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Simon Fraser University
Department of Mathematics
Burnaby Campus
MA TH 151-3, Spring 2005
Midterm I
February 9th,2005, 8:30 - 9:20 am
Last Name (please print):
First Name (please print):
Student Number:
Instructions:
- DO NOT OPEN THIS BOOKLET UNTIL TOLD TO DO SO.
- Fill in the above box.
- This exam contains 8 pages with a total of 7 questions. Once the exam begins please check to make sure your exam is complete.
- SHOW ALL YOUR WORK!
- If you fW1out of space in a problem, use the space on the back of the previous page and clearly indicate where the solution continues.
- Only scientific calculators are allowed.
- No book, paper, or device, other than the usual writing instruments, this booklet and a scientific calculator, shall be within reach of a student during the examination.
- During the examination, speaking to, communicating with, or deliberately exposing written papers to the view of other examinees is forbidden.
- Try your best!
Do not write in this table!
Question Marks
I (^17)
(^2) /
3 /
(^4) /
(^5) /
(^6) /
(^7 )
Total (^) /
I. Evaluate the following limits (if they exist). Show all steps in your working.
a) [2 marks] !~ (3X2 - 2x+ 1)' ~ [ :: \ Ch~ - '"2 "'- -<,) 14-
cvC-..r)S-'~~", lq.\N
= ( ~. \ ~ - '2- \ ~ \ \ ~
:: L'-+ <.
\b
b) [2 marks] lim.J x + 2 - 4 x~14 x-
~ - 4
-':>\ '+ ::( - '4-
_~--=-4
~A ::;-~ -\ 4)
~ (^) l\ cv-- ':;(ok'2 -- 16
(-::t- \4-) (j~k~ -t 4-)
""",-'>,+
'='. (^) '(.-, ~-:> \ '+
--=t- \ L+
C--:A-\ 4) ( .r~-k~ ~ '4-)
~'"""' ( j?. * 'J.. -:>-':> 11+ \ -r^ ~)
\ '-LO-;.(/""t <::. \ c~
~ ~
- Evaluate the following. DO NOT simplify your answers.
a) [3marks] Du(1+u2Y",= ':1 (',,""u'-)'2. ~ cl...t.A C-I.-lA'2.} cL.,,~~^ ~t^ ~
~ (\ ~ ~')1.. '":lV\
b) [3 marks] f'(y), where fey) = csc3y siny
~ \ (~') 0. d^ 'S~'"'^ ~^ -t^ C'S^ ~""'^ ~)^ ct^ c^ ~^ c-^ '2>~ ~ ~ <p",>\ J..u~ (^) \..-\ (.
C S c '3. ~. C <:> 5> ~ -t (-s ~'"' :::J) C - c s c ~ ~ C»~ '"? ~ 'J. ~ (~5) ~
c..S~ --:::.~
-:::-
~~""' Iu...z
- c Sc ~~. co'S. ':1 -t (s" ~) (- c'Sc ~::J Co ~ '3 '::::J'). ~
-x
J
de=- c) [3 marks] dx cosx
CO"; -=c ~ Ce-'X) - ~ ::
~ ci Co">::>l ~ ~~ ~'~,",r :<. I\A.-~ C€>.s ?"- -C( e c:l (-~) c""::>\
- ~~. (- s'" --:>l) c..L~"" ~~
::: (^) &OS -::(.)
... CDS ~
-:c (0)5 ~) ~>{. (-I) (^) i: ~ (- s:r- ~)
~~ <. =<-
- [4 marks] The distance x (in metres) of a particle from the origin at time t (in seconds) is
given by x(t) = t3 - 6t2 + 9t. When is the particle at rest?
,~ ~ M"'S~ _~\ QI"""\_ v~ ~o'--~~ ~\ (t'') ~ 0. ~\ (~) ~ J- (\:~ - b~? t ~«C) c\t ### ::. SI:::L- \2~ +~ ~\ (t-) ~ 0 ~L ~ 1:::'1 - ~ t -t 3 :: 0 ?~ '-- _\Q_ -l ~ l t - \) ( t - 3' ~ " (~) or ~~~ (5.). 4. [4 marks] A cube has sides of length _x_ (cm), and _x_ is increasing at a rate of 2 em/soWhat is the volume of the cube at the moment when the volume is increasing at a rate of36 em3/s? \[0W~12.... 'J^ -:0 -::f..?:. ~~ ,,\- cU. f"' (^) ,~/ _~'\.J_ ~t ### ":.. J.-.::J. 0\.2<- ~ -c d..(:- "2- 3~. '2.. WLQr- _d-..'\J_ - c\t -:.. (^) s ~ (c('r,3f;.') / :2:.b -;:^ ~ ?4..'<.. ' ":)(."2 "- Sb ~ b (;, ?~ ~ fb (^) _(r;:--~_ ~f(ct -'J~ \i::)C ~" (^) \.A) ~ '"c L-.., ,V,'"' \- _'\j_ -:. :::Jl~ "' 6 Sf; ( C~~) 6. [5 marks] Using only the definition of the derivative (i.e. from first principles), _findf'(x)_ for _I(x)_ = ~. \ Vf\r"oo, ( _S-::O;~~_ --r- b~ "-'> " ) ~' (?\"\ =-^ _\\n.--_ h -"":> - (^) ,,--, C'--- 'n-') 0 l\f'.-, n-') 0 - -::c ::: f to:>(-t ~') - (^) --+(?l ') ~ ## def ;hD/' ~f- d..~ \ \ '>J ~t\ '" «... ~~ -~ h [S:;:;\, n J--- f;;h .-\ _S;_ J"A-kh .+ ~~ ## (-X-t h) - A \-) ( s;;~ ~ s;. ) ~ " (~\-o ~~) ## \. ~v ,\\QI\\- 'CUt.:> ## \ ## k -t S-x -L -:t~ -::.. (^) U,.,..... " -") 0 =- (^) l\f'\.... \--\-'>" 7. (a) [3 marks] Suppose that _I_ is a continuous function on the closed interval _[a, b]._ List all possible conditions (involving _c)_ fori to have a global minimum at _x_ = _c._ (If your answer includes any concepts not mentioned in this question, explain their meaning.) ## c = ~ c ~ \,- -\ _'Cc')_ ~ 0 ( \ (c') ~"> ~.~\:^ e?t^ l S^ ~ (b) [4 marks] Find the minimum value attained by the function _I (x)_ == _2X2_ - _4x_ + 7 on the closed interval [0,2]. t (-~\ -,~^ vo,^ Ir\,^ ~u~^ ~r'\^ \0 I '2. F~ \e. Sot- oJj..^ f 0 \'" n 0'1-5^ (f'"' ( <>--') " ~ (0) =c '2 -0'"^ -^ 4- ,0^ ""'^ T^ -:r f _('J)_ ~ --:l __'2L_^ -^ l+ -""2- ~ 1-^ ,,1- -\' (-:t,\ == 4- -::k - L\- -f (-.~ ~ 0 ~ E">\.i S t-s _C?J~~~:_ ~-::>(-- 4 -=- 0 \«-^ cJ-^ ?\^ -=-^ \ ~ ( \) ::0 (^) ').\2 - 4.\ ~-=t (^) ~ 5 Co _rv-_ ~=V' \.~ ~ _(OJ)_ f (\) CV'J f( '2\ ./ '-.J="~ ~ _~t'_ z..cA ~ /"'\ \'\) ) _'1.')_ " s ~'\;-o.., lrv--. ~ 5,
