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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
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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.
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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 Rindgm 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 “iiiCd 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. Sudm “ital
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{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.
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[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 ShortAnswer 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
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am W «
ave00 S"W<x~3'><:sf _ 7T
(b) Evaiuate $337312“ (x ~ 5) tan :13.
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[gm (w ‘W/z) sum 5°]: gm M
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(c) Evaluate 11m w3e‘$+5.
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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 Ziaavxg‘g3: +25%?? “:0 d‘x.
.. ~23;  W2
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=9 35": 3L :3 8': L
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d’é‘ a DC Wm \
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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 execHa, m SDIU+L;W. Wad—2 xwrx~t ) 43m =' t3 'hwn m Inamgte Valve,
03' Quad we. Solohen i5 (JosiahVic. (M= “he. mknmi [DZ] War 9 5‘5 SJYI‘GHa iMmSe:~a rm— “!st semi (“‘0‘ 71“”
C<m 0W5 (ms; “flc axis 94* mow antrt. [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 ﬂSlin qu‘i' (‘5) Lug, 125% £015: DC “X.*~\. Nwtm 'teﬁ‘m QWLK $or Sucmssivo apfmmm'hans is 90c“)
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11‘ i “ W 2' l *‘ 3 t: MATH 159 {31 Page 6 of 13 58 (a) Let ﬂan} = :53 tan2(5$). 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 eatpm 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%. whyL of t=%. SW 0‘; 'hd. Qtvgg +0 q (Me U i un‘ v. *SSMt
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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 $1,000 and has an annuai interest rate of 5%. Assume no
additional deposits or withdrawals are made. {3} (a) if the interest is compounded annually. write a formuia for the baiance at the end oft
years. if 966) be “Here, amouri1L aﬁ‘o‘” "i ) me “i:
me :2 1000 L H i“) t
2‘ to%(i+ 0.06) [3] (b) If the interest is compounded semi~annuaiiy. write a formuia for the balance at the end
of t years. 0.05“ It [3] (c) if the interest is compounded continuously, write a formuia for the baiance at the end
of t years. 0051:
Mt) = i000 e Alf/3; TH 150 Page 10 of 13 Long—Answer Probiems: in questions 811, to receive fut? credit you must justify your answers
and show aii your work. 8. Considet the function f(x) :: 333g“?
{5] (3) Find and CiaSSify the Critical points of f as either local maximum. local minimum, or
neither. 19’th :: 3x16 _. « ‘5
“W + x3 e”. m = e. ‘” (3762—23)
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NEH—ﬂ... O 3 'F has nodkf a ﬂoat ml or Wm; 54 x=Q m... ‘3 has a Qou‘ mm; ed” 'x; =3“ [5] (b) f has three inflection points, find them. Indicate the intervals where the graph of the
function is concaveup and concave—down. axes C’tx) : ~c (32:52.3) + gﬁx+$((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+ .
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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?
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Wr‘hezsn' 23> he '31 c’m— Puna we: : Hh’(r38l
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‘.
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