MIdterm-Exam1-Solutions

MIdterm-Exam1-Solutions - Mathematics 115 — F2011 Midterm...

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Unformatted text preview: Mathematics 115 — F2011 Midterm Examination 1 —‘6 October 2011 o Read the instructions to each question carefully — be sure to fully answer the question that is asked. 0 Show all work and calculations. Provide all necessary reasoning. Use units where they are given. 0 You may use a calculator for arithmetic calculations. You may not use a calculator for any other purpose. If improper calculator use is evident in an answer, there will be no credit for that answer. Use 2 digits to the right of the decimal place when using decimal representation. Total points: / 100 ,m'“ .r i i L {ii 1 (1) Let me) = 1 m—l' fifla) Use the definition of the derivative to find 1” “it”; trewédwfl 2 «If? Let f be the function with this graph: M £7 (a) For What values of ac is f not continuous? [Give a reason for each value] / Lg m3; Eavmgtiiénkfi ‘2 "‘\\ 1’3 w/ 53 3 W ‘x. g i o F vb E n, diff“; . ; . w- ‘ ' W” R ~ 5:? “ :2. K g {:2 igfifig .(ikgkAw b «m {KM a” W $3 Qifiéfl 74%” *9 I 3,- / x. a; ' was if I ( {my}; :3: “a as; :6 {HM 4: a“ .fij/ y m _ H \M X”? {1, ,55 ’ n {I ,, 5 A m w ,,7 a. ' : $653 as m} Mais- ig n; , a W \ m VQLWQ § Qwfii‘ mi: *1 fiwg m4» gafiwié “iiéia’j‘g 6: 63.25;? K E} @1‘ I “k ' z ' é ‘ ( :‘ggéfigf/ Luz}, "3% g—éjfiwfifi, A; '3 E? ‘ Q}: V; : 7325 U éfiETa-Hb a, 3 u: V dd: (3) Find the derivatives of each of the following functions. Name the main differentiation rules used. cos a: (a) fly?) = 231: + sing: W}: \z X i ll) S Lama a, (a) ' Find all critical points of Show your calculations then place your final answer in the box: ‘1 Z” 1M” if; 315%.”) 7;: Q; 54“; :2" “i m" We “i . (4) (b) Determine the intervals where 9(23) is increasing and where g(:1:) is decreasing. Give your a??? reasoning and calculations then put your answers as indicated below: ‘g’iwll “v o W 92,) A G 1 l3 .4" {€12} i?» e 1: g(m) is increasing on: &”00 ‘“ l“) y ) ,5 \ a»; ‘ M g a \ \ gm My Jr Find all relative extrema of 5.; “”’ “a .E" g r ea 9 fl» .1 a ,.~ a, K” (d) Find the absolute maximum value and the absolute minimum value of g($) on [0, 2]. fr; 5 yfi) Suppose that f has derivative 1“ = 2x3 — 24:13. @\ l f (a) (a) Determine the intervals Where f is concave up and Where f is concave down. Give your / reasoning and calculations then put your answers as indicated below: i 1?“ (K7) 3; (5 Cigévrmcl $45 i“ W3 1; £2} \L a :2 4% W“ gr Fig 5% Kg A M a!) 1‘; AK ixvggiiikflil 2Q {xi {awfva im at” i" M {"‘iié'm‘ $6»; 3:? L} $65“) <5 (:3 r r ' “ l; W Z“ {it ‘33: Z” 431% x: f is concave up on: f is concave down on: (b) Find all inflection points of f Give your reasoning then place your final answer in the box: w gr Mi diff/$33 C hi1 C ire/roars 3i “we; m g IM ,1 ‘33: @ gm i“? (gm. :3: m 3 m ,: WM Wmfix 3’: "a m E} 2 ya I; (6) Suppose that an amoeba proteus is moving in a straight line and its position at time t hours is 3(t) = t + cost fif (a) Find the velocity function 11(t) and the acceleration function a(t). f x “a; r 3‘ NA é V r ,9er (15%} e": ’23“ ft my its”? 1.1 i i e {AM (b) Suppose the amoeba travels for 27r hours. What is the maximum distance of the amoeba from its starting point during this time? [Justify your answer.] it/ US} a: aw Etta i at to ; Z‘TT/i 5;) L (at _ ‘ g a C3; % 91 g gig! N «T N hiwgé i r Q; My Mn} fies; 3:3 3L} fix; fig om Lo! V ‘ 9 rm “6 air“ ; g» ‘ “i _ y Cgaatewig’r: iii—L ; a"; '2 i 3;“: f: j 3% r5; a {i t ' 9/ “YEW th‘zfigLfflkLMwWL digs! (136:5: LE; (C) What is the starting point of the amoeba? Does the amoeba ever move toward its starting point? [justify your answer.] HQ ‘ ,-~, at 1“) gig} :‘L‘fi g fir \w/ gift”) 2*: fags/dc 2:5,) £7 ., g, M Q i ,7,“ (VWVQAQLg/ZE a g"; L ‘xi i i; 3‘ e \M’” «Commie; a hum/Tr temer 3%??ng sf; gl 1 $2—31:+2 (7) Let Mm) = 2x2 _ 1 A6 (3) Determine if h(w) has a horizontal asymptote and justify your answer with a limit as 3: ——> oo. 4 em lit, ,3 ,, ,4, 5; 5i kw??? $23 w ’l Em (31‘, L“ if 89;}? fl ‘i M“ e?“ if R A - (b) List all vertical asymptotes of [Be careful!] 1 E Civulzv» mam-ea ludfii mmfiiéfgigtéwfilk ‘ Wt g; Q 53% 34% r >51» mMmm m fiftgtg m1 fl } M a: SS g; 1W 2 {24W 8 ‘3’ (8) With an unllmlted supply of the rlght nutrients, the mass of a type of tmchoderma (a mold) grows - ” exponentially With growth rate of 0.07, assuming the time units are hours. Let m(t) be the mass [mg] of trichoderma at time 15 [hours]. ' : ' ' I? l‘ f _- ,x (a) If 771(0) 25.35 mg then Whig: the mass at time 4 hours. 1 / a C} ' .N r mm a»; 2% 3i; L %l “’“ if: (c) What mass of trichoderma would you need initially to guarantee a mass of 30.00 mg after 24 fl hours? a“ KIAK 4:9“ .s f“ A oval t Mathematics 115 — F2011 Midterm Examination 1 — 6 October 2011 o Read the instructions to each question carefully — be sure to fully answer the question that is asked. 0 Show all work and calculations. Provide all necessary reasoning. Use units where they are given. 0 You may use a calculator for arithmetic calculations. You may not use a calculator for any other purpose. If improper calculator use is evident in an answer, there will be no credit for that answer. Use 2 digits to the right of the decimal place when using decimal representation. Total points: / 100 i 2 XL)» (1) Let W) = g. (a) Use the definition of the derivative to find f’ .9 x :7 I - (54%;? A} ta ta 2:. 3i W W" “’7‘” W m M : Egg“ w K W m an s Awe:sz my W. “as {up if», If it \‘5 .4 l ’E g ‘ \m/ Ki? (b) What is the equation of the tangent line to the graph of f at (2, f (2))? , x” , T \ W X} ’33 as ’ m is We Ea v (/19 . .i 5 (2) Let f be the function with this graph: £9 (a) For what values of a: is f not continuous? [Give a reason for each value] if: ‘ .2 I (25!? g M {52 6mg; {pf/gm,humfi m I, 3 Find the derivatives of each of the following functions. Name the main differentiation rules used. A a, 5 - 3 ~ r t :f 3 / ‘ f} , g (a) y = 1Ink/36”“) 3 if «4h é Em, itfish} *4 @er w WM e; I a c' 7 ‘ I 5'” g A s: / A (a ‘ I V’” 2/ “I! fix “Ear Q E J 1—,— Qffli.%~ x: a; :- l1 sin :3 a: + 2 cos m «~55! I X Z; A 6 \ 'vx ‘ I . gr w :2; '( Kalli/mi? [E Mistral; t i] ' ' >4+ ’2‘: of; {3% :2. mg {Mgr/er 3!; ~14" :2 Swiglillm‘lW {Karma 36;}: H V ixél: mfgwmcl 5;) a r « 5 F? (4) Let gm) = 53:3 — 3:135 with domain R. I (a) Find all critical points of Show your calculations then place your final answer in the box: m‘ a“ (b) Determine the intervals Where 9(37) is increasing and Where g(:v) is decreasing. Give your reasoning and calculations then put your answers as indicated below: gltegf} 34:; “l g, 3% if e {maxi} 1;: mi l (13: >43?” *1 (i) 6” all: x U; '2} anewu \l fig: NE W figsan w Z £3 fi, sew—«mg: w W (5) Suppose that has derivative = 1:3 — 2793. g (3) Determine the intervals where f is concave up and Where f is concave down. Give your :7. reasoning and calculations then put your answers as indicated below: Q i ’2 w \ W . W Qflflék) fig; rfia EQ'Mgie—e {mi} K “i Mi £5 ‘33 fl fgi‘xi) ' I M Mr K flair)“; a illcfiawm €19 ‘ (4WD 7; ._ .i‘ mMfiMWigwmfimnw.“ km“ 5. a“, «((71 WE: ‘” law z 4%” 42%) é: (swam E ii: A r M I 2 V {Kg if”; ME?) {Safaring ‘ Assn ("7 f is concave up on: xii“? i 1% Sigh f is concave down on: l «‘13. (b) Find all inflection points of f GiVe your reasoning then place your final answer in the box: if; i» 6 sf” (6) Suppose that an amoeba proteus is moving in a straight line and its position at time t hours is /,J 3(t) = t + sint + 1 [mm]. (a) Find the velocity function v(t) and the acceleration function a(t). i a «as = g M 2: i a: Limits: ‘5 “a Mti «:3 a“ if} 3 "ti/Era f- 5:1,? a; ” (b) Suppose the amoeba travels for 27r hours. What is the maximum distance of the amoeba a; a frOm its starting point during this time? [Justify your answer.] Lil/ti 7 it» *5? {a it; {at , :6 ft to A I ( A J aging) J L W \ flitéf ,x “#157 {Wraith diatm‘zrfi W W A K / r ‘ W " é, i f v “ «mm “" “3 ~ r- W M 2w its EtZfiTlg‘ZT/T’mi? “8’” ‘ Q (C) What is the starting point of the amoeba? Does the amoeba ever move toward its starting E” point? [Justify your answer.] ‘ W i M ii I IV I * I g} 1 (é , E K ("Vng 2% {r 3;ng :— i, is t a, Watt «p51 Q—mmmgmmégmm > EN “‘3 i C} “i z r; x ii k2, {f‘ a flaw} a. {+5 Eff; Q (g; j} f3 mam a. ,l t» 2 — 4 (7) Let h(w) = (a) Determine if h(:c) has a horizontal asymptote and justify your answer with a limit as a: —> 00. WW W x 15’ 4., » via—ng iwfi/xt tiff“ “73 k“ 0’ ,_ M372, ‘2 é [a Q i :3” i“ 7:3 \ gm at») .m {y} (b) List all vertical asymptotes of [Be carefull} g 2 tr 7‘“ Mn”? {W 43% a $523» x1972 1y wfiflwfléfi ,3, V "i i gt, 3% fitfigmé (Egafmgémifi. f3; 8 <8> K Lg / {/ is» f With an unlimited supply of the right nutrients, the mass of a type of trichoderma (a mold) grows exponentially with growth rate of 0.06, assuming the time units are hours. Let m(t) be the mass [mg] of trichoderma at time 15 [hours]. (a) If m(0) = 15.35 mg then What is the mass at time 6 hours? / \ in {g 7;: g: EM. Q? C 5 uéw£§// {i sit“; «a z? £2 2:1:be t (b) If 771(0) 2 20.00 mg then how long does it take for the mass to reach 70.00 mg? /" (c) What mass of trichoderma would you need initially to guarantee a mass of 40.00 mg after 12 (’ if hours? k 0 Lo! .1 . i 1,, _ my {am to i bng "WK it 3:: do} 0%“ i i I ’ 47': “is .3 I { .4" '9. W gm M ” EJW: 1%; l vs “ii” i flag a a in,» lav if?) 0 is)" ...
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MIdterm-Exam1-Solutions - Mathematics 115 — F2011 Midterm...

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