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Simon Fraser University
Department of Mathematics
Burnaby Campus
MATH 151-3, Spring 2005
Midterm 2
March 9t\ 2005, 8:30 - 9:20 am
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In'structions:
- DO NOT OPEN THIS BOOKLET UNTIL TOLD TO DO SO.
- Fill in the above box.
- This exam contains 8 pages with a total of 6 questions. Once the exam begins please check to make sure your exam is complete.
- SHOW ALL YOUR WORK!
- If you run out of space in a problem, use the space on the back of the previous page and clearly indicate where the solution continues. 6. Only scientific calculators are allowed (basic math/stat functions + memory key).
- 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 bestl
Do not write in this table!
Question Marks
Total /
(a) [3 marks] Let f(t) = In
(
~ J
. Apply laws of logarithms to simplify the functionf{t) 3t - before finding 1'(t). Do NOT simplify your answer. 1(
1
~l~)~ '\ (~1-)'-
l ~~-~ I( -:. ("\ C u. ~ - ~ "1 -
-:::
. -- ("1 t - "2 )
\ ~ (ll- t - "l ') - ~'""@t -'1.) -L. ~ '3..~-L
..L. -L y.. 'l. l4-~-"""t
( '-~ 0-.: '"' \.A .It) t'(-) '-
(b) [5 marks] Find dy dx in terms ofx only, given that y = (lnxyin x. Do NOT simplify your answer.
'''''-~R 0?~o..s ~:. "'"-S ~ <>-.s.~ ~',
(
'\ s,"" ?. '
\r"' ~ <; \r"\ _(\r"_ "'() )
~ ( 'S " ~ ') ~'1 C ~ ~I.)
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.l C'~)
~ C'~
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## ct-:>\ (\~ (\"" -x _))c-l_ d~ :s, --- ~ (f0:> _~\.A_ "-" \ .t) ~ (~"" (\" ")\~ CN:S ?\ ## (~- h00". \U.l~) ':. ~ (s,,",, -:0 -L ~ \,,">\ \ " -:>\. .,\.,~ "" ## ~s\'"" -=>\)(- -L \ ## "-\" ?\ ) ## :x. -T ( \r\ Gi"' ~ ') J Co~ :{ 2 3. [5 marks] The radius _r_ of a sphere and the edge length _x_ of a cube are changing in such a way that their combined volume is decreasing at a rate of _18n_ cm3Is and the edge length _x_ is increasing at the rate of _n_ cm/s. When _r_ = 3 cm and _x_ = 3 cm, is the radius _r_ increasing or decreasing and at what rate? (The volume ofthe sphere is _inr3_ 3 .) \'O~eV ...J ~ .( _j_^ "0^ ~,\,> ~ lr--t ~\~lk~ -t '?> -::( 1)\~~~~ '--0r t t '. cA'V o\t :. (^) 'l:t t1 - '1, 'l. _c\-!"'_ 3. ol~ -+ 2:>?l2.. ol.":(.- '*~ (\ ') ## w~ o-..s-R OJ '- 'oJ e/'\ .J...\J -\~\ ~t o J. (^) """~l ... \T ~c- v0' ~ \ ~3. (^) OV'.01. ~=-3 (^) / ~~'D:~\\;:'r' \'"" (^) ~\) ~v _Q.J_ -\ <6\1 -- '.:t l.1t11 _::::: a\ e- el\- -t- '2:,,:;-^ 1\ <:\,- ""'t -. -4-5 (^) - "\\ =-^ -^5 '1~ \l 4- 'S..-= _\...N_ \ ~ \'^ ~ ~^ <==-('--^ ~c:A (^) -:::>t~ 3. c..-- -' r<7-0 L\'--\. S' \" , \) _cAQ_ ~C?-s~'"'~ ~\:- (^) s- 4- ## c "" Is (a) [2 marks] The figure below shows the graphs _ofy_ = _x_ andy = 3 sin _x._ Mark on the x-axis the approximate location of each solution to the equation _x_ = 3 sin _x,_ and write down an interval containing the largest solution. y - 2 3 1 =^3 sin^ :x ### ~3\ x /://; -) // / ,// _1-_ 1 2 \ // ///// - v\)~~~ '" '"" ~ rv-- oJ _Vvtot_ dQ A0\~":~ s.-<::)~~~ ~~^ C~(<-\e1. p\r"' "'-.'" \-eJ^ _r_^ oJ^ _~o;~_ ~ \- S-:)~ .,)r- :1: _(~r_ <L?t CV'-- If '- <L)^ [~,^ ~6~ 5 5. (a) [2 marks] What does it mean for a functionf(x) to be _increasing_ on an open interval _(a, b)?_ ~ ("" Ov :::J ~ \ C::V-.), 0::>\-:L ~f"' (0<"-, \:. ") t r \ >0.")\,..." '" ~, <. ~ "- / ~ ~ \) .( ~(::t~) (b) [5 marks] Find and classify the critical points (minimum or maximum, local or global, or not ## an extremum) of f(x) = sinx-xcosx on the open interval (-5,5). f ~()l'1" Gc<; ~ - (\::( (^) ~::..<t cz><> -.:( + (co~~) ~')\"":J..') Co ~:::( - (- -:={ s', ""'?\ +- CO> ~ ) ## (p~ cA- """'t ') -:c -=. ?\ _$\1"'"_ :::(.- \r--¥C..rv ~ (^) (--s ,"S"'). c)\'-J'~ -:x.";.--\\ (^) / 0 ,. 11 l- ~'""''\v-~d (- S _J_ $''). Cr\\-l _c~_ f\) ~'""'tc:;: ~ (^) -::>l'" -Tl, 0, \\. \. ## i- f ## -"\ ## \ cRec \r---c 0 11 -+ ## \ \ _\.r--~_ ~c.. ~:j t'()t- ~\:-..r~-..J-( -te'-::'t f~~r lucc::v\ i??\..h~ ~c--/ f hQ<-~ ~ {\A>\...0 c..c:J Cd ,,-,~(V--I::>- (^) ~ oJ-o.-t ""<,-:>1... (^) -:;. -T\T\ ..., d-. Aer _e_ '> f'...c \- \,.-. cr..."<2 <>--r-" _Q_ ""-t ~ 1\0-."'''"' l- "'X ~ 0. Co rv- ~ ~ _-..N_ <~ _J_ ~ :zrt ~ C1<-~ ~"" I";;) ~ , t<; ~ AD '\.z..r iY\{~ "\'" kh _e_ CV'oO"--':Jl.(r--- \.r-- ~ ~ l _0"=>0.\_ ". _}L_ t (\l ') ;:: 11 ::! l' \4 ~-~ f (-'Il\~ -11 o! -'i"\4- ~s -~ I. )i. _K_ x ~ ( - '5 ') ~ - 1-.~ 'b ~ ( S ') ~ - '1,~~ ## i1 5 <;;;'v (^) t _\""'""o-~_ (^) C)\.\)~~ ~\- ~ ':t -.: -'\\ _I_ /' C?:) l c \::,.,..J ~ 0...')( 0--\.- ?l ': ~, r'\,0'\ C:> ~":olt~ 1".--1.,..\(\.. o..k :::(0;..0. '(-:\.\ ....;^ \..- "- e -::>-.,'S\-S \.1"'" \ (-..\ <;)?^ -;..1\:) s\f"'?\."<:. 0 ; 6. (a) [2 marks] Give an approximate expression for _~y_ (the change _iny)_ when _x_ changes by an amount ~ , where _y_ is a function of _x._ ~ ~ !:::. ti2') ~ ~ (b) [4 marks] A pendulum has length _x_ cm and a period of oscillation _T_ seconds, where _T_ = ~ £. Use your answer to part (a) to estimate to 3 decimal places the maximum error -vl in the calculated value of T if _x_ is measured to be 64 cm to within :t 0.5 cm. ### \~ ~~ -S\o ~^ _If...._ ## J-T -. _\ "l.". \ ?.. -'l. ~7>. ~o^1 fv ~, -!:::::. _~l)_ ~?\ WLQ/"- x "- ~4- ./ d-T ~o;( ## ~ '2". l _S\_ 0 11 g\.f -:, (^) '0- ## S.o ~"^ 1]^ ~-=-.. ~f\o \E=- (^) l~~ \ ~ o-s^ 'dev--.^ \ ~\\ ~ 12 _()'S-c!_ ## ~S\o MCl"'--Q\ (^) \U"" ~ ( l'"' (^) <-<::>oJe-vJ c-..\-e ~ (^) v ~ O'f- "'( "' ~PC\)~"-C\~ -::!: (^) 0- OG>'2 c" ". 8
