2018 calculus regional & key finalized

ACTM Regional Calculus Competition 2018

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.


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.


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.


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.


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.


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.¥


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.


TiebreakerQuestion1Name______________________School______________________Letf(x)beadifferentiablefunctionandcbeaconstantrealnumber.Let𝑔 𝑥 = 𝑐 ⋅ 𝑓 𝑥 .Completethefollowingstatement:𝑔, 𝑥 =_________________.Provethisresult.


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.


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.


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


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


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

.


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.

2018 calculus regional & key finalized

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