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Final

MATH 1157 Final: MATH 1157 Exam 3 Review


Department
Mathematics
Course Code
MATH 1157
Professor
All
Study Guide
Final

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MSLCMath1151
Exam3Review
1. Aparticleistravelingalongaonedimensionalpath(suchasanumberline).Thepositionoftheparticleisgovernedbythe
timefunction432
() 3 16 18 2xt t t t 
,wheretisinminutesand50
t
.Answerthefollowingquestions.
a) Atwhattimesistheparticlestationary?
b) Forwhichtimeintervalsistheparticlemovinginapositivedirection?Anegativedirection?
c) Whatistheparticle’smostpositiveposition?Mostnegativeposition?
d) Whatistheparticle’sdisplacement?Whatisthetotaldistancetheparticlehastraveled?
e) Whendoestheparticle’saccelerationundergoasignchange?Whatistheparticle’saccelerationatthetimes
whentheparticleisstationary?
f) Sketchagraph(onatxcoordinateplane)oftheparticle’spositionusingtheinformationabove.
 
2. Sketchthegraph
232
1
xx
y
x

usingyourknowledgeoflimits,functions,andderivatives(firstandsecondderivatives).
Findallthepointsofinterest(xandyintercepts,criticalvalues,maxandmins,concavities,etc).
3. Sketchagraphwiththefollowingproperties:
lim ( ) 0
xfx
and
7
lim ( )
xfx
,(2) (2) 1ff

, '(0) '(4) 0ff
'( ) 0fxon

0,4 ,0)(' xf on

,0 4,7 ,0)("
xf on
2,2,0)(" xf on
,2 2,7 
Domain

,7
4. Findtheareaofthelargestrectanglethatcanbeinscribedintheellipse
22
22
1
xy
ab
5. Supposearectangularboxistocontain10cubicfeet.Thelengthoftheboxistobetwicethewidth.Materialforthebox
costs$2.50persquarefootforthebottomofthebox,$1.50persquarefootforthesidesoftheboxand$2.00persquare
footforthetopofthebox.Findthedimensionsoftheboxthatwouldminimizethecost.
6. Findthemostgeneralantiderivativesofthefollowingfunctions.
a) 2
5
2
() 20fx x
x

b) 2
() 4sec sin
f
xxx
7. Findtheparticularantiderivativeofthefollowing.
a)()
x
f
xe, (0) 2F b)

3
2
() 3
x
gx
x
, (1) 3G
8. Usedifferentialstoapproximatesin(31 ).(Hint:Besuretoconverttoradians.)
9. Verifythefollowingintegralsgivetheindicatedresult.
a.
2
2
10
5
10
dx x x C
x
xxx
 
  b. 22
xx xx
ee ee
dx C


c.2
2ln 1 ln 1
1dx x x C
x

  d. 2
21
(1) 1
xx
xx
ee
dx C
ee

10. Estimatetheareaunderthecurve2
() 3
f
xx xontheinterval[1,9]using4rectanglesandrightRiemannSums.
11. Findthefollowinglimits.
a) 2
4
lim 3
x
xx

b)

1
lim x
x
xex
  c)

2
4
0
5
lim ln
x
x
x
d)
0
1
lim cot
xx
x



12. DeterminewhethertheMeanValueTheoremappliestothefollowingfunctiononthefollowinginterval,andifso,findthe
point(s)guaranteedbytheMeanValueTheorem.

5
2,3,6yx
x

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