MATH140 Exam 2 - Sample Test Solutionspenn.liontutors.com/uploads/MATH140_Exam_2... · MATH140 –...
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1 © LionTutors 2018 www.LionTutors.com MATH140 – Exam 2 - Sample Test 1 – Detailed Solutions 1. D. Create a first derivative number line 2 1 0 2 1 2 1 1 2 3 '( ) cos cos cos cos / f x x x x x x π = − = − = = = + − 4 π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 3 π 2 π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 2. C. Create a second derivative number line. Remember that a number is only an inflection point if it changes concavity at that point. 5 3 4 2 2 2 2 2 30 20 10 90 0 10 9 0 10 3 3 0 3 3 '( ) ''( ) ( ) ( )( ) , , f x x x f x x x x x x x x x x x = − + = − = − = − + = = = − + − − + -3 0 3 Since ''( ) f x does not change signs at 0 x = , it is not considered an inflection point. 3. B. For parts A & B, look at the first derivative number line. Parts C & D are false because the domain of the original function is all real numbers. Part E is false because the second derivative does not change signs around 0 x = . 10 6 0 10 6 3 5 / x x x + = = − = − + − + 1 3 3 0 0 x x = = -3/5 0
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Transcript of MATH140 Exam 2 - Sample Test Solutionspenn.liontutors.com/uploads/MATH140_Exam_2... · MATH140 –...
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MATH140–Exam2-SampleTest1–DetailedSolutions
1.D.Createafirstderivativenumberline
2 10 2 12 1
1 2
3
'( ) coscos
coscos /
f x xx
xx
x π
= −
= −
=
=
=
+ −
4π⎛ ⎞⎜ ⎟⎝ ⎠
3π
2π⎛ ⎞⎜ ⎟⎝ ⎠
2.C.Createasecondderivativenumberline.Rememberthatanumberisonlyaninflectionpointifitchangesconcavityatthatpoint.
5 3
4 2
2 2
2
2 30 2010 90
0 10 90 10 3 30 3 3
'( )''( )
( )( )( )
, ,
f x x xf x x x
x xx x x
x x x
= − +
= −
= −
= − +
= = = −
+ − − +
-3 0 3
Since ''( )f x doesnotchangesignsat 0x = ,itisnotconsideredaninflectionpoint.
3.B.ForpartsA&B,lookatthefirstderivativenumberline.PartsC&Darefalsebecausethedomainoftheoriginalfunctionisallrealnumbers.PartEisfalsebecausethesecondderivativedoesnotchangesignsaround 0x = .
10 6 010 6
3 5/
xx
x
+ =
= −
= −
+ − +
133 00
xx
=
= -3/50
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4.E.Tofindtheabsolutemaxandmin,findthecriticalnumbersthatarewithintheinterval.Takethecriticalnumbers,andtheendpointsoftheintervalandplugintotheoriginalfunction.
2
2
2 2
2
2
2
2 2 12
2 42
42
( )( ) ( )( )'( )( )
'( )( )
'( )( )
x x xf xx
x x xf xx
x xf xx
− −=
−
− −=
−
−=
−
Thecriticalnumbersare 0 2 4, ,x = howeverweonlyconsider 4x = sinceitistheonlynumberintheinterval.
3 9( )f =
4 8( )f = ßMinimum
7 49 5( ) /f = ßMaximum
5.D.Simplifybymultiplyingbytheconjugate
.
2 22 2
2 2
2 2
2 2 2 2
2 2
4 9 4 3lim 4 9 4 3 *4 9 4 3
4 9 (4 3 ) 12lim lim4 9 4 3 4 9 4 3
12 12 12lim lim 32 2 44 4
x
x x
x x
x x x xx x x xx x x x
x x x x xx x x x x x x x
x xx xx x
→∞
→∞ →∞
→∞ →∞
+ + −+ − −
+ + −
+ − −=
+ + − + + −
= = =++
3©LionTutors2018
6.E.UsethedefinitionoftheMVT.
2
2
2
2
16 116 1
1 1 11615
15116
15
1 116
164
( ) ( ) '( )f f f x
x
x
x
xx
−=
−
− −=
−−
=
− −=
=
=
(Notethatyoudonotneedtoconsiderx=-4becauseitisnotintheinterval)
7.D.Thisisanoptimizationproblem.Wewanttominimizethedistancebetweenthegraphandthepoint.Firstsolvefory,thenusethedistanceformula.Thentakethederivativeofthedistanceandsetitequalto0.Rememberthatyoucansquarethedistanceequationbeforetakingthederivative,tomakeiteasier.Thiswillnotaffecttheanswer.
2 2
2
56 8
56 8
y x
y x
= −
= −
2 2 2
2 2
2 2 2
2 56 8 0
2 56 82 56 8
0 2 2 160 2 4 1614 42 7
( ) ( )
( )( )( )
/
d x x
d x xd x x
x xx xx
x
= − + − −
= − + −
= − + −
= − −
= − −
− =
= −
4©LionTutors2018
8.C.Tofindhorizontalasymptoteslookatthepowers.Sincethehighestpoweristhesameontopandbottomthendividethecoefficients(don’tforgetthesigns).Tofindverticalasymptotes,SIMPLIFY,thensetthedenominatorequalto0.
4 1 13 1
4 13
( )( )( )( )( )
( )( )
x xf xx x
xf xx
− +=
+ +
−=
+
3 03
xx+ =
= −
9.B.Uselongdivisiontodividethenumeratorbythedenominator.
2
2
3 81 3 5
3 38
( )
xx x x
x xx
+− +
− −
10.B.Todeterminewhichgraphthisislookatintercepts,horizontalasymptotes,verticalasymptotes.
ThegraphhasaH.A.aty=0,soweknowthepoweronbottommustbebigger.ThiseliminatesanswerchoiceD.
ThegraphhasV.A.atx=1andx=-1.ThiseliminatesarechoicesC&EbecausethosefunctionsonlyhaveaV.A.atx=1(remembertosimplifythefunctionandsetthedenominator=0tofindV.A.)
Thefunctionhasay-interceptatx=0.Thereforewhenyouplugina0forx,youshouldget0fory.ThiseliminatesanswerchoiceA.
Therefore,byprocessofeliminationthisgraphhastobeanswerchoiceB.
5©LionTutors2018
11.D.Findthelinearization,thenplugin2.1
10 3 210 3 63 163 16
2 1 3 2 1 162 1 9 7
y xy xy xL x xLL
− = − −
− = − +
= − +
= − +
= − +
=
( )
( )( . ) ( . )( . ) .
12.E.Tofinddy calculatethedifferential.Tofind yΔ calculatetheactualchange.
88 2 58
'( )( )( )( )(. )
dy f x dxdy x dxdydy
=
=
=
=
22
2 5 2
54 1 4 2 12
24 159
( . ) ( )
( )
y f f
y
yy
Δ = −
⎡ ⎤⎛ ⎞ ⎡ ⎤⎢ ⎥Δ = − − −⎜ ⎟ ⎣ ⎦⎢ ⎥⎝ ⎠⎣ ⎦
Δ = −
Δ =
13.A.ThisisanapplicationoftheMeanValueTheorem.
10 810 8
10 3 72
10 3 1410 17
( ) ( ) '( )
( )
( )( )
f f f x
f
ff
−≥
−
−≥
− ≥
≥
6©LionTutors2018
14.C.Thisisanapplicationofthe2ndDerivativeTest.Plugeachofthecriticalnumbersintothe2ndderivative.Ifyougetapositivevaluethenitisalocalminimum.Ifyougetanegativevaluethenitisalocalmaximum.
0 3''( )f = − ,sothereisalocalmaximumatx=0
273 24
''( / )f = ,sothereisalocalminimumatx=3/2
2 5''( )f − = ,sothereisalocalminimumatx=-2
15.B.Thisisanoptimizationproblem
200200
xy
yx
=
=
1
2
2
2
2
2
2 42002 4
2 8002 8008000 2
8002
2 80040020
'
P x y
P xx
P x xP x
x
x
xxx
−
−
= +
⎛ ⎞= + ⎜ ⎟
⎝ ⎠
= +
= −
= −
=
=
=
=
20020
102 20 4 1080( ) ( )
y
yPP
=
=
= +
=
16.C.Firstsimplify,thentaketheantiderivative.2 2
22 2 2
1 1 1sin sin cscsin sin sinx x xx x x+
= + = +
Sotheantiderivativeis cotx x C− +
7©LionTutors2018
17.B.Taketheantiderivativeandthenusethepointtosolveforc.
Theantiderivativeof2 sinx x− is 2 cosx x C+ +
2
2
2
5 0 05 1
4
42 2 2
42 4
( ) coscos( )
cos
f x x x CC
CC
f
f
π π π
π π
= + +
= + +
= +
=
⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞= +⎜ ⎟
⎝ ⎠
18.
Thisisanoptimizationproblem.Findaformulaforvolume,thentakethederivativeandsetitequalto0.
2
2
2
18 2 18 218 218 2 1 2 18 2 218 2 4 18 2
0 18 2 18 2 40 18 2 18 69 3
( )( )( )( )
' ( ) ( ) ( )( )( )( )' ( ) ( )( )[ ]( )( ),
V l w hV x x xV x xV x x xV x x x
x x xx x
x x
= × ×
= − −
= −
= − + − −
= − − −
= − − −
= − −
= =
However,xcannotequal9.Sincethereisonly18inchestoworkwith,youcannottakeaway9inchesfromeachside.
19.TRUE.ThisisthedefinitionofRolle’sTheorem.Alsonotethatsince𝑓isdifferentiable,thenitmustalsobecontinuousaccordingtothedefinitionofdifferentiability.
20.FALSE.Anabsoluteminimumcanoccurattheendpointoftheintervalyouarelookingat.
21.FALSE.Thesecondderivativealsoneedstochangesignsatthatpoint.Lookatthefunction𝑓 𝑥 = 𝑥!asacounterexample.
8©LionTutors2018
22.Ithelpstocreatethe1stand2ndderivativenumberlinesbeforetryingtograph.
a
03
02
23.
a)x-Intercept(s):(0,0) b)y–Intercept:(0,0) c)VerticalAsymptote(s):NONEd)HorizontalAsymptote(s):y=0(sincethepoweronbottomisbigger)
e)Increasing:(-1,1) − + −f)Decreasing:(-∞,1)U(1,∞)g)Relativemaximum(s):x=1 -1 1h)Relativeminimum(s):x=-1
− + − +i)ConcaveUp:(- 𝟑,0)U( 𝟑,∞)j)ConcaveDown:(-∞,- 𝟑)U(0, 𝟑) - 𝟑0 𝟑k)Inflectionpoints:x=- 𝟑,0, 𝟑
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MATH140–Exam2-SampleTest2–DetailedSolutions
1.C.Rememberthatfindingthelinearizationjustmeansfindingtheequationofthetangentline.Findthepointbyplugging0intotheoriginalfunctionandfindtheslopebytakingthederivativeandpluggingin0.
Point: 105
( , )
Slope: 1 25 /( ) ( )f x x −= +
3 2
3
1 52
1 1 102 5 2 5 5 5 10 5
/'( ) ( )
'( )( )
f x x
f
−= − +
− − −= = =
Line: 1 1 05 10 5
( )y x−− = −
15 10 5
15 10 5
15 10 5
( )
xy
xy
xL x
−− =
= −
= −
2.A.Usedifferentialstoapproximatedy ,thechangegoingfrom10to10.3.Rememberthatinthiscaseyourdx=0.3.
Addthisto 10( )f togettheanswer.
108 32 4
'( )(. ).
dy f dxdydy
=
= −
= −
10©LionTutors2018
10 3 1010 3 20 2 410 3 17 6
( . ) ( )( . ) ( . )( . ) .
f f dyff
≈ +
≈ + −
≈
3.C.Takethederivative,usingthequotientrule.Thensetthetopandbottomequalto0.Rememberthat𝑥 = 0cannotbeacriticalnumbersinceitisnotinthedomainoftheoriginalfunction.
2 3
1 31 3 2 3
2 2 1 3 2 1 3 2
2
22 2 12 22 2 1 2 233 3 32 2
xf xx
x x xx x x x x xf x
x x x x x x
−−
−=
⎡ ⎤− − −⎢ ⎥− − − − + −
⎣ ⎦= = = =− −
/
// /
/ /
( )( )
( ) ( )( ) ( ) *'( )
( ) ( )
1 12 0 2 63 3
2 0 2
x x x
x x
− = ⇒ = ⇒ =
− = ⇒ =
4.D.Findthecriticalnumbersandsetupafirstderivativenumberline.
3 22 30 1 30 1 3
'( )( )( ), ,
f x x x xx x x
x x x
= + −
= − +
= = = −
− + − +
-3 0 1
5.E.Findthecriticalnumbersintheinterval.Takethosenumbersandtheendpointsoftheintervalandplugintotheoriginalfunction.
2 20 2 12 0 1
3 32 2 2
'( ) sin cos coscos (sin )
cos sin
,
f x x x xx x
x x
x xπ π π
= +
= +
= = −
= =
0 0
32
3 12
2 0
( )
( )
f
f
f
f
π
π
π
=
⎛ ⎞=⎜ ⎟
⎝ ⎠
⎛ ⎞= −⎜ ⎟
⎝ ⎠
=
11©LionTutors2018
6.E.Usethesecondderivativetesttoanswerthisquestion.Wheneverthe2ndderivativeispositiveatacriticalnumber,thereisalocalminimum.Whenthe2ndderivativeisnegativeatacriticalnumber,thereisalocalmaximum.Ifthe2ndderivativeisequaltozero,thenyoucannottell.
0 0''( )f =
2 8 2 16''( ) ( )f = − = − ßlocalmaximum
6 24 6 144''( ) ( )f = = ßlocalminimum
7.C.Tofindinflectionpointslookatthe2ndderivative.Rememberthatconcavitymustchangeatthatpointinordertobeaninflectionpoint.
3 5
2 4
2 2
2
5 20 1260 60
0 60 10 60 1 10 1 1
'( )''( )
( )( )( )
, ,
f x x xf x x x
x xx x x
x x x
= + −
= −
= −
= − +
= = = −
-++-
-1 0 1
( )f x hastwoinflectionpoints(atx=1andx=-1)
8.A.AnswerchoiceAisthefalsestatementbecausex=-4isnotinthedomain,soitcannotbeconsideredacriticalnumber.
9.B.UsethedefinitionoftheMVTtoanswerthisquestion.
12©LionTutors2018
1 01 0
4 5 11 2
1 212
14
( ) ( ) '( )
( ) ( )
f f f x
x
x
x
x
−=
−
− − −=
=
=
=
10.E.TofindVAsetthedenominatorequaltozero.TofindHAlookatthehighestpowerontopandbottom.
V.A. 22 2 0( )x + =
2 2 02 2
1
xxx
+ =
= −
= −
H.A.Thepowersarethesameontopandbottomsowetakethecoefficientsanddivide.Thecoefficientontopis-2andthecoefficientonbottomis4(becausewehave2squared).-2/4=-2.SotheH.A.is 1 2/y = −
11.C.Totakethelimitasxgoesnegativeinfinity,lookatthemostdominanttermsontopandbottom.
3 9
3
3
3
82
2 12
lim
lim
x
x
xx
xx
→−∞
→−∞=
12.A.TofindV.A.,setthedenominatorequalto0.
4 04 04
xxx
− =
− =
=
13.D.Takethesecondderivativeandsetupasecondderivativenumberline.
13©LionTutors2018
3
03
2 2
'( ) sin''( ) coscos
,
f x xf x x
x
x π π
= − +
= −
− =
=
− + −
2π
32π
14.B.Firstsimplifythefunction,thentaketheantiderivative.
3
3 3
3
1 2
2
( )
( )
tf tt t
f t t−
= −
= −
Sotheantiderivativeis2
22t t C−
− +−
Whichsimplifiesto2
1 22
t Ct−
− +
15.D.Taketheanti-derivativetwice,solvingforCeachtime.
14©LionTutors2018
2
2
2
3 2
3 2
6 43 4
4 3 1 4 14 1
5
3 4 52 5
2 0 0 02
2 5 23 27 18 15 23 26
a t tv t t t C
CC
C
v t t ts t t t t C
CC
s t t t tss
= −
= − +
= − +
= − +
=
= − +
= − + +
= − + +
=
= − + +
= − + +
=
( )( )
( ) ( )
( )( )
( )( )( )
16.TRUE.Firstnotethatsincethederivativeexists,thefunctioniscontinuous.Ifthederivativeisnon-zero,thenitmeansthefunctionisalwaysincreasingoralwaysdecreasing.Thereforeitcannothavethesamey-valuetwice.
17.TRUE.Everyfunctionhasanabsolutemaximumandminimumonaclosedinterval.
18.FALSE.Functionsdonotnecessarilyhaveanabsolutemaximumonanopeninterval.
19.FALSE.If𝑓and𝑔arebothincreasing,itmeansthat𝑓! 𝑥 > 0and𝑔! 𝑥 > 0,howeveritdoesnotnecessarilymeanthat𝑓and𝑔aregreaterthan0.Whenyoucomputer𝑓𝑔,youhavetousetheproductrule,soiftheoriginalfunctionislessthan0,thentheproductmayalsobelessthan0,andthusnotnecessarilyincreasing.
20.Thisisanoptimizationproblem.Overallwewanttominimizecost.
15©LionTutors2018
2
2
2
888
82
2
( )
Vx y
yx
y
y
=
=
=
=
=
21.Takethelimitsasxgoestoinfinityandnegativeinfinity.
2 2
2 2
39 4 9 3 13 1 3 3 3
39 4 9 3 13 1 3 3 3
lim lim lim lim
( )lim lim lim lim
x x x x
x x x x
xx x x xx x x x
xx x x xx x x x
→∞ →∞ →∞ →∞
→−∞ →−∞ →−∞ →−∞
+= = = =
+
+ −= = = = −
+
SotheH.A.arey=1,andy=-1.
22.
2 2
22
2
2 1
2
2
2
3
3
5 1 4 386 12
966
6 9612 96
960 12
9612
12 9682
( )( )
'
C x x xy
C x xx
C xx
C x xC x x
xx
xx
xxx
−
−
= + +
⎛ ⎞= + ⎜ ⎟
⎝ ⎠
= +
= +
= −
= −
=
=
=
=
16©LionTutors2018
a)Intercept(s):(0,0) b)VerticalAsymptote(s):NONEc)HorizontalAsymptote(s):y=0d)Graphcrossesthehorizontalasymptoteatx=0(settheequationtotheasymptotetofindthis)
e)Increasing:(-1,1) _+_f)Decreasing:(-∞,-1)U(1,∞)g)Relativemaximum(s):(1,1/2) -11h)Relativeminimum(s):(-1,-1/2)
i)ConcaveUp:(- 3 ,0)U( 3 , ∞)_+_+
j)ConcaveDown:(−∞,- 3 )U(0, 3 )
k)Inflectionpoints:x=- 3 ,0, 3 - 3 0 3
23.Lookingatthegraphofthederivative,createafirstderivativenumberlinebylookingatwherethegraphispositiveandnegative.Thatis,wherethegraphisaboveorbelowthex–axis.
17©LionTutors2018
+ + − +
-1 1 4
Tocreatethesecondderivativenumberline,lookatwherethegraphisincreasinganddecreasing.
−+-+
-103
increasing:(-∞,-1)U(-1,1)U(4, ∞)decreasing:(1,4)
concaveup:(-1,0)U(3, ∞)concavedown:(∞,-1)U(0,3)
localmaximum:x=1localminimum:x=4inflectionpoint:x=-1,0,3
24.Thisisanoptimizationquestion.Overallyouaretryingtominimizecost.
2 2
2 2
22 2
2 1
2
2
2
3
3
2
10*2 8*220 16 20
20 2020 16
20 320320' 40
3200 40
320 40
320 4082
20 5(2)
C r rh V r hC r rh r h
C r r hr r
C r r
C rr
rr
rr
rr
r
h
π π π
π π π π
π π
π π
ππ
ππ
ππ
π π
−
= + =
= + =
= + =
= +
= −
= −
=
=
=
=
= =
