2018 Calculus Regional & Key Finalized
Transcript of 2018 Calculus Regional & Key Finalized
ACTM Regional Calculus Competition 2018
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Work themultiple-choicequestions first, choosing the singlebest response from the choicesavailable.Indicateyouranswerhereandonyouranswersheet.Thenattemptthetie-breakerquestionsattheendstartingwithtiebreaker#1,then#2,andfinally#3.Turninyouranswersheetandthetiebreakerpageswhenyouarefinished.Youmaykeepthepageswiththemultiple-choicequestions.Figuresaren’tnecessarilydrawntoscale.Anglesaregiveninradiansunlessotherwisestated.
1. 2
2
4 16lim 162x
xx®
æ ö-=ç ÷-è ø
.Bythedefinitionofalimit,thereisapositiverealnumberd suchthat
24 16 16 0.42
xx-
- <-
if0 2x d< - < .Thelargestvalidvalueofd is
A.0.02B.0.05C.0.1D.0.2E.0.5
2. ( ) ( )0
4 5limsin 2 cos 3x
x xx x®
æ ö+ =ç ÷ç ÷
è ø
A. UndefinedB. 1C. 2D. 4E. 9
3. Whichofthefollowingindicatesthepresenceofahorizontalasymptoteforthegraphofy=f(x)?A. ( )
4lim 3xf x
®=
B. ( )lim 3x
f x®¥
=
C. ( )limx
f x®¥
=¥
D. ( )3
limxf x
®=¥
E. Eachoftheotheranswersisincorrect.
4. Thereisastackofnewspaperswhoseweightisgivenbyw(t)wheretistime.Amatchisthrowninthestackandwenoticethatthefireisincreasinginvigorattimet=2.Whichofthefollowingmustbetrue?
A. w'(t)>0andw"(t)>0B. w'(t)>0andw"(t)<0C. w'(t)<0andw"(t)>0D. w'(t)<0andw"(t)<0E. Onecannotdeterminethesignsofthesederivatives.
ACTM Regional Calculus Competition 2018
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5. ( ) ( )0
' 'limh
f x h f xh®
+ -=
A. DoesnotexistB. 0C. f(x)D. f'(x)E. f"(x)
6. ( )( )( )ln sinddx x =
( )( )( )( )
( ) ( )( )( )
1
1
A. ln cos
B. cos ln
C. cos
D. sin
E. cot
x
x
x
x
x
x
7. ( )3 4 5d
dx x x x+ + =
( )2
2
3
3 2
A. 3ln 2 5
B. 2 5
C. 5
D. 3 4 5E. Each of the other answers is incorrect.
x
xx
x x
x
x x
+ +
- + +
- + +
+ +
8. Thedepthofthewaterxfeetfromtheendofaswimmingpoolisgivenby ( ) 21
803h x x= + forx∈[0,20].Whatistheaveragedepthofthewateronthisintervaltothenearesttenthofafoot?
A.3.7B.4C.4.3D.4.7E.Eachoftheotheranswersisincorrect.
9. Aregionisboundedbythecurvesx=2,x=4,y=x4,andy=4x.Computetheareaoftheregion.Roundyouranswertotwodecimalplaces.
A.24.00B.25.28C.52.25D.78.40E.Eachoftheotheranswersisincorrect.
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Forproblems10and11.Followingisatableofvelocitiesandtimessincemidnightforavehicle.
thours 1 3 5 7 9 11 13 15 17
vmiles/hour 25 45 53 55 60 62 61 53 42
10. Givethebestestimateoftheinstantaneousaccelerationexactlyat7:00am.A.1mi/h2B.1.75mi/h2C.2.5mi/h2D.7.86mi/h2E.55mi/h2
11. Usethemidpointrulewith4intervalstoapproximatethetotaldistancetraveledfrom1:00amto5:00pm.
A.860miB.828miC.430miD.203miE.180mi
12. Whatistheequationofthetangentlineto ( )2 11
xf xx+
=-
atx=2?
A. y=7–xB. y=x+5C. y=5−xD. y=2x+5E. Eachoftheotheranswersisincorrect.
13. ( )( )2sinddx x =
A. ( )2sin x B. ( )sin 2x C. ( )2sin x D. ( )2cos x E. Eachoftheotheranswersisincorrect.
ACTM Regional Calculus Competition 2018
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Forproblems14and15.Followingistheflowrateofapollutantinalakeinliters/hourasafunctionoftimeinhours.
14. Estimatethetotalamountofthepollutantwhichenteredthelakefromtimet=2tot=5.A. 5litersB. 10litersC. 15litersD. 20litersE. 30liters
15. Howfasttheflowratechangingatt=4.4?
A. Decreasingat1.5liters/hour/hourB. Decreasingat2.5liters/hour/hourC. Decreasingat3.5liters/hour/hourD. Increasingat2.5liters/hour/hourE. Increasingat3.5liters/hour/hour
16. ( ) ( )( )sec tanddx x x = A. ( ) ( )32sec secx x- B. ( )sec x C. ( ) ( )3sec tanx x
D. ( ) ( )( )
2 2
3
cos 2sincosx x
x-
E. Eachoftheotheranswersisincorrect.
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17. Ifp(3)=4,p'(3)=0andp"(3)=2whichofthefollowingmustbetrueaboutthegraphofs(x)?A. Thegraphhasalocalmaximumat(3,4).B. Thegraphhasalocalminimumat(3,4).C. Thegraphhasaninflectionpointat(3,4).D. Thereisaholeinthegraphat(3,4).E. Eachoftheotheranswersisincorrect.
18. ( )( )3sin 5xddx e =
A. ( ) ( )3sin 52 315 cos 5 xx x e
B. ( )3sin 5215 xx e C. ( )3cos 5215 xx e D. ( )2cos 15xe E. Eachoftheotheranswersisincorrect.
19. Acontainerisintheshapeofasquarepyramidwiththevertexatthebottom.Itsbaseis12meterson
eachside,anditsheightis4meters.Itisbeingfilledwithwateratarateof9m3/min.Howfastisthedepthofthewatergrowingwhenthedepthis2meters?Thevolumeofapyramidisgivenby𝑉 = #
$𝐵ℎ.
A. 2meters/minuteB. 1meter/minuteC. 0.5meters/minuteD. 0.25meters/minuteE. Eachoftheotheranswersisincorrect.
20. Thereisalinegoingfromtheorigintoapointonthegraphof 2 3 , 0xy x e x-= ³ .Ofallsuchlines,
whatistheslopeoftheonewiththelargestslope?A. 1
3e B. 1
3 C. 1
2 D. 1
9e E. Eachoftheotheranswersisincorrect.
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21. TheregionRboundedbythegraphsof32y x= , 1y = ,and 4x = isrevolvedaroundthey-axistoforma
solidofrevolution.Thevolumeofthissolidisgivenbytheintegral
A.4 9
4
1
1x dxpæ ö
-ç ÷è øò
B. ( )4
3
1
1x dxp -ò
C.216 3
2
1
1x dxpæ ö
-ç ÷è øò
D.3 16 4
9
1
1 y dypæ ö-ç ÷
è øò
E.8 4
3
1
16 y dypæ ö
-ç ÷è øò
22. Hereisatableofvaluesforafunction ( )y f x= :
x 2.9 2.99 2.999 2.9999 3 3.0001 3.001 3.01 3.1
( )f x 0.67872 0.67932 0.67984 0.67998 23 0.64002 0.64021 0.64235 0.64467
Thevaluesinthistablesuggest
3lim ( )xf x
®=
A.23B.0.68C.0.64D.0.66E.Thelimitdoesnotexist.
23. 7 2
7
3 7lim5 3 7
x
xx
xx
-
®¥
æ ö+=ç ÷- ×è ø
A. 13
-
B. 1147
-
C.0
D. 35
E.¥
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24. sin( ) sin( )limx a
x ax a®
-æ ö =ç ÷-è ø
A. cos( )a- B. ( )sin 1 C.0 D. cos( )a E.Noneoftheotheranswersiscorrect.
25. Thefunction ( )g x hasaderivative '( )g x thatiscontinuousoveraninterval [ ],a b .Thedefiniteintegral
'( )b
a
g x dxò canbeinterpretedas
A.Thenetareabetweenthegraphof ( )y g x= andthex-axisbetweenx=aandx=b.B.Theaveragerateofchangeof ( )y g x= betweenx=aandx=b.C.Theaveragerateofchangeof '( )y g x= betweenx=aandx=b.D.Thenetchangeinthefunction ( )y g x= betweenx=aandx=b.E.Thenetchangeinthefunction '( )y g x= betweenx=aandx=b.
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TiebreakerQuestion1Name______________________School______________________Letf(x)beadifferentiablefunctionandcbeaconstantrealnumber.Let𝑔 𝑥 = 𝑐 ⋅ 𝑓 𝑥 .Completethefollowingstatement:𝑔, 𝑥 =_________________.Provethisresult.
ACTM Regional Calculus Competition 2018
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TiebreakerQuestion2Name______________________School______________________
Considerthefamilyoffunctionsoftheform ( ) axf xbx c
=+,wherea,b,andcareallnon-zerorealnumbers.
Answerthefollowingintermsofa,b,andc.Justifyallanswers.a)Identifyanydiscontinuitiesofthefunction.Determinewhetherthediscontinuitiesareremovableornon-removable.b)Find '( )f x .c)Find ''( )f x d)Determinetheintervalswhere ( )f x isincreasingordecreasing.
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TiebreakerQuestion3Name______________________School______________________Thefollowingtablegivesvariousvaluesofafunctionanditsderivatives.x ( )f x '( )f x ''( )f x 0 1 2 42 5 0 14 11 6 3Furthermore, ''( )f x iscontinuousforallrealnumbersx.
Isitpossiblefortheline𝑥 = 3tobeaverticalasymptoteforthegraph ( )y f x= ?Explain.
ACTM Regional Calculus Competition 2018
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Solutions1 C2 C3 B4 D5 E6 E7 C8 D9 B10 B11 A12 A13 B14 E15 C16 A17 B18 A19 D20 A21 E22 E23 B24 D25 D
ACTM Regional Calculus Competition 2018
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TiebreakerQuestion1SolutionLetf(x)beadifferentiablefunctionandcbeaconstantrealnumber.Letg(x)=cf(x).Completethefollowingstatement:g'(x)=cf'(x).Provethisresult.Proof:
( ) ( ) ( )
( ) ( )
( ) ( )( )
( ) ( )
( )
0
0
0
0
' lim
lim
lim
lim
'
h
h
h
h
g x h g xg x
hc f x h c f x
hc f x h f x
hf x h f x
ch
c f x
®
®
®
®
+ -=
+ -=
+ -=
+ -=
=
DefinitionofDerivativeDefinitionofg(x)DistributiveProperty[ca+cb=c(a+b)]LimitProperty:
( )( ) ( )( )lim limh a h a
c p h c p h® ®
é ù=ë û
DefinitionofDerivative
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TiebreakerQuestion2Solution
Considerthefamilyoffunctionsoftheform ( ) axf xbx c
=+,wherea,b,andcareallnon-zerorealnumbers.
Answerthefollowingintermsofa,b,andc.Justifyallanswers.a)Identifyanydiscontinuitiesofthefunction.Determinewhetherthediscontinuitiesareremovableornon-removable.
b)Find '( )f x .c)Find ''( )f x d)Determinetheintervalswhere ( )f x isincreasingordecreasing.
a)Sincefisarationalfunction,itiscontinuousthroughoutthedomain.Theonlydiscontinuityisatthe
zeroofthedenominator, cxb
= - .Atthisvalueofx,thenumeratorwillbe 0acb
- ¹ sinceaandcareboth
notzero.Thus, lim ( )bxc
f x®-
doesnotexist,sofhasanon-removablediscontinuity.
b)QuotientRule: ( ) ( )( ) ( )2 2'( )
bx c a ax b acf xbx c bx c+ -
= =+ +
or
ProductRule ( ) ( ) ( ) ( )( )
2 1 22'( ) ) acf x ax bx c b a bx c bx c abx a bx c
bx c- - -= - + + + = + é- + + ù =ë û +
c) ( ) ( )( )
2 33
2''( ) 2d abcf x ac bx c ac bx c bdx bx c
- -é ù= + = - + = -ë û +
d)Thedenominator ( )2bx c+ ofthederivative '( )f x willbepositiveforallrealvaluesofxwith
cxb
¹ - .Thusthesignof '( )f x willdependonthesignoftheproductac.
• Ifaandcarebothpositiveorbothnegative,then '( )f x willbepositiveandthus ( )f x willbe
increasingontheintervals , cb
æ ö-¥ -ç ÷è ø
and ,bc
æ ö- ¥ç ÷è ø
.
• Ifaandchavedifferentsigns,then '( )f x willbenegativeandthus ( )f x willbedecreasingonthe
intervals , cb
æ ö-¥ -ç ÷è ø
and ,bc
æ ö- ¥ç ÷è ø
.
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TiebreakerQuestion3SolutionThefollowingtablegivesvariousvaluesofafunctionanditsderivatives.x ( )f x '( )f x ''( )f x 0 1 2 42 5 0 14 11 6 3Furthermore, ''( )f x iscontinuousforallrealnumbersx.
Isitpossibleforthelinex=3tobeaverticalasymptoteforthegraph ( )y f x= ?Explain.
No.Thecontinuityof ''f impliesthat 'f isdifferentiable,andthuscontinuous,forallx.Repeatingthelogicgivesfiscontinuousforallx.Sincefiscontinuousforallx,therecanbenoverticalasymptote.