Related Rates Section 2.6 Read Guidelines For Solving Related Rates Problems on p. 150.
Lesson 13: Related Rates of Change
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Transcript of Lesson 13: Related Rates of Change
. . . . . .
Section2.7RelatedRates
V63.0121.034, CalculusI
October19, 2009
Announcements
I Midtermaverage57.69/75(77%), median59/75(79%),standarddeviation11%
I Solutionssoon.
. . . . . .
“Isthereacurve?”
I MidtermI Meanwas77%and
standarddeviationwas11%
I Soscoresaveragearegood
I Scoresabove66/75(88%)aregreat
I Forfinallettergrades,refertosyllabus
. . . . . .
Whatarerelatedratesproblems?
Todaywe’lllookatadirectapplicationofthechainruletoreal-worldproblems. Examplesofthesecanbefoundwheneveryouhavesomesystemorobjectchanging, andyouwanttomeasuretherateofchangeofsomethingrelatedtoit.
. . . . . .
Problem
ExampleAnoilslickintheshapeofadiskisgrowing. Atacertaintime,theradiusis1kmandthevolumeisgrowingattherateof10,000literspersecond. Iftheslickisalways20cmdeep, howfastistheradiusofthediskgrowingatthesametime?
. . . . . .
A solution
Thevolumeofthediskis
V = πr2h.
WearegivendVdt
, acertain
valueof r, andtheobjectis
tofinddrdt
atthatinstant.
. .r.h
. . . . . .
Solution
SolutionDifferentiating V = πr2h withrespecttotimewehave
dVdt
= 2πrhdrdt
+ πr2����0
dhdt
=⇒ drdt
=1
2πrh· dVdt
.
Nowweevaluate:
drdt
∣∣∣∣r=1 km
=1
2π(1 km)(20 cm)· 10, 000 L
s
Convertingeverylengthtometerswehave
drdt
∣∣∣∣r=1 km
=1
2π(1000m)(0.2m)· 10m
3
s=
140π
ms
. . . . . .
Solution
SolutionDifferentiating V = πr2h withrespecttotimewehave
dVdt
= 2πrhdrdt
+ πr2����0
dhdt
=⇒ drdt
=1
2πrh· dVdt
.
Nowweevaluate:
drdt
∣∣∣∣r=1 km
=1
2π(1 km)(20 cm)· 10, 000 L
s
Convertingeverylengthtometerswehave
drdt
∣∣∣∣r=1 km
=1
2π(1000m)(0.2m)· 10m
3
s=
140π
ms
. . . . . .
Solution
SolutionDifferentiating V = πr2h withrespecttotimewehave
dVdt
= 2πrhdrdt
+ πr2����0
dhdt
=⇒ drdt
=1
2πrh· dVdt
.
Nowweevaluate:
drdt
∣∣∣∣r=1 km
=1
2π(1 km)(20 cm)· 10, 000 L
s
Convertingeverylengthtometerswehave
drdt
∣∣∣∣r=1 km
=1
2π(1000m)(0.2m)· 10m
3
s=
140π
ms
. . . . . .
Solution
SolutionDifferentiating V = πr2h withrespecttotimewehave
dVdt
= 2πrhdrdt
+ πr2����0
dhdt
=⇒ drdt
=1
2πrh· dVdt
.
Nowweevaluate:
drdt
∣∣∣∣r=1 km
=1
2π(1 km)(20 cm)· 10, 000 L
s
Convertingeverylengthtometerswehave
drdt
∣∣∣∣r=1 km
=1
2π(1000m)(0.2m)· 10m
3
s=
140π
ms
. . . . . .
Outline
Strategy
Examples
. . . . . .
StrategiesforProblemSolving
1. Understandtheproblem
2. Deviseaplan
3. Carryouttheplan
4. Reviewandextend
GyörgyPólya(Hungarian, 1887–1985)
. . . . . .
StrategiesforRelatedRatesProblems
1. Readtheproblem.
2. Drawadiagram.
3. Introducenotation. Givesymbolstoallquantitiesthatarefunctionsoftime(andmaybesomeconstants)
4. Expressthegiveninformationandtherequiredrateintermsofderivatives
5. Writeanequationthatrelatesthevariousquantitiesoftheproblem. Ifnecessary, usethegeometryofthesituationtoeliminateallbutoneofthevariables.
6. UsetheChainRuletodifferentiatebothsideswithrespecttot.
7. Substitutethegiveninformationintotheresultingequationandsolvefortheunknownrate.
. . . . . .
StrategiesforRelatedRatesProblems
1. Readtheproblem.
2. Drawadiagram.
3. Introducenotation. Givesymbolstoallquantitiesthatarefunctionsoftime(andmaybesomeconstants)
4. Expressthegiveninformationandtherequiredrateintermsofderivatives
5. Writeanequationthatrelatesthevariousquantitiesoftheproblem. Ifnecessary, usethegeometryofthesituationtoeliminateallbutoneofthevariables.
6. UsetheChainRuletodifferentiatebothsideswithrespecttot.
7. Substitutethegiveninformationintotheresultingequationandsolvefortheunknownrate.
. . . . . .
StrategiesforRelatedRatesProblems
1. Readtheproblem.
2. Drawadiagram.
3. Introducenotation. Givesymbolstoallquantitiesthatarefunctionsoftime(andmaybesomeconstants)
4. Expressthegiveninformationandtherequiredrateintermsofderivatives
5. Writeanequationthatrelatesthevariousquantitiesoftheproblem. Ifnecessary, usethegeometryofthesituationtoeliminateallbutoneofthevariables.
6. UsetheChainRuletodifferentiatebothsideswithrespecttot.
7. Substitutethegiveninformationintotheresultingequationandsolvefortheunknownrate.
. . . . . .
StrategiesforRelatedRatesProblems
1. Readtheproblem.
2. Drawadiagram.
3. Introducenotation. Givesymbolstoallquantitiesthatarefunctionsoftime(andmaybesomeconstants)
4. Expressthegiveninformationandtherequiredrateintermsofderivatives
5. Writeanequationthatrelatesthevariousquantitiesoftheproblem. Ifnecessary, usethegeometryofthesituationtoeliminateallbutoneofthevariables.
6. UsetheChainRuletodifferentiatebothsideswithrespecttot.
7. Substitutethegiveninformationintotheresultingequationandsolvefortheunknownrate.
. . . . . .
StrategiesforRelatedRatesProblems
1. Readtheproblem.
2. Drawadiagram.
3. Introducenotation. Givesymbolstoallquantitiesthatarefunctionsoftime(andmaybesomeconstants)
4. Expressthegiveninformationandtherequiredrateintermsofderivatives
5. Writeanequationthatrelatesthevariousquantitiesoftheproblem. Ifnecessary, usethegeometryofthesituationtoeliminateallbutoneofthevariables.
6. UsetheChainRuletodifferentiatebothsideswithrespecttot.
7. Substitutethegiveninformationintotheresultingequationandsolvefortheunknownrate.
. . . . . .
StrategiesforRelatedRatesProblems
1. Readtheproblem.
2. Drawadiagram.
3. Introducenotation. Givesymbolstoallquantitiesthatarefunctionsoftime(andmaybesomeconstants)
4. Expressthegiveninformationandtherequiredrateintermsofderivatives
5. Writeanequationthatrelatesthevariousquantitiesoftheproblem. Ifnecessary, usethegeometryofthesituationtoeliminateallbutoneofthevariables.
6. UsetheChainRuletodifferentiatebothsideswithrespecttot.
7. Substitutethegiveninformationintotheresultingequationandsolvefortheunknownrate.
. . . . . .
StrategiesforRelatedRatesProblems
1. Readtheproblem.
2. Drawadiagram.
3. Introducenotation. Givesymbolstoallquantitiesthatarefunctionsoftime(andmaybesomeconstants)
4. Expressthegiveninformationandtherequiredrateintermsofderivatives
5. Writeanequationthatrelatesthevariousquantitiesoftheproblem. Ifnecessary, usethegeometryofthesituationtoeliminateallbutoneofthevariables.
6. UsetheChainRuletodifferentiatebothsideswithrespecttot.
7. Substitutethegiveninformationintotheresultingequationandsolvefortheunknownrate.
. . . . . .
StrategiesforRelatedRatesProblems
1. Readtheproblem.
2. Drawadiagram.
3. Introducenotation. Givesymbolstoallquantitiesthatarefunctionsoftime(andmaybesomeconstants)
4. Expressthegiveninformationandtherequiredrateintermsofderivatives
5. Writeanequationthatrelatesthevariousquantitiesoftheproblem. Ifnecessary, usethegeometryofthesituationtoeliminateallbutoneofthevariables.
6. UsetheChainRuletodifferentiatebothsideswithrespecttot.
7. Substitutethegiveninformationintotheresultingequationandsolvefortheunknownrate.
. . . . . .
Outline
Strategy
Examples
. . . . . .
Anotherone
ExampleA manstartswalkingnorthat4ft/sec fromapoint P. Fiveminuteslaterawomanstartswalkingsouthat4ft/sec fromapoint500ftdueeastof P. Atwhatratearethepeoplewalkingapart15minafterthewomanstartswalking?
. . . . . .
Diagram
.
.m
.500
.w.w
.500
.s
.4ft/sec
.4ft/sec
.s =
√(m + w)2 + 5002
. . . . . .
Diagram
.
.m
.500
.w
.w
.500
.s
.4ft/sec
.4ft/sec
.s =
√(m + w)2 + 5002
. . . . . .
Diagram
.
.m
.500
.w
.w
.500
.s
.4ft/sec
.4ft/sec
.s =
√(m + w)2 + 5002
. . . . . .
Diagram
.
.m
.500
.w.w
.500
.s
.4ft/sec
.4ft/sec
.s =
√(m + w)2 + 5002
. . . . . .
Diagram
.
.m
.500
.w.w
.500
.s
.4ft/sec
.4ft/sec
.s =
√(m + w)2 + 5002
. . . . . .
Expressingwhatisknownandunknown
15minutesafterthewomanstartswalking, thewomanhastraveled (
4ftsec
)(60secmin
)(15min) = 3600ft
whilethemanhastraveled(4ftsec
)(60secmin
)(20min) = 4800ft
Wewanttoknowdsdt
when m = 4800, w = 3600,dmdt
= 4, and
dwdt
= 4.
. . . . . .
Differentiation
Wehave
dsdt
=12
((m + w)2 + 5002
)−1/2(2)(m + w)
(dmdt
+dwdt
)=
m + ws
(dmdt
+dwdt
)Atourparticularpointintime
dsdt
=4800 + 3600√
(4800 + 3600)2 + 5002(4 + 4) =
672√7081
≈ 7.98587ft/s