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Name (Last'^ First)
Signature
Lecturer Section
UNIVERSITY OF MASSACIIUSDTTS AMIIERST DEPARTMENT OF MATIIEMATICS^ AND^ STATISTICS
Math 131 Exam^3 November^ 28, 7:0G8:30 (^) P.m.
Instructions
. Thrn ofi all cell phones^ and watch^ alarms!^ Put^ awav iPods, etc . There are six (6) qu€stiotrs. . Do all work in this exarn booklet.^ You mav coatinue^ work to ihe backs of pages (^) and the bta.nk page at the end, but if vou do so indicate wh€re . (^) Do not us€ any other paper except this exam booklet^ ard the one-page^ "cheat sheet" that You Preparcd
o Organize 1":rf work in an unambiguous order- Show all necessarysteps'
. (^) Answers given without supporting^ ri'ork^ Inay^ receive^ 0 creditl . (^) If 1ou use your calculator to do numerical calculations,^ be sure to show the s€tup leading to what you are calculating. . (^) Be ready to show J()ur tltrdass ID card when you hand in youl exam booklet'
QUESTION PER CENT^ SCOR,E
I 16
I l
4
Fre€ 3
TOTAL r
rD#-
r. (a) (4%)Find a.llc tical numbersof the fu nct;,otI@) :^ x t/ r xz.
, .,x^ \
c(,)=v.;=-)]ffi] =jr-r.^ r-
ffi 0""- ..o^
q,(i3
QXuo..> gqco (^) 6u ' -J;
C(i\ i \o-\ ) u'h\oq( : I
^=-(--G,
(b) (12%) what are the absolute (th-at is, global) maximum value and the absolute (that is, global) midmum ralue of l(r) on [0, U, add at whidr r in [0,1] are those mlues reached? (Use appropriate methods lrom calculllr, not eBti(oatesobtained by graph- ins bhetunction.)
Cotn (^) € O,f^ g
o i,-or ' O
(al =^ (-, J \ - __ -:-l-\ 6C--z
o
Ll\o\oq
^1o+
Vo,-\ ^ e 0r\d
Gtouc.\ Aic^ rlat v'e-
C!-+ (^) t = /az
c|.\ ()6*1^ x:^ or^ \
oti
5+
"/t-;, r*..ges^ \ro\ (^) Llg:) (^) c\cn
ffo.cY.'ed (^) ) ti:^ +nt \ 3y^
0,\ \d^ /
:)\ -\ (:g^ ^g (^) R!v\ (^) L\i or1 ia
C6r1+i !-\ .^- G ^.= o,-\ -i._ q-\obed
\n*es -q-\ S.r \o\
J* E x+*^ ewrg
\T.\q! -{heox^ e {y,\ {-."l(nx / fi-\ \ i-\
d.L (^) o++A"1L4.
(3 x 5% :^ 15%) Use appropriate methods of calculus to find the €coct lues of the following limits. (Do not uae your calculator to estimate the lillits.)
G) li-.^
lnr -r,". A^u) : \61)^ :o
*"(; i.l, (^) *.,co.) - ,o-1+) : o
\
3" L' \osi:o"^ A3q\q"
V (^) ,.^- '/'
#"'i (^) --u,.*,to.o)
= E;(E)
= (^) - .:
(b) qin_
\
J
= J'^ flr x-)o
tc) lim l: I
x-a o'
_t+
Jih(:"r.r-x (^) ) =^ o-o
J"n\ (^) f- = 6
5o L AoF1+ o'.^ A1$iq-
Cc,3(x)-\ --t^ -:o^ b- ll"E$t:' aRl !e-
.--0"((t)) L
e
r" t;)
a x ('./,r -r.t"i
e
-iri-;:
M!" J'HoP'to'a
{ -r\
!9 o*
h(t x -t o'
-xrr,.tx)
e
e,
e
'xio+ (^) /
a
e
=^4 -t--1>
- (16%) The R€d Sox are going to construct a wooden case to proudly display their World Seri€€trophy. This box will hare a square^ back and an open front, and it will have a volume of 4,000 cubic ioch€s. What dimeruions for the box will minimize the total amount of mate als used for its sides and base? Follow this outline to find your sohltion:
(a) (2%) Identify the reriables invoh€d (maybe dlaw a picture to help).
(b) (4%) Det€rmiEe what frnction (of a single wriable) is to be miDimiz€d and otr what domain.
(c) (8%) Determineat what numberthat tunction takesits minimum lalue.
Be sule to justiry wby the fu.trctionactually doestake its minimum^ there!
(d) (2%) Ans*.er the original qu€stion: what are the midimizing dimeDsions?
6. (a) (E%)Find the lineerizatior,(r)^ :^... ol (/ixa=16.
/.t
x (^) '(4) (^) ;: \,
\ (x) = X
{'(x) =
Ltx) =^ 't6')^ b-^ c-)
.t =::i^ 3a
_L (^) ( X \a (^) )
--
3a
x
u(x) = ;! x
(b) (6%) Use thir tinarizor;on to &pp!@.im.tp9f47. ci* l.oo, *".o
a decim6lround€d to 3 dtatts t.o the rlght of th€ decimal point.""
(.tvote: The apprqiDation J,oufrd rced not be the sameas the value
rour calculatorgiveafor fiIA.)
\r.e< e-€ o". g
V-r kJt"^
i i.'
-x r-
qJY\g r\
Nam€ (Last, First) rD#
Signature
Section # (^) -
UNIVERSITY OF MASSACHUSETTS AMHERST DEPARIMDNI' OF NlATHEMAIICS AND STATISTICS
Math 131 Exam 3 November 29,
7:00-8r30p.m.
Instructions
. T\rrn otr all cell phones and watch alarms!
Put awaycellphones.iPods,etc.
. The'e are.ix (6) qup<rion<. . (^) Do il work in this exam booklet. You may continue work to the backs of pages and the blank page at the end, but if you do so indicate where. . Do not use any other paper except this exam booklet and the one-page (^) "cheat sh€et" that you prepaxed. . (^) Orgarize your work in an unambiguous order. Show all nec€ssaxysteps. . (^) Answers given without suppo*ing work (^) rnay receive 0 credit! . (^) If you use your calcnlator to do numedcal ca}ftlatioN, be surc to show the setup leading to what you arc calculating. . (^) Do r!o, wdte a.nythingin the table below. . (^) Be prepared to show your UMass ID card when you hand in your exan booklet.
QIJESTION PER CENT^ SCOR.E
Free 4 4
TOTAL 100
2. Q x 4% = 16lo^ tJrf,^ spFopriste metbodsof calculusto 6nd the €tact values
of th€ folwbg limits. (Do nol useyour calculator^ to €siieat€ thc lioits.)
ln(2r)
l8l tllrr ' (^) r-112 s\lznit-
flfi ^ ("x)=".
(!) : o
,{f" ti..^
(zt ) = =i'^ L-tt)^ :^ o
J u*tto'"AW\rE
X i^ 't z^ 6ftl 03: (^) arrorgLrr) T(
ol ;,g,.954;g-g^ $f, (co:r'tx)- co"lo)^
= ..- ' -- o
ufl_":o 'so 3frlv.+or
AR\e-s
-.{ eln lqx)^ + 9:..'{91) -^
o
Jir^
.1,i.b-
J,lagt+cl^ 4p? lic^
ag a.' /
-,oca:l{y) +g^ c, =(cr) (^) - - rt'IJ^ -+
o-\
k) liE o E-L
i- Ce\e.t. \ na-i (^) e-
J,!\o1rr.q[ !-ro!,^
q?P\ie (^) s
-l.rir.- 'arr-(7)X_t oc
(d) m,*fl) (^) -,,'o.o'-
\r.,
= "a," l99l -) o
x+@ (v,) ) o
= +co5(*tt^ t) -r *65(")^
= *^1
3. (s) (6%)nad the lh€sdrstion.L(l,)= ...^ (/.Niata:32.
c,^) = " _r,a^
+l.z) =a ,(x)-;x ,[:z)-t.br" =tF^ _5.
L(xl='tq)(r-a) ++(a)
= ;5tr-?z)
r (^3)
-- !v^ - 33 + lvo 60 Eo^ gA
-- (^) -.'l..- x (^) -
\aB 80 BO
*
L(xtr 6*
_;
(b) (6Tol Urc thit liaanzatiott to spprdmst€ {6. Gi* }o* "o
t
decimsl with at lerst 3 dfuli! to tb€ rtght of tLe d€cieal point.
(/t/otc; Tha spprodmstion )ou 6nd lced mt be the ca,meas the \ralue
your calculotorgivesfor {m.)
, K <---
scx- -,^ &^
J^
'.c< (^) x (^) nt-6-<. 34,
-r1-:O |<. (^) tf, teo)
- A rocket is rising lertic.lly at a coDsta.ntspeed of 300 ft/sec. A cyclist is tral€ling along a straight roa.dat a steady 20 ft/s€c. wlter the cyclist pa.cs€s under lhe rocket, it is 600 fed abow her.
(a) (4%) Draw a diagran depictingtbe6ituation. carefiily Labelingall lleri- able oua.ntities.
(b) (^) (12%) Hov/ fa.st is tie distance betw€€n the observer and the rocket in- creasing 1 second later?^ (Beein by statiIlg cl€arly, in terms of the l"xiabks you used, what rates arc gi\€n atrd what rate is to be found.) You may give your final €,Eswerin exact form or round it to two decimal places. At! *rco.O )^ I^ -^ _"
X'+5" = a
7xx' +db"S'^
= 4zz'
o."d q=^ @+3oo=1@
[ao)lzo) +^ troo^ uoo) = (1oo ea)z'
z'=
5oo , 3+
co/3c
10o -E?
zd * 1oo' q6c', ea
- (16%) Aa engineer is designing a clos€d cylindrical can that v.'ill c,ontain 56ncmr of soup. What should the dimmsions of the can be iD order to mitrimize its co€t? lgnore any metal n€eded to join^ the top a,nd bottom to the side. (You may round ),our fillal aBr^,€r to two decimal plar.es.) Follow this outline to find your solution:
(a) Id€ntify the \ariables involved (maybe draw a pictue to help).
(b) Determine what flnction (of s sirgle riabl€) th&t is to be minimized and on what domain.
(c) Detersine at what Dumber that fullctioD takes iis minimum €.lue. Be sure to justify^ why the function actually does take its minimum there!
(d) Answer the original questiotr: what dimensions rDidmize the can's co€t?
l. (s) (6%)Find all criticslnuebelaof the firDctio !(c:z+ !, -- x r'{X-'
q t
{'($=-air = -..r^ - =^ x3- g ? dce> (^) rot ex:'\ d.+ (^) X =^ O
Otu- a.\5^ ?€,< o a\ (^) X : (^) a
\
(b) (14%) what are the (ab$hrte) (^) rnsximum ldue ldd the (abslute) (^) :nini aum value of l(c) on [l/2,4], and at which c h (^) {1/2,4l arc tboee valuee resfhed? Use appropriat€ methods ftota calculus, ip, €stimat€s obteiaed by groph- ing the fmction.
Sti) =^ :,. +^^
=!^ +\u (^) =\v.s
q t?) = -*t
:
{i.t) =^ ^ +^ +- = r.zs
a\ X^ ='lz
NYI:o\e+e r^ .^ rr-o.\we- o+ 3
0"t x=^?
Nb eo\u..e^ ff.o^.x^ \rA\ !!e^ OQ^ t U. 5
q-p
2. (ax5%=2O%)'lt'e derivative ti (c) of a certair fufttion l(r) is givenby:
f'(,)=4x3-xa = x'(r-r)^ :[j;il
Use methods of calculus to answer the following urftouf finding a formda for l(r) itselL Show nork to justify^ your ansluers!
(a) Where is /(c) increasing? Where is it decreasing? o. (-"''o)^ J:;'," 1
j (^) l''-o e { decsea-:i '\
on (-oo' o)
o" [o"r)
-:i:"
]
e {'^ >o^9 Q^ i^c'eqs'n3^ o^^
(o't)
"jJfi- -:.':
e q'<o^ =)^ e dec.eavl o^ r.r,c")
(b) Where is /(o) concar€ upward? Wlere is it concal,edormward?
Q"[1) = ra1: -cx- = 3S t3l:)^ et*"""'^ te<o
('t\er\ (^) )(:o 'x:
Atl'. 1-?'ir\vc^ {os^ x^
,i x+o .e"rq+ive qo<^ x > 3
{ cc^co-ve^ u.P^ 6'^ (-*,o)^ u^ lo'e)
t concave^ do-.. (^) o ''\ (e.^ oo)
(c) At which c, if any, does I have a.d iniection point?
i^!ec+i.on
Ro.\
a* x=
(d) At which ,, if ary, does / ha1€ a local maximum? A loca.l minimum?
!6c!or\ c^A x^ A^ X^
\oqc\ !-niva^ o'^ X^
= O
(2 x Mo: r27o\
(a) CoDplet€ the sedencebelow so aa
Mean Value Theoreo:
to give s correct sbtement of the
If t fu contiBuouson tbe clo€edid€Ival [0,6l and diftre[tiable on the
open interval (c, D),then:
Iotfiplete the aetnenccl
+^L<e gxis:^ CLr\
€!(Cl.\ +v\o_*
x- !-o\\r-g^ C^ e Lq .\o)
c e Lt."l
= r.^(a) =^ o\o?e^ ol^ c
{'L c)
+ts\ - { La-)
b (^) -o-
5 d...g^ .v\g<^ e^
qx I 5 +
tnLz) - rnt't
(b) Apply tbe MeanValueTleorcm to J(c) = lno on [I,2l to prov€:
^rri
-rv,.e. Av T
qo< (^) u-h\ cY\
{'(c)
[\or-p 9v^ e(^ I
Q'tx)
oq \qn5e.r+ \\qS
Yx is^ o.^ &asea!^ n3^ Q"c1i6'.l
^o.+ {iygb \^e^ s\o€^ e'
a+ a;5 x- \ra^ l.'.e,. -
g\oRe 5 uJ\\c\in \e '\e'c\roJ
L!21 \ie^ \ce\wae6^ -1 o,.. A^ 1,-. 1r'.e<e.\ (^) o< e (^) I,/, (^) r (^) -,. r \
. 2 ^ \Y^ \2_)^ (^ \
-x
a
-:e'
. .os
L I
* _-^
/' f' (16%) A bargeis I ravelingar a srcadyspFedof 3 mil"s per ho/ aionga st raight ril€rbank when it pass€sa village there.^ Fi!€^ minutes latE'r a ferry leaves^ a dock directly acro€s^ the river fiom^ the village and head-sio'i{,ards the village^ ^{a ^
* .{' - Z
at 1.5 mil€s per hour. The ri1€r is 1/2 mile wide. How fast is the distance (^) .r.--,-l " - betw€en the barge and the fery changing five minutes aftlltlbetlelry sta4ed k$ ,! li.- ty cr*sing the rile;?
crGsmg rtre nlerr t--J_-- .x -l-^ 1'1::.^**:^_.^Nl
(Identify the \uiables you us€and what they represent.Indicate in tenn
"t ,^ "ilf:d^
i'\
theserariableswhat ir givenandwhat is to b€ found.)
v, =j^ \ )^ "'^ ,,N'
't'. (^) q/=. \ (^) --? 7
xl (^) \J '^ \ ,\o'$
.1-,..,oqe *l--g-.)F'"'-^ ..l ,,*-^ *o;P96c{^ ^ --? ' --^ jj---.v---------./-
(i-d (^5) )*( (^)? )- s)^ =^ J,
,ir\€n X=
a,-.\ 1--
{> q.s
\2 E.i | "'1<
d.-
J
(.,!4 ?t )
t
bz
L?\ r-2)
