150final-1077-key

150final-1077-key - 1. 2. 7. instructions: SllViON FRASER...

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Unformatted text preview: 1. 2. 7. instructions: SllViON FRASER UNlVERSlTV DEPARTMENT OF MATHEMATlCS Finai Exam MATH 150 Fall 2007 instructor: (ClRCLE ONE) Dr. Mulholland & Dr. Goddyn December 13, 2007, 7:00 - 10:00 pm Solutions Name: (please print) family name given name SFU ID: student number SFU~email Signature: J l... MW Do not open this booklet until told to do 50. Write your name above in block letters. Write your SFU student number and email “3 on the line provided for it. Write your answer in the space prOVided below the ques~ tion. If additional space is needed then use the back of the previous page, Your final answer should be simpli- fied as far as is reasonable. Make the method you are using clear in every case un— less it is explicitly stated that no explanation is needed. This exam has 11 questions on 13 pages (not including this cover page). Once the exam begins please check to make sure your exam is complete. No calculators, books, papers, or electronic devices shall be within the reach of a student during the examination, Leave answers in "calculator ready" expressions: such as 3 +111 7 or efl. During the examination. communicating with, or deliberately exposing written papers to the view of, other examinees is forbidden. Maximum Score 1 12 I. p...‘ O H O H O H O 11 MATH 156' Page I of 15’ i2] 13 (a) State the intermediate Vaiue Theerem, cieariy identifying any hypotheses and the conciusion, :1? t is cmiinuou$ m m, stoma mtumi Leak: mi. N i; Gina amber Rkuem glad one/i lwkerc. 9(a)} =75 fiéb‘) 'i’kevx “Hive; W58 0k viz/miner“ C. m I (Ono) Such ‘Hnoci “(3(6) :2 N. i [2} (b) State the definition of derivative of a function f at a number a. The (fienuq'ii‘m 0/? a Rind-gm a? (1+ 0. number a is ’F/(a‘) " flaeh} “- imao h" "Hat‘s gain} eat/{$45 [2] (c) State the definition of a critical number of a function f, R “iii-Cd number is (x nv‘mbet’ C limit!» oi S3 50014. mi Moi =0 or FCC) does no’v exisir . MA TH Z50 Page of [2} (d) State the Mean Value Theerem, clearly identifying any hypetheses and the conclusion. 91‘? ‘l’3 is a. Sud-m “ital Cl) "l3 {5 (amalznluouf. m m— Cbfl “MW! £6753 (ll) ‘5: l5 dimerenhelolae, we “lie, (7)9991 inlerval COVE») 4.1mm ‘l’kere {s a mmleei’ C. “A (cue) 5t;an W‘l’ 59, Rb) Mlle.) (6') 2 b - ex. {2] (e) Give an example of a function f and a value x = a such that f”(a) = 0 but f does not have an inflection point at a: :2 a. ll «R20; 2 x gl/ I Z 1' ‘l‘ QICXJ “:3 l2. I, 50 (W) 0 k Nola Wm :2 0 K “m” “P 0* 5°“ “law‘s 000,0) a-wl (0)60) hence. docs nol' haw-e, an m olnl' x: O . [2] (f) Give an example of a function defined everywhere on the closed interval [0,2] but with no absolute maximum on that interval. MA TH E 50 Page 3 of 233’ Questions 2 ~ 7' are Short-Answer Questions. Each part of each question is worth 3 marks, but not 33! questions are of equal difficuky. Uniess otherwise stated, it is not necessary to simpiify your answers for these question. To receive fuli credit you must justify your answers. [3] 2. 1.» 2 2w ,2 (a) Evaiuate 11m rfi—oo 5+35w3cc4 E» __ am W « ave-00 S"W<x~3'><:sf _ 7T (b) Evaiuate $337312“ (x ~ 5) tan :13. ‘ . ,a o In L’“ . 5:81 + C1"Wz) €05 7K [gm (w- ‘W/z) sum 5°]: gm M X”? 772, Cost, “‘30 1—9797, “SW”; 5 I + O ..._. .3. -— l "' I (c) Evaluate 11m w3e‘$+5. IHOO "7"0 Z 3 L44 32, gm; 1 a? ’6'»: “fi %' 5° W‘s L'” 3'» ounce 67"5 9c-boo 81'; \>°° ha Ll” . X'J’W x-S \ %L,“ aaasfl "3 flm Q xpaoo 823$ MA TH E 50 [3] 3. {3) Find the equation of the tangent line to the curve 3/ r: (2 + m)e‘x at the point (0,2) 1 “x «at. g *2 e we {Mm} 4- &3er 840%) 3 Stage. cl «5°43;er at (0,2,) '. —~ QQGCMC) r: ml ‘l’finragn’l' QWLC‘ '4 8—»2 :2 CW3) Cx..o‘) Pageé of 13 2333+3m2+3x+6 [3] (b) Find an equation of the slant asymptote to the curve y : 2 :5 +10 + 1 21c.+l 1.1+ x“ 2£+3x1 +32; +9: 2.7} + 2x?” +2x Slan‘l' aSm'l‘Olnl xz+x+g 6'0 IZ+X+'l 5' [3] (c) Find the point on the curve x2~ my + y2 = 11 which lies in the first quadrant and has tangent line parallel to the line y = ~I~ 2. Izwry+y1=ll Zia-avxg‘g3: +25%??- “:0 d‘x. .. ~23; - W2 28‘ 7C. WW" %=‘l =5 3-27; .3 "\ 26—1, =9 35": 3L :3 8': L ‘ l (46L! ’Pluaama «Vs-’8 Hal-e *1“. 0731 ll :2) 3;: H ’X?‘”I;'L+9L = 73 3’21": “13 +1.5 who; m5 ‘ch com um <9va 9; fiATI-Z‘ 1563 i3} “Ebb 4. at "—— (11,: Page 5 of (a) if y : (cosxV, find the derivative dy/dx. Ema :1 "x, 9m £60396) fig“ EE 2? Qdcm 2:3) “to x - gig (“5"‘x‘j d’é‘ a DC Wm \ 3;: 2 % C an ((05%) Meosm ) it n x .3. xSthx. M .., (Cost) ( Qn (£0526) (OW; (b) Show that the equation 3:4 z x + 1 has exactly one solution in the interval [1, 2]. 22+ Mn 14 50mm to saw newt-:0 has exec-Ha, m SDIU+L;W. Wad—2 xwrx~t ) 43m =' t3 'hwn m Inamgte Valve, 03' Quad- we. Solohen- i5 (Josiah-Vic. (M= “he. mknmi [DZ] War 9 5‘5 SJYI‘GHa iMmSe:~a rm— “!st semi (“‘0‘ 71“” C<m 0W5 (ms; “fl-c axis 94* mow ant-rt. [3} (c) Use Newton's Method with the initial approximation :31 1 to find 352, the second approximation to the root of the equation 364 2 at + 1. ‘t flS-lin qu‘i' (‘5) Lug, 125% £015: DC “X.*~\. Nwtm 'tefi‘m QWLK $or Sucmssivo apfmmm'hans is 90c“) grave-i ’2 xx ’- “‘7‘” fian) '1 Clack-1 Theresa-c, “ 9a.:i I”! "i :1. 11‘ i “ W 2' l *‘ 3 t: MATH 159 {31 Page 6 of 13 58 (a) Let flan} = :53 tan2(5$). Find f’(x}. $220 '2' 3x} théSx) 4— :eg fi<+mfl5£33 3( atmwzb- fifinC§Xz3> it an 3x} “fitnlCS’x/E + 3x?" +6.46 1(1ng ‘5’" x3 L affair! (SW/3 ' S€Cz(5x> i .s—u -— 3% {ran Legacy + to 78 +an(5m3 sd‘CS’x‘) , [3] (b) Suppose f is a function satisfying f(5) : 2 and f’(5) :7 4. Using a linear approximation to f at a: 2 5, find an approximation to f(4.9). I Rm) It 425) + £65) (‘1.‘1 ~ 5’) “—"‘— 2 + L! C - o. i) 1' Q "' 0‘1 : LC; . d2}; [3] (c) Find a function f such that the curve y = satisfies a? 212$, passes through the 9, point (0,1), and has a horizontal tangent there, d1 ‘53 = ‘1’“ fig; 2. 0m _ i MA TH E 5 0 Bi m : 33in(t), y : 5cos(t), Page 7 of 13 6“ (a) Sketch the curve which is given by the parametric equations 0 5 t5 37r/2. Cieariy iabei the initiai and tetminai points and describe the motion of the point ($(t),y(t)) as t varies in the given intervai (Le. use arrows to indicate the direction the point is traveiing). [N N m p mmhu’ (SJ? 1 ) Cum Qiés ow “hm- eat-pm Z “L. C%3+C%i 2‘ 3% (b) Find the slope of the tangent iine to the curve in part (a) at the point (1,3,1) : ( ME) 2=2~ compounds +0 11%. why-L of- t=%. SW 0‘; 'hd. Qtvgg +0 q (Me U i u-n‘ v. *SSMt 3Cos-t 4% 3 ( V23 "' S 5”) (793) Bees (‘73) -5 43 MATH 1750 Page 8 of 15’ {3] (c) The graphs of the equations (i) r :2 3sin (26) (ii) tr = 1 sin(8) are drawn below in pofar coefdinates. An extra graph is drawn as well. Match the equation with its corresponding graph and write your Choice (either (ii) or neither) in the box under the graph. MATH 156? Page 9 of J5? 7. A bank account initiaiiy contains $1,000 and has an annuai interest rate of 5%. Assume no additional deposits or withdrawals are made. {3} (a) if the interest is compounded annually. write a formuia for the baiance at the end oft years. if 966) be “Here, amour-i1L afi‘o‘” "i ) me “i: me :2 1000 L H i“) t 2‘ to%(i+ 0.06) [3] (b) If the interest is compounded semi~annuaiiy. write a formuia for the balance at the end of t years. 0.05“ It [3] (c) if the interest is compounded continuously, write a formuia for the baiance at the end of t years. -0051: Mt) = i000 e Alf/3; TH 150 Page 10 of 13 Long—Answer Probiems: in questions 8-11, to receive fut? credit you must justify your answers and show aii your work. 8. Considet the function f(x) :: 333g“? {5] (3) Find and CiaSSify the Critical points of f as either local maximum. local minimum, or neither. 19’th :: 3x16 _. «- ‘5 “W + x3 e”. m =- e. ‘” (3762—23) _: (is-7:4}; PLZLBax-j)_ _ Cnhud going 130,3 4” .+_ + 5.. NEH—fl... O 3 'F has nodkf a float ml or Wm; 54 x=Q m... ‘3 has a Qou‘ mm; ed” 'x; =3“ [5] (b) f has three inflection points, find them. Indicate the intervals where the graph of the function is concave-up and concave—down. axes C’tx) -: ~c (32:52.3) -+ gfix+$((px "3791) = em§( ~378+x3 + (ox ~36) aims- ‘5 C (78* GL1 + (are) -115 = e 1(xz“6x +Q3 VCR/3 =0 :3 36:6 or xzvéx’rc, :10 «3 7C: GiJ3G’fl'E 20:25 MATH 159 Page 21 of If? {10} 9. A swimming pool is 15 m long and 5 m wide its depth varies uniforméy from 1 m at the sheilow end to 4 m at the deep end. Suppose that the pool is being Wed at the rate of 10 IDS/min. At what rate is the depth of the water at the deep end increasing when the depth there is 2 m? (The figure shows a cross section of the pooL) 1m ii V be '31» voim Q"; 0+ . kt LL be, M Nléhho’f” +1“- wa‘hu' a? we. (Rebhmshcp V 5'ng K, V' ow gg ,oUL ovc 2 1 2‘“ dk 4}; a as he}; OH: 0% MATE I50 I age 2? of 13 {10] 10. A stozage container is to be made in the form of a right circular cylinder and have a volume of 2871’ m3. Material for the top of the container costs $5 per square metre and material for the side and base costs $2 per square metre. What dimensions will minimize the total cost of the container? lull-fl ii C Lem malol'lim cmlmgy, é‘xSl C 1:: Sfiprfeko? 4* Q.[Arm cl: Side] Km 2 gyr'L+Qe(ch-K§+;L(Trrl) '= 327762“ + H,er Volwm. cl CM’l'wW3 V r: 2,8?!" Wr‘hezsn' 23> he '31 c’m-—- Pun-a we: : Hh’(r3-8l FL (‘2- Mtnlm Whit“. r: 9 5 wk h= 7. cm :2 23w + 5m «elm.- CLF‘) t: +00 (woo 13/1/33?le Page 2'3 of {O} 11. BONUS E4 points} A number a, Es calied a fixed point of a function f if f(a) : a, Show that if f’(:c) : 1 for 3H rea§ numbers x, then f has at most one fixed point. SUFFOK a» and, b owe, ££w° 90:43 , M“ £8 m“ W“ WW W MC— Cb ox C» L"; [we] Suck m+ ’9‘. 7:: b‘q ‘ x / S‘lnw ajé: Grfl baa paws PM» m5 Mina;ch 1a,. mem‘ts- “MW, We, Cs OJ” Mod" one. P0541!" ...
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150final-1077-key - 1. 2. 7. instructions: SllViON FRASER...

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