Mlti 142SFi.s 2009 @na$6
nn
PageI
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Math 142 - Erem 3 Revlew
NOTE:Exam3 coven sections5.4-5.6,6.1, 6.2, 6.4, 6.5, 7.1, nd 7.2. This rwiew is intendedto highlightthe material coveredon Exam 3 but shouldnot be usedas your sole sourceof practice. Also refer to your instructor'slecture
notes,previousweek-in-reviews,suggestedhomework,zupplementalhomewor{<,
andthe online homeworkasadditional
preparation.
sourcesfor reviewand exao
l. Findtheareaboundedbetween/(.r) : f andS(r):r.
M^=J_i
+Jo'
U-x')&>
Q3-x)Lr
-- (^T^v({-X r Xr -t, o) r l-T*t(*-^?' x ' D't)
=
L4 +
I
.+
,-lT_1
l-:r-)
2. The price{emand equationfor a certainproductis givenby p : D(x) : 75 - O.lt dollarsper itern, aridthe pricesupplyequationfor this productis p : .S(.r): 15"0'ooz'dollarsper item.
(a) Find the marketequilibrium point. Roundthe equilibrium quantity to the near€stitem, and mund the equi-
libriumpriceto thenearest
cent.
15_0,tX
Ir--15-0'IY
\t__
7 ,
(nkrsd
i1'".'11"i*1" \ G'"pl'1 Atc'
-/
X= 4Vt, U57q+ --t 4tDtkms
.4= r t 4. O V47ob 4 f i44. o2 g. r iP - n V--4to
i,tmt.
F4u;tt tpy
p.i.^+--V- r,' U
- .dl
(,
lt
owndid\
'
'
(b) Usingyour roundedanswersin (a), find the producers'surplusat the equilibrium price lwel. What doesthis
numb€rreDresent?
(a+ot' lTd'*'") d*
P5=Su'o
(g'+'tl-15/#*,x, 0,4t0)
= h',IorL
{41n'-,,
\ W 4a i tD pruilu0r
^
+Al
<-JhrD raoYosnt<
u'l4tod^!- wl'ba{ tD
ffiWkW-l#y(=$not
Page2
Mdl 142SFi.8 2009@rn dta R|nry
3. Acme Media Companyhas a uniform annualdemandfor 21,6ffi DVDs. It costs$O.50to storea DVD for one
yearand $294to setup the machinerythat producestheir DVDs. How menytimesper year shouldAcme produce
DVDs to rninimizethe total storageand setupcosts?
prvtilttva( ia<a'cl' /Yfuch'hrhh
cou*ol i K**=t1'+vturtt*x6O',pt
{+ettbt,'t
-------=o'n.ru-rlrs pr gnc.
1lpvdtca
dor 5=*..zvttt*,+br
fi\.^;$iLc-tns+trCt'()= 0,50(5) + N44
Y! =\tvo'
I
ChrdL
\
,rl4t t )ytt,rp]
= o' zs - izt-t4o6*:z-
l"#)o'(x)
q'7s*2:b":Dqoo
-- D'3:i--also.r4)
=
\@ifth>
I-thlIr0"f\@i
V-'4.a6r1
=--o
D
t'
*^o,'dp.a,uz
lf:y - *'LavL 4
.,n
.a
l ?YDt
..
'
0.ZSX"-tp?S-OrrUU= O
'-i;="2e4otvoo| y?
gcar
tur
I
l$bE+od.is| 'vrJvlvv'.>
+ry
.5"4ll
t{?+o
r r bool{ rbq 4 :r
_x _-rs D 4o
l * U rc, , a
4. Findtheabsolute
extemaof /(.r) : (5 -.r)(r+ 7)2oo (+A!-LX -SlrV, A^.4,^\
\.r
(a) [-9, -s]
P'(t) - (-i (x+-l)L -f-(s-x; (z) (x+z)r ( r )
= (xr7)f_t gr-r) rd,6-x>]
= (xt1)
(-x -l
+ I o*;ey<)
5b eab;mO+
= (x +t) t-Axtv) =,e
X =- 1
O
-3r*4-o
- 4x = - V
x-- I
/
.+ abg,nt''t't"
4O
/
D^W-duw-|t-q'+f ,'\fu ahtxun-
wwrtturrr\\tLtu* rdfii:'is
?9:Y,?-\=atNY"W
;s Oa-st
vnivrrrMlar fi-L,t/,.Q- e,*l
$
a*/ - - - 1
'rTtLrualt
?'(x)=(x+'DGbr+V)
f" (*): (r)(-?r+a)+ Cx+?)(-3)
f "t x ) -
- 3 ' { + ) - Zv - 2l
f,'(*>-- - \ox - lB
f "( D- - - lo - tt= - 0 ,4 1 0
fi
Slnald
- ,
(Eeu-6
zovae./urlv.izs+.
AWw
iU.ttZ ( o,a\,..ti"'a
0n-.Un,
u-a-e4
f{ fr.) L5ag"
wur,{ir,r,|l,r,rn
alrtilottttrsa* *--l: k l W
on{'d:
**ffi.* nnin.
.-flcs.rt,ll^r
fo)--]stt
Md
142SFirg 209 Ollalts
Page3
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5. The U.S. PostalServiceconsidersa packageto be regular-sizedif its lengthplus gir$ (distancearound)doesnot
exceed84 inches.
end3
(a) Find the dimensionsof a rectangularbox with square
volume. What is the maximumvolume?
-
ftat satisfiesthis restrictionand has maximrml
&irtt"= 4\ +:J= 84 tt
+
l=84-4X
YOfutna--- XLtt, -- V(:O
sv...(0,ztJ
rlWrrLE rl*.,,,tl.Vci> = X'(94-"tx)
Don"aivn
V(x)=g4xz-4x?
x --t4 4,4 --84-4 tt,l) =2 l
V'(x)=lu SX- lzd- = D
Y 2zo
lax(lLl- X) 2D
! z.o
t.
ttl -4{7zo
-4y,>41
x! , l
lLN.o
x--o
btr rJ i-{A M'$u'nry-
il"
t4-X.0
vohl-*L
x --rq
S l on-J
hM dlltVlt
x l4Lrtv.-28Lrt,U,*
lLlr.n
-)L
d l I v/x)
o
nt'vv, vlU'nl
W{i
i S €4EB Ltq3,
14 | s+tcba'bs'vao4
1 - r\ o
(b) Find the dimensions(radiusandheight) of a cylin&ical containerthat satisfiesthis restiction and hasmaximum volume. What is the maximumvolume?
G i" t t ^ =9 rr
+ h --9 4 ->
Vt-? * -?trr
yol.uw€-7IrLV = Vrr)
(r>= ntGi_e,
wlwiniu-x'!
_
)
vrf --*lir( - lt'r3
DrYtui^
Y2z0
h>zo
.ar=fr=Sh=s4'aV(a
=lbtra- l-orrrf' =o
V'Cr)
bnr(ot - T r) -- o
t,
t4'enr>.P
[.ttr--o
r: 9
-Tr >->2
-l-tr( >z-8,4
( <Fxv.1ttc1g
D
4A
'tl
UL
n
i
0-8-nn =o
r =f
<-thc,
bqs-1.57$te
n
^--rr
llnttdirntnsittts4#q
ufrlaiPr
tAU,^A.ricaQ
*.<,1\11 uftg" rroairut't*
rr)
f^t+
4L_l
an lo, i-j
\\fu/i^-t a*-
rvll,t =#
idas a^tr l\tiqhv-rrta
lruw*tw.n fu-.^is 1t1,t1.548a\^5.
Fage4
Md 142S!rhs2009@rE.eR.ey
6. Find the equationof the curvethat pasesthroughthe point ( I , 9) if its slopeis givenby
t
* : c€ - sr +4 for all.r I o.
A+LlrL=q
l ' , l ( t f -s* t+u)dr
y -- E(4*)* 52alxt +t)x +c
",
-- A\4 -5&'t\f,\ vctx+c
tt)+C : J
g ' t,->- ltr ) a-{ 4 ^ lt\ +4
following
orthe
bYhand'
lrt9!$,fft
l) x
L
-sf)a'
@)J(e-iktu
--5 x
U'-e
r n/t0t
,n
^\,
, ;J@'"^-t^r)
(roe'''-ro*)61y
au--(o/"'- toD"t^
Lr I u d u
+,l.+) +L= L(r"*- s tf +c
otl^-:!*
=J(u*r)(i)a*
U= LnYf ?
du-- \
'rx
-,.f ,,tdu
--avkrs
t** fi*l -')a
+btr) -r) t+
SQi''+st-1
-L
ttl
,n
tV
l
+
o1zt"')rs(1{"'"
)
lC
Md rdsFi'.al09oB&rdr
Page 5
{o fr'#l^n-t^-'Y-
- S,2(rn'- 1ru,') a.^
-(7. M- '\tt4t\t'
-b
'/ll
--(\,f rt^-')1,"
'-(r(D'tt(q'') -(k,,i +2t,t')
4rpt _52 _ lV1;)
ttoz
A
a>l,'{v*
, o1 -t/"
= V Jo 4=7^
, is-o
,
dx
l t,rg _J
q-- 5-e
1X
-t d'&x
6.i|.=
u = 5 .-{ ' o)= t-t@
ldhtn x=o,
tltln |=)
I u--5- c
t(L)
u-€:d
r
lJ,t #du
---i b,,url'*
pu+) =1!nw-"1r1"!^t
=(+ulr-eD*@
@tzQ+t1|a'
=Je(u)"du
L t --zt 4
tut--db
=g6a-+;utla
)ge-t{)au
- %(tH8 IC
=iu'ofi,,f+c +(t +u)''
Md
Page6
!42 Sp.ing 2009 OE .i[Fr RElt
8. The pricedemandequationfor a certainit€m is given by p : D(r) : -0.002(.r+ 100)2* 7(X)0dollars per item,
where.r is the numberof items that can be sold at a price of $p. If the currentprice per iten is $4,580,find the
consumers'surplus.What doesthis numberrepresent?
r-o
C,=S (oo>-p)**
=f**(yr -,l5so)l.y
=f^xn+(Y,
- 4*ot x, orIDD)
u.uul
$ l,szz,v6?,?3
fr:n/a- I
L+ 1oo
l,--*0.(Datxt-ho)
o
1L--45t0
CvtLph, ca-l.t
' {'der*g+
x--tooo ,7 - 4sso
.4,usrulfeVnts lU- fok4 M)n\q5 lo (mSu"rrrS
]JJt1rpa,b
witli.n$ lD p44 rnnov^dr., *+, sto ptr it?n
abfehby
latprd.uc( hr $+sqo,
Inuha-re-s6Ll
marginalcostfor its dinettesetsis givenby C t(r) : 3eo'or'1
9. AcmeFumitureCompany's
ff
eachmonth.
where: is thenumberofdinetGsetsproduced
aommp". ,"f
resultsfrom goingfrom a productionlevelof 100to 150dinettesetsper
1a)Findrh(chan:G totafcolDnhat
month.
rOA^2a u.^ALY ils dl.viv,
"bu^I
-
W\Lcm\,If (i"o^*+ ff)a*
+ns/n )x )l0o,.so)
--F^ur(tt'o'*
r1-------.---1
tI
--\$^a-t4.t"
(b) If Acme'sfixed cost for producingdinettesetsis $l,700, find a modelfor total cost.
n/
-4r
^nrv
=
4(x) J'Q"*''^+ |1s* " )dt,
.L
cN)- *r/'o'" +zts(?x')tc
A OIY
=
{ x +C
Ca) 3ood,","i TsrD
o=
t100
CLo) booe+-EDJD+L
lldb
3oo + c
l'-{00
U 0\
. X
K^i.s#4
ll40D
doUnrs
e
=
300"''''^
t75?'JT
x)
MNr+<
f6 d,*t* tL+s(trofuLu/ 'La-cbu
;;;
tt
MrI[ 142sFi.t 2(x,9 @,H.dE
Page7
lelt
10. Bob invested$3,000into an accountpyng 7.3o/oper year compoundedcontinuouslyat the beginningof 2003.
Findtheaverage
accountbalanceduring
- th@-nri!ffi)f
iiT;-;=-u^
A&\ = Pe'+ o.or3L
2003.
n
Ac")=3oooe
(edv us t-o 4
I l. Calculatethe following definiteintegralsby referringto the graphof/(x) andindicatedareabelo$'.(courtesyJenn
Whitfield)
y61a,1u1
loo
Qtl" flia, =
- 21o3
+,1,'fx\dv
Jf ftr.ltx +Joftrtolr
3 + (- a,20") | 5'1s1 = 5 , 114
1,5?
<">
[ rav.
'a,1-0b+5,181 -l'uoq
l,1 '75
61l)y61a"
= --1;'l fur,tdx
(- I,boq)
l,boq
M!|i la2 SFiig 2004 @n d.r
PageI
rrb.y
12. Appmxinate the areaunder/(.r) -- r/r=l +l on the interval [l,Z] using a midpoint sum with 3 rectanglesof
equd widlh. Includean appropriateslcetchwith your answer.
r , ' ilt { ' =1 :aLI '- 9-4- -_
r,/
rrletl"'l
l
HaPa=
l(Po> + fu) + ftu)\
= A(r+6, 172-l rb, zvb t)
T>.q vb+
f^.!M.d.2-
Ivt,t-pan*VW
A',rc*
7t s--to
\) idA
2-
L
ltD =5
1
u
l 1t ) - '5' 191
flu\ -'lt ztvl
.-t--
7
ll .4tr+>
/
+ lL ,417>
73,qvt'+
13. lf f(x) in the previousproblemrepresentsthe ra0eat which a swimming pool is flling with water and is given
in gallonsper minute, where.r is the numberof minutes since5:30pm,what doesthe ap'proximatedareaabove
rcpresent?
x'ai&\
\^oiah'V
Lnuk) x (nti ,+/c5) -- g'Uo^S
lVltons
'J
4r,tt
I
IJ
',rt
lrtw1',TlpM+o6',b1p^tfu vU--r* SIuJdfur 44t SlDio'rnhq
J
aa at>sv*VZ 9'Zts42g-ozswts
,d'i^ur-.va
tl. fida tneareaUetrveen
/(.r) : O.Oa)--r andg(.r):9.2111 onthein#al [-7,7].
\_...^\,.,--_-_-a
!,---==_t_L
Jr
-lz
P.*--t;
art 6wu$dr
/+,ogr*l)
-1,-t) +f"drt (y-y, -srr)
-Y2
- h,fu(Y,
,2x1
,X,
s04
= a(
+r E
5ryh, 2lcealt.6)ortzck
(no+:hoontotde\
-@\
-l t5 \
Mdh 142SFie 2m9 €tHltlE
Page9
RDr.t
aJ t1
462* + 3950
i,t, \
nd. f"(x)=ffi,
-t54x
15.civenthat/(.r):"ffi,
f'@:61ft,
(a) Find the domainof /(.r).
QotvtLlrt',(-a,-a)u (-€,e) v (s, a)
y+-pf =o
(U1ralruunfo'rs otwt
Y 2t{
x = ts)
O) Find dl interceptsof /(.r).
X - DYtt>
fn)--o --7 /*+t
--o
ItD f,-La{{xupl5
t4-Lnt:
4x'---v
r\ L-)
-- -L
9zo\= Y-:o'zg
xL=_Z
vID(b)6bfx\
(c) Find all asymptotesof /(.r).
voftlLt,Lt leoW utvS]t*'n l first!
bxL+>
fcut'-(x+B)tx-5)
,l
.
t
-A
$0frbN\W A8rrlr,t
Slttll'
t. ^ ^ 4 1 ,7+ u '
'
2
*'!b -t -zs= i=z
aagu@
!=n ,t lr,orie.
tuholaryrq+oes"
'*=-6 4.*A 7<--1
(d) Find all int€n"alswhere/(.r) is increasingand all interv"alswhere/(r) is decreasing.
o'^!^Jil -ts'+Y --o
X --o
('L*\=o
f'{v-)=
ffi)rTNE
ft'A\
,7
7/.\">,
t't
(--,,-<)
fuitx'-tS
!
9'ti;y:-L:J42+-*)
'
. &tx\it\t\ulal^g r>'-
g!.
ll
t**"
dwviq
Y'lt ln++"do^o;n
(e) Find the coordinatesofall local extema.
hca.l"il^tu[i{I,t/r'rvr
(0,-?)
11
2<g
--rt
&
-r4
rl
^4(5o),
'f ry..)is dtavona\^-
(o,o)a-"|/I a).
Md
Pagel0
142SFingr00' @llcdt6lrE
(0 Find all intervalswherc/(r) is concaveupwardand all intervalswhere/(-r) is concavedownward-
L 5\so'9
f "tx)-- o 6r .LXi i ^4 b 7 N +
4bzxL=-3gEo
irnz*r+bgsn
n.,
tNL
f"tx)=ffi
*'---?.
^
r@g t&-'4.
v.,.o
W N,=LS
-b I nb//,z-l
O 1-tt1/u>s
-55
4\ t
nD+ in dolnai n
(g) Find the coondinates
ofall inf,ectionpoints.
1'trtw t,'-t5
. f tr) iscvMunu*ntd, ox
(-nr-s)c"^l l t,-).
dWtmd
"(*) is otvugue
ttYr(-s, S)
Aiz WDI i A
1)nt/tun)n, /trl Lw \!),rfunm
graph
Sketchthe
@0.
ptlt+s
of /(r).
probabilitydensityfimction
randomvvariable
with associated
S.nflJ that.ris a continuous
AD+cNe2l,
ttnorvUWia!.vno
{Z ir.r}r
=,f ''' L*"d
?(l.xz-l,s\
= l^(,rv('/^t ,X,1,
'5\--A