math1151 final exam review

8/12/2019 MATH1151 Final Exam Review

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MSLCMath1151

FinalExamReview

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1. Considerthefunction ( )y f x asdefinedbythegraphbelow.

a) Evaluatethefollowing:

i)lim ( )x

f x

ii)6

lim ( )x

f x

iii)2

lim ( )x

f x

iv)3

lim ( )x

f x

v)6

lim ( )x

f x

vi) lim ( )x

f x

b) Whatisthedomainof ( )f x inintervalnotation?

(Assumethefunctiongoespastthevisibleportion.)

Whatistherangeof ( )f x ?

c) Whereis ( )f x discontinuous? Whereis ( )f x

continuous?

d) Whereis ( )f x notdifferentiable?

e) Estimate '(1)f andgivetheequationofthetangent

linethere.

f) Estimate '( 3)f andgivetheequationofthetangentlinethere.

2. Findthefollowinglimitsorstatethatthelimitdoesnotexist. (Donotuseyourcalculatortohelpwiththese!)

a)

2

22

4lim

6z

z

z z

b) lim

x

xe x

c)

2

0

6lim

sinx

x x

x

d)

2

20

sin (3 )limt

t

t t e)

0

1 cos(2 )lim

sin cosx

x

x x

f)

2

2lim

3u

u

u u

3. Let2

2( )

2 1

xf x

x

. Usethedefinitionofthederivativetocalculate '(1).f Whatistheequationofthe

tangentlinethere?

4. Differentiatethefollowingusingderivativerules.

a)3 3( ) sin sin( )f x x x b)

22 5 ln164( ) 1 xg x x e c)

3 5 42( ) 2

x x

h x x x

5. Let

32( 1) cos

( )(csc )

x xz x

x x

. Find '( )z x withlogarithmicdifferentiation.

6. Suppose3 2 0x y xy . Findtheequationsofthetangentlinesatthepoint(1,1).

7. Useimplicitdifferentiationtofind 'y if 2cos( ) 3xy xy y .

8. Findthepoint ( , )x y onthegraph xy nearestthepoint (4,0) .

9. Let

2 1 on 0, 2( )

1 on 2,

ax bxf x

x

Findaandb sothat ( )f x iscontinuousanddifferentiable

everywhere.


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MSLCMath1151

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10. Determinethehorizontal(orslant)andverticalasymptotesof

a)3 1

( )1

xt x

x

b)

2

34)(

2

x

xxxz

11. Sketchthegraphof3 22 9 20x x usingyourknowledgeoftheinformationobtainedfromtheoriginal

functionandthefirstandsecondderivatives. Besuretoindicateallimportantfeatures. (Includesincreasing,

decreasing,max,

min,

concavity,

intercepts,

etc)

12. Sketchagraphof ( )f x withthefollowingproperties:

( 7) ( 4) (1) (6) 0f f f f , ( 2) 4f , (4) 2f , '( 2) '(4) "(0) 0f f f

lim ( ) 2x

f x

, lim ( ) 2x

f x

5

lim ( )x

f x

5

lim ( )x

f x

'( ) 0f x on ( , 5) ( 5, 2) (4, ) , '( ) 0f x on ( 2,4)

"( ) 0f x on ( , 5) (0, 6) , "( ) 0f x on ( 5, 0) (6, )

13. Findthemostgeneralantiderivativesofthefollowingfunctions.

a) 8 2

( ) 17ef x x x

b)2 1( ) 4 cscf x x

x

14. Findtheparticularantiderivativeofthefollowing.

a) ( ) xf x e x , (0) 2F b)

35

12( )

3

xg x

x , (1) 3G

15. Find y and dy if , 4, .11

xy x x

x

. Then,usedifferentialstoestimatey(4.1).

16. a) Expressthefollowingsumintheform

1

n

i

i

a

.3 3 3 3

5 5 10 5 15 5 5 52 1 2 1 2 1 2 1

n

n n n n n n n n

b) TheaboveisarightRiemannSumforthedefiniteintegral 2

b

f x dx . Findbandf(x).

c) Findthevalueoftheabovesumas n .

d) TheabovesumcanalsobearightRiemannSumforthedefiniteintegral

0

( )

b

f x dx . Find thenew

functionandboundsandcomputethevalueofthesumas n .

17. Expressthedefiniteintegral

3

2

1

x dx asarightRiemannSumandcomputethesumusingsigmaalgebraandthe

limitas n . VerifythatthesumyoucomputeisthesameascomputingtheintegralusingtheFundamentalTheoremofCalculus.

18. Suppose

5 0 5

0 2 3

( ) 10, ( ) 3, ( ) 6f x f x f x . Then3

2

( ) ?f x


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19. Findthefollowingintegrals:

a.

2

6

sec

tan

xdx

x b. cos 2 2 sin d c.

cos xdx

x

d.2

( 1)( 1)

xx e dx

e.

2 2

2

(ln )dx

x x f.

2

2

1 3x x dx

g.

5

1

2 1x dx h. 1

63 4

0

4 5x x dx i.2

2

1x dx

20. a. Let 51

cotx

g x t dt . Find 'g x .

b.Let

22

3( ) sin(2 )

xt

xf x e t dt . Find '( )f x .

21. Considerthe

integral

20

( ) cos 2 .

x

f t dt x A Determine

( )f t and

the

constant

A.

22. Aboatisbeingpulledtowardadockbymeansofaropeattachedtothefronttipofthebow. Initiallythereare

30feetofropeoutandtheropeistaughtandbeingreeledinbyacirculardevicethetopofwhichis10feet

higherthanthepointwheretheropeisattachedtotheboat. Thiscirculardevicehasaradiusof1footand

turnsattherateofonerevolutioneverypiseconds. Howfastistheboatmovingalongthewaterwhenthere

are15feetofropeout?

23. Abaseballdiamondisasquarewhichis90feetoneachside,andthepitcher'smoundisatthecenterofthe

square.Ifapitcherthrowsabaseballat93milesperhour,howfastisthedistancebetweentheballandfirst

basechangingastheballcrosseshomeplate.

24. Considerarectangleofperimeter12inches.Formacylinderbyrevolvingthisrectangleaboutoneofitsedges.

Whatdimensionsoftherectanglewillresultinacylinderofmaximumvolume?

25. Supposeacarpenterhas20feetofwindowtrim. Theplanscallforawindowtobebuiltsuchthatthewindow

isrectangularwithanequilateraltriangleatthetop. Findthedimensionsofthewindowsuchthattheareaof

thewindowismaximized. Whatistheareaofthewindow?

26. SupposethatafunctionT(t)isadifferentiablefunctionthatmodelshowthetemperaturechangesoverone24hourperiod. Lett=0atmidnight,withtmeasuredinhours. Refertothefollowingtable:

t 0 2 4 6 8 10 12 14 16 18 20 22

T 50 47 43 42 55 64 73 77 83 82 75 65

a) Isthereatimebetween6amandnoonwhenthetemperatureis61degrees?

b) Findtheaveragerateofchangeofthetemperaturefrom6amtonoon. Isthereatimebetween6amand

noonwhenthetemperatureischangingatthisrate?

math1151 final exam review

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