lesson 13: related rates of change
DESCRIPTION
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.TRANSCRIPT
. . . . . .
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