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07exam1sol

# 07exam1sol - SOLUTIONS Math 41 Autumn 2007 First Exam...

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Unformatted text preview: SOLUTIONS Math 41, Autumn 2007 First Exam —— October 16, 2007 Page 2 of 14 1. (8 points) Depicted is a graph of the function f. Sketch the graphs of the following functions. Label clearly the coordinates of the graphs’ endpoints as well as both the “maximum” point and “minimum” point. (You don’t have to keep the same length scale as above, and no additional justiﬁcation is necessary.) (a) 9(96) = 3f (596) E l E l E i (% )3) Math 41, Autumn 2007 First Exam —— October 16, 2007 Page 3 of 14 For easy reference, here again is the graph of the original function f: Math 41, Autumn 2007 First Exam — October 16, 2007 Page 4 of 14 2. (20 points) Find each of the following limits, With justiﬁcation. If there is an inﬁnite limit, then explain Whether it is 00 or —00. h (a) lim————-—~—- = hm W h—>O (2 —I— h)2 _ (2 — h)2 [1—90 ’44‘ﬂwli2— (”AM“) 1490 3L [Iv->0 3 (b) lim —.——3"“ : li 3”“ x—>4" \$2 — 2x — 8 Xma‘l" (x—HXHZ) n Numera’l'b‘f‘ mfproacl/mg »! (i0) wlur/e ole/Io; 50 #15 li’mi‘f inVo/wes’ Men/717 n 5101/! (SP ”(Qt/H"): 5 1(er X ”gait! " S’X ‘5 [email protected]) ° ”‘4 is nejatw ) (21401 °' X+2 is lows/hm) 3d : (x4) (x12) [I’M Xb-ﬁ Lf" Miwa’fior approaches 0 ‘ / J n I .' . . \$.\ ' scan «Tl/m" X<L{’ (7:9. )(vaL/ j ) Math 41, Autumn 2007 First Exam — October 16, 2007 Page 5 of 14 (c) ggrggsin (1n (3)) 52212 mama were (me i l L» , ‘ Snag, XL)”; X '0) M GLOVE IS €7U2Va/eml ”f0 yin/8+ Smﬁg’fﬂ)- gu‘f‘ “M 247 = “00 J Yaw Se #1 limhL we’ve «”4371 win 3 {,5 a ’v! TL + hm . ( , u: men 0 14.9- Sim/(l ‘ 77,15 //ym+ I/OES 140+ 620's?! ) Since ’H’LQ Value 010 52ml osci/Lr/eg ihﬁmkle/y £72)“ .4de 4%; I as + increases wfféaj beam/rag) am/ 0/0615 1437‘ (d) hm 6—1/7” sin (—21) :12—>O+ \$ ”f/qual: 4 1/4/09 \ [ ey bseercrlf/Om‘. ‘BrsTL ’chor [e 6‘) Is Sim/7) (WI/e Seam/5X is Lays/105900 W9 H mm 1‘0 use S7oeez£7Zéomm ] For arr/v >90J we. have I" ,7” i [M e = {M -ey x270+ y—a‘bo : O 0on [M ”K 3 If ,. X90+ g y-éyroQ 87 ﬂ 0 a 47m) Hm COW/”70”" "10 ﬁt S7UQEZ'9/ib 0mm lye/Mi) 0.1M" so we Ma), apply H‘ +0 23014601036 {x321 éJ/XS/rl \$219530/ as ave/L Math 41, Autumn 2007 First Exam —— October 16, 2007 Page 6 of 14 3. (5 points) Show that 1im(7 — 2x) = 3 by ﬁnding a 6 > 0 such that (1;-—>2 {(7— 2:0) — 3| < 6 Whenever 0 < I1: — 2| < 5. We, [we I(7~&’x)‘~3’ ’~' [W‘Rx ( : [Rx—J” = m c) 59 f(7»ax)~3I<& 502144046: {10 QIX~52I<£ lie. 6?an (MIX :1“ (*le < 5/; . ’Uws [STE/i} Works) Since whenever 0", We 1m (Ma-‘3! = arm; < a x-2/<8=%) “36f: E) as aésrmi. Math 41, Autumn 2007 First Exam -— October 16, 2007 Page 7 of 14 4. (12 points) Let f(x) = 656:2. (a) Find the domain of f 00mm”; ('09 In 63%) [a 32 0Q) (lie. all )(quo’z) [49 Is a 700+Iém£ mp ’fwo ””JIOMS {1’0"} “"8 6,96%! g ‘0‘“‘5’70003 ”(2 So W [7960’ ”if as}: "D e ’99ch 4;! game X >6 ﬁn 4216‘) 8 ”1:0 {for x21" 9) So 4515 is only Va/w Mo‘f" in (Ismail/1.] (b) Find the equations of all vertical asymptotes of f , or explain Why none exist As jus— tiﬁcation for each asymptote m —— a calculate both the one—sided limits lim f ( ) and :12—>a+ lim f ( ), Showing your reasoning. III—’04— I10 4‘ has 01 Wr‘flcal as m 51%;) ”Hoe“ ’His Wouid? twp/>1 Marl Mimi’s am a £01» Nbgl’ 11;? Q") ”Mi/3’r x12”? 61') inVOiL/es F’V‘ﬂm’b/ )- 1'" [Mum/er], «(9 “WM Lev Aﬂoat/mus at x=a 56!”E ”P is a 91116me15 a? timﬁ'omg COVV’l’IW/ws 014 4/1/51} don/112mg) So up ﬁlm” is also (oMmL/ouj 52%: (3er Palﬂ‘l‘ in 1+3 Jammy“ This leaves Ma!) Q as 1%: only paw/LEM}, ”(gr an M’eri‘ke (“mat and J . 50 We COMFeri W 593‘ [495 a :40 01L ’ ‘ 1" ["2 Xq‘eﬂ‘f dug . er 0r efproaelmj 36’, 1‘ ’0 (\$0) an; R dmwuha’i’or 573/)" 012(le €312“ El "' 11]”; ’9th i9? into/Me 5419,1147‘ We all’mjy know [Xx/Q1 a, if ti Wr‘ﬁm/ [/lSyM/‘k‘fQ mop! Molly/1’ ‘6? X Hear [mg WW1 X<Ing) have ng >0 mo? eﬁa<o so Sim 1 lop/y’ﬂlﬁ Math 41, Autumn 2007 First Exam — October 16, 2007 Page 8 of 14 (c) Find the equations of all horizontal asymptotes of f, or explain Why none exist. Justify each asymptote With a limit computation. We, 14,105+ comfu’iq/ +de [1114th 411‘ Iﬁnt’fyf 17:05; ;0 Mi [E/ am M4 AoNZowfa; asym/oﬁfizs" If? ”p (d) It is a fact that f is a one—to—one function. Find an expression for the function f‘1(x), the inverse of f. SWf+Cle xKyj Swim ”€3,— y: X 59] yﬁ Egg—.1 Wee—a X: 7: EJak (Beak Math 41, Autumn 2007 First Exam — October 16, 2007 Page 9 of 14 5. (8 points) Let f(a:) = «\$172. derivative. Show the steps of your computation. *YHL —-F‘x) ,F/(X) 2 hag + i ,. ,L. I’M tower V¥+3L [1—90 A Find a formula for f’ (m) using the limit deﬁnition of the ﬁlm" J» ”—J'“ [1.30 h(\’7€+h+1 ”(+3 [W W“ «MM “'90 x+2. ‘ Jx+h+z s hm (2:5, ” ‘X+h+“2. («Ix-r2; + Jx+ln+Z h-EG LWVMMZ “rt/wz + \«’>«:+L.+6L {W (W?) —~ (x'w‘ +2) D L—‘VG ﬂmq’x’f-h+z " (137734?!“ VX‘H’HZ) Math 41, Autumn 2007 First Exam — October 16, 2007 Page 10 of 14 6. (6 points) The graph of the function g is given below, as well as the graphs of the function’s ﬁrst and second derivatives, g’ and g”, respectively. Indicate which graph belongs to which function, and give your reasoning. _1() ‘___J . V.” r...‘ .. L..- We, “M‘f'eﬁlﬁcvwg ‘5 3) 19m ”Jaséeaellcum is 3”) anal {ﬁg 5:)“ curve is j" RWWEB (”this is Omlyas‘muplﬂj of many Vaiiol Wa/ys’fo argon) S » 1% "pitted” curve is ﬁlm/yr We 1%; mm; 4/703) if we mm taint: a? kt as a aim/m 0? SW 49mm t, as can, t Ww/J Le iwcmmsmj awrytuﬂ4m_ But ‘Haés 15 +08 011‘ ”6%” 0+“ 7% WW: 4ch CUWE’S amt/m) whit means 792d ﬁe Maﬁa! curve MOS‘TL he 3 5 H" emirf he rj’ or a”) £660,059. each 59 ‘Maﬁe, is 7%: dew {we 5? awﬁﬂw— depic’faj curs/13,, o Oﬁé‘e MN: {:ng 3‘) [7": 8:23] ﬁn fie}: of} which of ‘Me Nit/min}? CUM/BS is 3/ : Mme/y) ”the SO/U’I/M curt/t, MSbi/fc curve is Peer/”Ive Oil; ﬁxacﬁf 1% Same, Various 010»: where fl am! So on . 9 By PMCESS#0C”(JIM(ha%18M ) ”fl“ ﬂé‘éej 09"”— iS‘ a” [Bull also) its POSI’hl/Y Qde‘iy WW 3 is 0314(3th UP”) dc) i5 (”CW/1:7) Math 41, Autumn 2007 First Exam —-— October 16, 2007 Page 11 of 14 7. (8 points) Let h(ac) : \$4 — 4:133 + 2x2 + x. (a) Find h’ (1:) using any method. L'IX) r: Lie/12x24» 55% I ‘ (b) Show that there is a number a between 0 and 1, satisfying the property that the tangent line to the graph of h at the point (a, h(a)) is horizontal. We lame/1c: rhaw w +1me on 31 En [g 1] ma +4.21 Mpg-:0 Bat whee tar £470): [>OJ at] W): Myth-l <0) Wt @130 +11% (2’60 is Carma] Me (A? a loo/yam i, 4203/ 4t. Maw 010 an jamt‘rk WW, mm m Sa‘l'ig‘ppeij 50 we may afP/y HI +0 avmzimég {ﬁajl ‘ﬂrzem {3” some, 35 in [(5)1] 141% £174) :0) exact/7 wéanL we Mata at 550m Math 41, Autumn 2007 First Exam — October 16, 2007 Page 12 of 14 8. (16 points) Differentiate, using any method you like. You do not need to simplify your answers. 2+\/E x2+ﬁ ’2“ V: Z V (a)f(\$) m — 3 :2 L+L,L Xi ,J—‘L r 5/ 5 ﬂ 5 #34722 ‘ 5X*éLXé"X 3’X 332—1 x4+\$ 37X]: (ZxXXﬁxg (([XBW i) guy-X (C) 9(rr) = (d) P(t) = 62 —— 3cost+tsint Wmamﬂwww. E) Math 41, Autumn 2007 First Exam — October 16, 2007 Page 13 of 14 9. (7 points) Candidates in a Presidential primary believe their support in a certain small state is affected by the number of different advertisements they make and broadcast on local TV. Let V(a:) be the number of voters (in thousands) that end up voting for a candidate who uses :1: ads during the campaign. The following table gives values of V(\$) for certain 35. ----| V(m)| 16 20 28 34 37 (a) Estimate the value of V’ (200). What are its units? - V __ V’[aoo) : [1M 5") W200) .. Y—ao’loo x— 2.00 . V5210‘~V , . 6r >626UGI) {he Al‘ﬁgéf‘é’mee 7796!?‘3A r’s { I: (200): O C '31 :zlw‘feis . 6* X: Flag #1 {{er WK? fl! W 2 0 3/ “illewsnvdzer; ~10 ‘ m4 .. (b) What is the practical meaning of the quantity V’ (200)? Give a briefone— or two—sentence explanation that is understandable to someone who is not familiar with calculus. When, Hm came/Jerk is using 1200 ads by Ark/tier Camfm‘jm) alt-[y [mirage in 41% :4va 010 unis w)” [Maj wheel and increase, in 4%. umber 010 My; vat/’13 ’Per him/Aer) at a m’ﬁe 6P 0.7 ﬂown/item grace (be. 700 writers per not) 3 +th is ) ’ﬂie caMfc/ﬁe sling/(J expect 4‘63 90;” alw+ 700 Mars «gr ML, m, mi m? (as :5 ’an "(mew «Wren/14419024” principle) , Math 41, Autumn 2007 First Exam —— October 16, 2007 Page 14 of 14 10. (10 points) Sketch the graph of a function f with all of the following properties. Be sure to label the scales on your axes. o The domain of f is all real numbers, and f is continuous everywhere 0 f is an odd function 0 f has a local maximum at (—2, 2) o f is decreasing for |m| < 2 o f is increasing for 2 < |:z:| < 3 o f’(m) = —3 if lxl > 3 0 lim f’(:r) = ——00 arr—>0 o f is concave down for -—3 < a: < 0 o f has an inﬂection point at (0,0) Z l /3+au3,gn+ i5 V3147“?! all 0 a . ’ " 4%: éMPL Symn/id‘ric, alﬂov+ 01‘; in) because ‘P is (it? (So amfh is macaw “f ﬁr.— O<‘><< 3) ml 413:) 2‘32) etc) (Slope lg) ...
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07exam1sol - SOLUTIONS Math 41 Autumn 2007 First Exam...

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