RatesofChange
Letfbeafunction.Givenaninterval[x1,x2],
wedefinetheaveragerateofchangeoff
withrespecttox overtheinterval[x1,x2] to
betheratio
f(x2)–f(x1)
,
x2–x1
assumingthatf isdefinedatx1 andx2.
Thursday,January19,2017
RatesofChange
Ex:Supposeyouhavethefollowingdataregardingthe
humanpopulationoftheworld.Whatistheaveragerate
ofchangeinhumanpopulationbetweentheyears1950
and1990?
x
(years since1900)
P(x)
Population
(inmillions)
0
1650
10
1750
20
1860
30
2070
40
2300
50
2560
60
3040
70
3710
80
4450
90
5280
100
6080
110
6870
Well,bydefinition,theaveragerateof
changebetweentheyears1950-1990is
givenby
P(1990)–f(1950)
1990−1950
=
5280−2560
1990−1950
=
2720
40
=68millionpeopleperyear.
unitofmeasurement
RatesofChange
Population(inmillions)
Ex:Supposeyouhavethefollowingdataregardingthehuman
populationoftheworld.Whatistheaveragerateofchangein
humanpopulationbetweentheyears1950and1990?
(1990,P(1990))
P(1990)-P(1950)
(1950,P(1950))
1990-1950
yearssince1900
Anotherwaytothink
aboutthevalueofthe
averagerateofchangein
humanpopulation
betweentheyears19501990isastheslopeof
thelinethroughthe
points(1950,P(1950))
and(1990,P(1990))
RatesofChange
Ex:Supposeadynamiteblastkicksupa
rockintotheair.Thetablebelowgives
theheight(infeet)oftherockattimet
(inseconds).
t
(inseconds)
h(t)
Height oftherockfrom
theground
(infeet)
0
0
2
256
4
384
6
384
8
256
10
0
Wecanplotthedatapoints:
RatesofChange
Ex:Supposeadynamiteblastkicksupa
rockintotheair.Thetablebelowgives
theheight(infeet)oftherockattimet
(inseconds).
t
(in
seconds)
h(t)
Height of
therock
fromthe
ground
(infeet)
0
0
2
256
4
384
6
384
8
256
10
0
Havingplottedthedatapoints,wecantry
andfindafunctionh whichaccurately
modelsthedatapoints:
Wecanplotthedatapoints:
h(t)
RatesofChange
Letfbeafunction.Givenapointa,wedefinetheInstantaneousrateof
changeoff withrespecttox ata tobethelimit
limx→a
f(x)–f(a)
,
x-a
assumingthelimitexists.
Note:Anyarbitraryx valuecanbewrittenasx=a+h,forsomevalueh.Then,
sayingx goestoa isequivalenttosayingh goesto0.Thus,wecanequivalently
definetheInstantaneousrateofchangeoff withrespecttox ata tobethe
limit
limh→0
f(a+h)–f(a)
h
Noticethisispreciselytheslopeofthetangentlineatx=a!
Thursday,January19,2017
RatesofChange
Ex:Supposeadynamiteblastkicksupa
rockintotheair.h(t)givestheheight(in
feet)oftherockattimet (inseconds).
Estimatetheinstantaneousrateof
changeindistance(i.e.,velocity)atx=6
•
Thevelocityatx=6isgivenbytheslopeofthe
tangentlineL ofh atthepoint(6,384).
•
WeknowthatwecanapproximatetheslopeofLby
theslopesofthesecantlinesgoingthroughthepoint
(6,384).
Namely,wecanusethesecantlineL1 goingthrough
(4,384)and(6,384),andthesecantlineL2 going
through(8,256)and(6,384),togetlowerandupper
boundsfortheslopeofthetangentline.
L1
(4,384)
(6,384)
h(t)
L
(8,256)
•
TheslopeofL1 is0,andtheslopeofL2 is-64.
•
SincetheslopeofLisbetween0and-64,wecan
approximateitbytakingtheaverageof0and-64,
whichis-32
•
Thus,theslopeofLis-32,whichmeansthatthe
velocityatx=6 is
L2
-32feetpersecond(ft/s).
Warning:Don’tforgetyourunits!
RatesofChange:ClassProblems
Derivative
f(a+h)–f(a)
Givenafunctionf wehavethat limh→0
,equivalently
h
f(x)–f(a)
limx→a
,givesustheslopeofthetangentlineatx=a andthe
x-a
instantaneousrateofchangeoffata.
Asthecourseprogresseswewillseethatthislimit,whenitexists,encodesalot
ofinformation.
Formally,limh→0
denotedbyf’(a).
f(a+h)–f(a)
iscalledthederivativeoff ata.Itisoften
h
Derivative
Ex:Givenf(x)=4x– 3x2,findf’(2).
Or
Derivatives:ClassProblems