. . . . . .
Section4.7ApplicationstoBusinessand
Economics
Math1aIntroductiontoCalculus
April2, 2008
Announcements
◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323◮ MidtermII:4/11inclass
..Image: FlickruserPulpolux
. . . . . .
Announcements
◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323◮ MidtermII:4/11inclass
. . . . . .
Outline
Lasttime
Cost
Marketing
UtilityMaximization
Nexttime
. . . . . .
Theorem(L’Hopital’sRule)Suppose f and g aredifferentiablefunctionsand g′(x) ̸= 0 near a(exceptpossiblyat a). Supposethat
limx→a
f(x) = 0 and limx→a
g(x) = 0
or
limx→a
f(x) = ±∞ and limx→a
g(x) = ±∞
Then
limx→a
f(x)g(x)
= limx→a
f′(x)g′(x)
,
ifthelimitontheright-handsideisfinite, ∞, or −∞.
L’Hôpital’srulealsoappliesforlimitsoftheform∞∞
.
. . . . . .
Theorem(L’Hopital’sRule)Suppose f and g aredifferentiablefunctionsand g′(x) ̸= 0 near a(exceptpossiblyat a). Supposethat
limx→a
f(x) = 0 and limx→a
g(x) = 0
or
limx→a
f(x) = ±∞ and limx→a
g(x) = ±∞
Then
limx→a
f(x)g(x)
= limx→a
f′(x)g′(x)
,
ifthelimitontheright-handsideisfinite, ∞, or −∞.
L’Hôpital’srulealsoappliesforlimitsoftheform∞∞
.
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SummaryForm Method
00 L’Hôpital’sruledirectly
∞∞ L’Hôpital’sruledirectly
0 · ∞ jiggletomake 00 or
∞∞ .
∞−∞ factortomakeanindeterminateproduct
00 take ln tomakeanindeterminateproduct
∞0 ditto
1∞ ditto
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Outline
Lasttime
Cost
Marketing
UtilityMaximization
Nexttime
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Someeconomicsterms
DefinitionThe (total)cost functionofagoodistheamountthatmustbespentto produce x itemsofagood.
Thisisincontrastto
DefinitionThe price functionofagoodistheamountthatmustbespenttopurchase anitemofagoodif x itemsaresuppliedtothemarket.
Theunitsofcostaredollars(orothercurrency), whiletheunitsofpricearedollarsperitem(orothermeasureofquantity).
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Cost
Let C(x) bethecostofproducing x unitsofaproduct. Whatshapeshouldthegraphof C have?
. . . . . .
Since C′ > 0 always, what’smoreinterestingisthe averagecostfunction
c(x) =C(x)x
.
Assumingthatwecanselleverythingwemake, wewanttominimizethecostsofproduction.Whenistheaveragecostminimized?
c′(x) =xC′(x) − C(x)
x2=
C′(x) − c(x)x
,
so c′(x) = 0 ifandonlyif C′(x) = c(x). Inotherwords,
TheoremAtthepointofminimalaveragecostofproduction, theaveragecostequalsthemarginalcost.
. . . . . .
Since C′ > 0 always, what’smoreinterestingisthe averagecostfunction
c(x) =C(x)x
.
Assumingthatwecanselleverythingwemake, wewanttominimizethecostsofproduction.
Whenistheaveragecostminimized?
c′(x) =xC′(x) − C(x)
x2=
C′(x) − c(x)x
,
so c′(x) = 0 ifandonlyif C′(x) = c(x). Inotherwords,
TheoremAtthepointofminimalaveragecostofproduction, theaveragecostequalsthemarginalcost.
. . . . . .
Since C′ > 0 always, what’smoreinterestingisthe averagecostfunction
c(x) =C(x)x
.
Assumingthatwecanselleverythingwemake, wewanttominimizethecostsofproduction.Whenistheaveragecostminimized?
c′(x) =xC′(x) − C(x)
x2=
C′(x) − c(x)x
,
so c′(x) = 0 ifandonlyif C′(x) = c(x). Inotherwords,
TheoremAtthepointofminimalaveragecostofproduction, theaveragecostequalsthemarginalcost.
. . . . . .
Since C′ > 0 always, what’smoreinterestingisthe averagecostfunction
c(x) =C(x)x
.
Assumingthatwecanselleverythingwemake, wewanttominimizethecostsofproduction.Whenistheaveragecostminimized?
c′(x) =xC′(x) − C(x)
x2=
C′(x) − c(x)x
,
so c′(x) = 0 ifandonlyif C′(x) = c(x). Inotherwords,
TheoremAtthepointofminimalaveragecostofproduction, theaveragecostequalsthemarginalcost.
. . . . . .
Since C′ > 0 always, what’smoreinterestingisthe averagecostfunction
c(x) =C(x)x
.
Assumingthatwecanselleverythingwemake, wewanttominimizethecostsofproduction.Whenistheaveragecostminimized?
c′(x) =xC′(x) − C(x)
x2=
C′(x) − c(x)x
,
so c′(x) = 0 ifandonlyif C′(x) = c(x). Inotherwords,
TheoremAtthepointofminimalaveragecostofproduction, theaveragecostequalsthemarginalcost.
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ExampleLet C(x) = 16, 000 + 200x + 4x3/2. Find
(a) thecost, averagecost, andmarginalcostataproductionlevelof1000units,
(b) theproductionlevelthatwillminimizeaveragecost
(c) theminimumaveragecost.
Solution
(a) 8000(27 + 5
√10
), 8
(27 + 5
√10
), 200 + 60
√10, or
342, 491, 342.491, 389.737
(b) 400
(c) 300
. . . . . .
ExampleLet C(x) = 16, 000 + 200x + 4x3/2. Find
(a) thecost, averagecost, andmarginalcostataproductionlevelof1000units,
(b) theproductionlevelthatwillminimizeaveragecost
(c) theminimumaveragecost.
Solution
(a) 8000(27 + 5
√10
), 8
(27 + 5
√10
), 200 + 60
√10, or
342, 491, 342.491, 389.737
(b) 400
(c) 300
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Outline
Lasttime
Cost
Marketing
UtilityMaximization
Nexttime
. . . . . .
Marketing
Let p(x) bethepricefunction. Whatshouldtheshapeof p be?
DefinitionThe revenue functionisthetotalamounttakeninfromthesaleofx itemsofagood.
WehaveR(x) = x · p(x)
andthe profitP(x) = R(x) − C(x).
Clearly, profitismaximizedwhen P′(x) = 0.
TheoremAtthepointofmaximalprice, themarginalrevenueequalsthemarginalcost.
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Marketing
Let p(x) bethepricefunction. Whatshouldtheshapeof p be?
DefinitionThe revenue functionisthetotalamounttakeninfromthesaleofx itemsofagood.
WehaveR(x) = x · p(x)
andthe profitP(x) = R(x) − C(x).
Clearly, profitismaximizedwhen P′(x) = 0.
TheoremAtthepointofmaximalprice, themarginalrevenueequalsthemarginalcost.
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Marketing
Let p(x) bethepricefunction. Whatshouldtheshapeof p be?
DefinitionThe revenue functionisthetotalamounttakeninfromthesaleofx itemsofagood.
WehaveR(x) = x · p(x)
andthe profitP(x) = R(x) − C(x).
Clearly, profitismaximizedwhen P′(x) = 0.
TheoremAtthepointofmaximalprice, themarginalrevenueequalsthemarginalcost.
. . . . . .
Marketing
Let p(x) bethepricefunction. Whatshouldtheshapeof p be?
DefinitionThe revenue functionisthetotalamounttakeninfromthesaleofx itemsofagood.
WehaveR(x) = x · p(x)
andthe profitP(x) = R(x) − C(x).
Clearly, profitismaximizedwhen P′(x) = 0.
TheoremAtthepointofmaximalprice, themarginalrevenueequalsthemarginalcost.
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Marketing
Let p(x) bethepricefunction. Whatshouldtheshapeof p be?
DefinitionThe revenue functionisthetotalamounttakeninfromthesaleofx itemsofagood.
WehaveR(x) = x · p(x)
andthe profitP(x) = R(x) − C(x).
Clearly, profitismaximizedwhen P′(x) = 0.
TheoremAtthepointofmaximalprice, themarginalrevenueequalsthemarginalcost.
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ExampleA manufacturerhasbeenselling1000televisionsetsaweekat$450each. A marketsurveyindicatesthatforeach$10rebateofferedtothebuyer, thenumberofsetssoldwillincreaseby100perweek.
(a) Findthedemandfunction
(b) Howlargearebateshouldthecompanyofferthebuyerinordertomaximizerevenue?
(c) Ifitsweeklycostfunctionis C(x) = 68, 000 + 150x, howshouldthemanufacturersetthesizeoftherebateinordertomaximizeitsprofit?
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Solution
(a) Tofind p(x) wehave p(1000) = 450, p(1100) = 440. So
p(x) =p(1100) − p(1000)
1100− 1000(x− 1000) + p(1000)
= − 110
(x− 1000) + 450 = − 110
x + 550
(b) Thesalesfunctionis R(x) = xp(x) = − 110
x2 + 550x. Marginal
revenueistherefore
R′(x) = −15x + 550.
Revenueismaximizedwhen R′(x) = 0, orx = 550× 5 = 2750. Now p(2750) = −275 + 550 = 275,sotherevenue-optimizingrebateis$275.
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Solution
(a) Tofind p(x) wehave p(1000) = 450, p(1100) = 440. So
p(x) =p(1100) − p(1000)
1100− 1000(x− 1000) + p(1000)
= − 110
(x− 1000) + 450 = − 110
x + 550
(b) Thesalesfunctionis R(x) = xp(x) = − 110
x2 + 550x. Marginal
revenueistherefore
R′(x) = −15x + 550.
Revenueismaximizedwhen R′(x) = 0, orx = 550× 5 = 2750. Now p(2750) = −275 + 550 = 275,sotherevenue-optimizingrebateis$275.
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Solution(Continued)
(c) Nowwehave
P(x) = R(x) − C(x) = − 110
x2 + 550x− (68, 000 + 150x)
= − 110
x2 + 400x− 68, 000
Marginalprofitis
P′(x) = −15x + 400
whichiszerowhen x = 2000. Thiscorrespondstoarebateof$100.
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Outline
Lasttime
Cost
Marketing
UtilityMaximization
Nexttime
. . . . . .
UtilityMaximization
A studentderivesutility(happiness)fromconsumingburritosandchipsaccordingtothefunction
u(x, y) = 10x1/2y2/5
where x isthenumberofburritosconsumedand y isthenumberofbasketsofchipsconsumedeveryweek. Burritoscost$5apieceandbasketsofchipscost$2apiece.Ifthestudenthasafixedbudgetof$18withwhichtobuyhisMexicanfood, whatquantitiesofburritosandchipsshouldhebuytomaximizehisutility?
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SolutionWeneedtomaximize u subjecttotheconstraintthat5x + 2y = 60. So
u(x) = 10x1/2(18− 5x
2
)2/5
dudx
= 1022/5
{12x
−1/2(18− 5x)2/5 + x1/2 25(18− 5x)−3/5(−5)
}=
10
22/5x−1/2(18− 5x)−3/5 {1
2(18− 5x) − 2x}
=10
22/5x−1/2(18− 5x)−3/5 (
9− 92x
)So u′(x) = 0 when x = 2. Then y = 4.
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SolutionWeneedtomaximize u subjecttotheconstraintthat5x + 2y = 60. So
u(x) = 10x1/2(18− 5x
2
)2/5
dudx
= 1022/5
{12x
−1/2(18− 5x)2/5 + x1/2 25(18− 5x)−3/5(−5)
}=
10
22/5x−1/2(18− 5x)−3/5 {1
2(18− 5x) − 2x}
=10
22/5x−1/2(18− 5x)−3/5 (
9− 92x
)So u′(x) = 0 when x = 2. Then y = 4.
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Outline
Lasttime
Cost
Marketing
UtilityMaximization
Nexttime
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NexttimeNewton’sMethod
