“???Fédiftm ,_ f5 "C or i r m i i M S i. HUB 1‘er0. 95’7")?“ j ._ .géi-ti a . x f‘ CE) 1:“ (it 5‘ Gﬂ‘ WSSES TIN? L‘tiﬁYiE :isxirzk RI _ _' awe; Si ( 3w”; i* @ 'i'es'i’ Ci“ 15> Cardiff. +3.) gm , Timed: M3 N1 new a .- {The} (v. v- b F- d —} A l ‘ ) m f o) W ‘9 e \J-Q if? ‘1‘ (2:: M if), a “H 39 4 I . t- U i t MA.“ g» 1 r I St’ms‘dﬂ X {’1’ J ‘ X . )2 {Via} (Egan? ' at : to v "ems rm" “v 7'. EM X . M I E” ‘ {m w B 4‘ ‘ [to y" -::r it it Law] Li" “W 0) Graph both f(x) and f""(x) on the same-coordinate system and explain how the graphs are relatedﬁﬂhm Giant d) Find the domain and range of f‘ioc) . Hint: You can figure out the domain and range of f“'(.x:) knowing that of f(x). 1):) Mai m (ﬂ :- ROW/18% \$06} 7 (10 points) Solve the following equations: a) in(x+ 3)-«Inx: 21n2 QM EEK) ‘12. EM '2‘2 “5”le “’3 PE 3 (7/) Y 5 » W L; -- ~1~ ‘- x 10/0 \ I , a “fluke. 2% MS 1;}: p) (Km. 5) (x .1, D Mﬁw% wa \$®rmk3 ﬂ a _. ‘12.. MSQ, b 0?th ~33{0i/-‘33 ...
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Soln. Mid 1. Version2, 2-9-09 - Solidirn Math 3 Midterm...

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