lesson 7: the derivative
DESCRIPTION
The derivative is a major tool for investigating the behavior of a function. Since functions are ubiquitous, so are their derivatives. Velocity, growth rates, marginal costs, and material strain are all examples of derivatives. We motivate and define the derivative and compute a few examples, then discuss how features of a function are manifested in its derivative.TRANSCRIPT
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Section2.1TheDerivativeandRatesofChange
V63.0121.027, CalculusI
September24, 2009
Announcements
I WebAssignmentsdueTuesday.I OfficeHourstoday3-4. SeeSectionCalendarforup-to-dateOH.
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RegardingWebAssignWefeelyourpain
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Explanations
Fromthesyllabus:
Graderswillbeexpectingyoutoexpressyourideasclearly, legibly, andcompletely, oftenrequiringcompleteEnglishsentencesratherthanmerelyjustalongstringofequationsorunconnectedmathematicalexpressions. Thismeansyoucouldlosepointsforunexplainedanswers.
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Rubric
Points DescriptionofWork3 Work is completely accurate and essentially perfect.
Workisthoroughlydeveloped, neat, andeasytoread.Completesentencesareused.
2 Work is good, but incompletely developed, hard toread, unexplained, or jumbled. Answers which arenotexplained, evenifcorrect, willgenerallyreceive2points. Workcontains“rightidea”butisflawed.
1 Workissketchy. Thereissomecorrectwork, butmostofworkisincorrect.
0 Workminimalornon-existent. Solutioniscompletelyincorrect.
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Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
. . . . . .
Thetangentproblem
ProblemGivenacurveandapointonthecurve, findtheslopeofthelinetangenttothecurveatthatpoint.
ExampleFindtheslopeofthelinetangenttothecurve y = x2 atthepoint(2, 4).
UpshotIfthecurveisgivenby y = f(x), andthepointonthecurveis(a, f(a)), thentheslopeofthetangentlineisgivenby
mtangent = limx→a
f(x) − f(a)x− a
. . . . . .
Thetangentproblem
ProblemGivenacurveandapointonthecurve, findtheslopeofthelinetangenttothecurveatthatpoint.
ExampleFindtheslopeofthelinetangenttothecurve y = x2 atthepoint(2, 4).
UpshotIfthecurveisgivenby y = f(x), andthepointonthecurveis(a, f(a)), thentheslopeofthetangentlineisgivenby
mtangent = limx→a
f(x) − f(a)x− a
. . . . . .
Graphicallyandnumerically
. .x
.y
..2
..4 .
x m
3 52.5 4.252.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphicallyandnumerically
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.y
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..3
..9
x m3 5
2.5 4.252.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphicallyandnumerically
. .x
.y
..2
..4 .
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..2.5
..6.25
x m3 52.5 4.25
2.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphicallyandnumerically
. .x
.y
..2
..4 ..
..2.1
..4.41
x m3 52.5 4.252.1 4.1
2.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphicallyandnumerically
. .x
.y
..2
..4 ..
..2.01
..4.0401
x m3 52.5 4.252.1 4.12.01 4.01
limit 41.99 3.991.9 3.91.5 3.51 3
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Graphicallyandnumerically
. .x
.y
..2
..4 .
.
..1
..1
x m3 52.5 4.252.1 4.12.01 4.01
limit 41.99 3.991.9 3.91.5 3.5
1 3
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Graphicallyandnumerically
. .x
.y
..2
..4 .
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..1.5
..2.25
x m3 52.5 4.252.1 4.12.01 4.01
limit 41.99 3.991.9 3.9
1.5 3.51 3
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Graphicallyandnumerically
. .x
.y
..2
..4 ..
..1.9
..3.61
x m3 52.5 4.252.1 4.12.01 4.01
limit 41.99 3.99
1.9 3.91.5 3.51 3
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Graphicallyandnumerically
. .x
.y
..2
..4 ..
..1.99
..3.9601
x m3 52.5 4.252.1 4.12.01 4.01
limit 4
1.99 3.991.9 3.91.5 3.51 3
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Graphicallyandnumerically
. .x
.y
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..4 .
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..3
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..2.5
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..2.1
..4.41 .
..2.01
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..1
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..1.9
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..1.99
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x m3 52.5 4.252.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Thetangentproblem
ProblemGivenacurveandapointonthecurve, findtheslopeofthelinetangenttothecurveatthatpoint.
ExampleFindtheslopeofthelinetangenttothecurve y = x2 atthepoint(2, 4).
UpshotIfthecurveisgivenby y = f(x), andthepointonthecurveis(a, f(a)), thentheslopeofthetangentlineisgivenby
mtangent = limx→a
f(x) − f(a)x− a
. . . . . .
VelocityProblemGiventhepositionfunctionofamovingobject, findthevelocityoftheobjectatacertaininstantintime.
ExampleDropaballofftheroofoftheSilverCentersothatitsheightcanbedescribedby
h(t) = 50− 5t2
where t issecondsafterdroppingitand h ismetersabovetheground. Howfastisitfallingonesecondafterwedropit?
SolutionTheansweris
v = limt→1
(50− 5t2) − 45t− 1
= limt→1
5− 5t2
t− 1= lim
t→1
5(1− t)(1 + t)t− 1
= (−5) limt→1
(1 + t) = −5 · 2 = −10
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 15
1.5 − 12.51.1 − 10.51.01 − 10.051.001 − 10.005
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5
− 12.51.1 − 10.51.01 − 10.051.001 − 10.005
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.5
1.1 − 10.51.01 − 10.051.001 − 10.005
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1
− 10.51.01 − 10.051.001 − 10.005
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.5
1.01 − 10.051.001 − 10.005
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01
− 10.051.001 − 10.005
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01 − 10.05
1.001 − 10.005
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01 − 10.051.001
− 10.005
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01 − 10.051.001 − 10.005
. . . . . .
VelocityProblemGiventhepositionfunctionofamovingobject, findthevelocityoftheobjectatacertaininstantintime.
ExampleDropaballofftheroofoftheSilverCentersothatitsheightcanbedescribedby
h(t) = 50− 5t2
where t issecondsafterdroppingitand h ismetersabovetheground. Howfastisitfallingonesecondafterwedropit?
SolutionTheansweris
v = limt→1
(50− 5t2) − 45t− 1
= limt→1
5− 5t2
t− 1= lim
t→1
5(1− t)(1 + t)t− 1
= (−5) limt→1
(1 + t) = −5 · 2 = −10
. . . . . .
UpshotIftheheightfunctionisgivenby h(t), theinstantaneousvelocityattime t0 isgivenby
v = limt→t0
h(t) − h(t0)t− t0
= lim∆t→0
h(t0 + ∆t) − h(t0)∆t
. .t
.y = h(t).
.
..t0
..t
.∆t
. . . . . .
Populationgrowth
ProblemGiventhepopulationfunctionofagroupoforganisms, findtherateofgrowthofthepopulationataparticularinstant.
ExampleSupposethepopulationoffishintheEastRiverisgivenbythefunction
P(t) =3et
1 + et
where t isinyearssince2000and P isinmillionsoffish. Isthefishpopulationgrowingfastestin1990, 2000, or2010? (Estimatenumerically)?
SolutionTheestimatedratesofgrowthare 0.000136, 0.75, and 0.000136.
. . . . . .
Populationgrowth
ProblemGiventhepopulationfunctionofagroupoforganisms, findtherateofgrowthofthepopulationataparticularinstant.
ExampleSupposethepopulationoffishintheEastRiverisgivenbythefunction
P(t) =3et
1 + et
where t isinyearssince2000and P isinmillionsoffish. Isthefishpopulationgrowingfastestin1990, 2000, or2010? (Estimatenumerically)?
SolutionTheestimatedratesofgrowthare 0.000136, 0.75, and 0.000136.
. . . . . .
Derivation
Let ∆t beanincrementintimeand ∆P thecorrespondingchangeinpopulation:
∆P = P(t + ∆t) − P(t)
Thisdependson ∆t, sowewant
lim∆t→0
∆P∆t
= lim∆t→0
1∆t
(3et+∆t
1 + et+∆t −3et
1 + et
)
Toohard! Tryasmall ∆t toapproximate.
. . . . . .
Derivation
Let ∆t beanincrementintimeand ∆P thecorrespondingchangeinpopulation:
∆P = P(t + ∆t) − P(t)
Thisdependson ∆t, sowewant
lim∆t→0
∆P∆t
= lim∆t→0
1∆t
(3et+∆t
1 + et+∆t −3et
1 + et
)Toohard! Tryasmall ∆t toapproximate.
. . . . . .
Numericalevidence
r1990 ≈ P(−10 + 0.1) − P(−10)
0.1≈
0.000136
r2000 ≈ P(0.1) − P(0)
0.1≈ 0.75
r2010 ≈ P(10 + 0.1) − P(10)
0.1≈ 0.000136
. . . . . .
Numericalevidence
r1990 ≈ P(−10 + 0.1) − P(−10)
0.1≈ 0.000136
r2000 ≈ P(0.1) − P(0)
0.1≈ 0.75
r2010 ≈ P(10 + 0.1) − P(10)
0.1≈ 0.000136
. . . . . .
Numericalevidence
r1990 ≈ P(−10 + 0.1) − P(−10)
0.1≈ 0.000136
r2000 ≈ P(0.1) − P(0)
0.1≈
0.75
r2010 ≈ P(10 + 0.1) − P(10)
0.1≈ 0.000136
. . . . . .
Numericalevidence
r1990 ≈ P(−10 + 0.1) − P(−10)
0.1≈ 0.000136
r2000 ≈ P(0.1) − P(0)
0.1≈ 0.75
r2010 ≈ P(10 + 0.1) − P(10)
0.1≈ 0.000136
. . . . . .
Numericalevidence
r1990 ≈ P(−10 + 0.1) − P(−10)
0.1≈ 0.000136
r2000 ≈ P(0.1) − P(0)
0.1≈ 0.75
r2010 ≈ P(10 + 0.1) − P(10)
0.1≈
0.000136
. . . . . .
Numericalevidence
r1990 ≈ P(−10 + 0.1) − P(−10)
0.1≈ 0.000136
r2000 ≈ P(0.1) − P(0)
0.1≈ 0.75
r2010 ≈ P(10 + 0.1) − P(10)
0.1≈ 0.000136
. . . . . .
Populationgrowth
ProblemGiventhepopulationfunctionofagroupoforganisms, findtherateofgrowthofthepopulationataparticularinstant.
ExampleSupposethepopulationoffishintheEastRiverisgivenbythefunction
P(t) =3et
1 + et
where t isinyearssince2000and P isinmillionsoffish. Isthefishpopulationgrowingfastestin1990, 2000, or2010? (Estimatenumerically)?
SolutionTheestimatedratesofgrowthare 0.000136, 0.75, and 0.000136.
. . . . . .
UpshotTheinstantaneouspopulationgrowthisgivenby
lim∆t→0
P(t + ∆t) − P(t)∆t
. . . . . .
Marginalcosts
ProblemGiventheproductioncostofagood, findthemarginalcostofproductionafterhavingproducedacertainquantity.
ExampleSupposethecostofproducing q tonsofriceonourpaddyinayearis
C(q) = q3 − 12q2 + 60q
Wearecurrentlyproducing 5 tonsayear. Shouldwechangethat?
ExampleIf q = 5, then C = 125, ∆C = 19, while AC = 25. Soweshouldproducemoretoloweraveragecosts.
. . . . . .
Marginalcosts
ProblemGiventheproductioncostofagood, findthemarginalcostofproductionafterhavingproducedacertainquantity.
ExampleSupposethecostofproducing q tonsofriceonourpaddyinayearis
C(q) = q3 − 12q2 + 60q
Wearecurrentlyproducing 5 tonsayear. Shouldwechangethat?
ExampleIf q = 5, then C = 125, ∆C = 19, while AC = 25. Soweshouldproducemoretoloweraveragecosts.
. . . . . .
Comparisons
q C(q)
AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4
112 28 13
5
125 25 19
6
144 24 31
. . . . . .
Comparisons
q C(q)
AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112
28 13
5
125 25 19
6
144 24 31
. . . . . .
Comparisons
q C(q)
AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112
28 13
5 125
25 19
6
144 24 31
. . . . . .
Comparisons
q C(q)
AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112
28 13
5 125
25 19
6 144
24 31
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q
∆C = C(q + 1) − C(q)
4 112
28 13
5 125
25 19
6 144
24 31
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q
∆C = C(q + 1) − C(q)
4 112 28
13
5 125
25 19
6 144
24 31
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q
∆C = C(q + 1) − C(q)
4 112 28
13
5 125 25
19
6 144
24 31
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q
∆C = C(q + 1) − C(q)
4 112 28
13
5 125 25
19
6 144 24
31
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112 28
13
5 125 25
19
6 144 24
31
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112 28 135 125 25
19
6 144 24
31
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112 28 135 125 25 196 144 24
31
. . . . . .
Comparisons
q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4 112 28 135 125 25 196 144 24 31
. . . . . .
Marginalcosts
ProblemGiventheproductioncostofagood, findthemarginalcostofproductionafterhavingproducedacertainquantity.
ExampleSupposethecostofproducing q tonsofriceonourpaddyinayearis
C(q) = q3 − 12q2 + 60q
Wearecurrentlyproducing 5 tonsayear. Shouldwechangethat?
ExampleIf q = 5, then C = 125, ∆C = 19, while AC = 25. Soweshouldproducemoretoloweraveragecosts.
. . . . . .
Upshot
I Theincrementalcost
∆C = C(q + 1) − C(q)
isuseful, butdependsonunits.
I Themarginalcostafterproducing q givenby
MC = lim∆q→0
C(q + ∆q) − C(q)
∆q
ismoreusefulsinceit’sunit-independent.
. . . . . .
Upshot
I Theincrementalcost
∆C = C(q + 1) − C(q)
isuseful, butdependsonunits.I Themarginalcostafterproducing q givenby
MC = lim∆q→0
C(q + ∆q) − C(q)
∆q
ismoreusefulsinceit’sunit-independent.
. . . . . .
Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
. . . . . .
Thedefinition
Alloftheseratesofchangearefoundthesameway!
DefinitionLet f beafunctionand a apointinthedomainof f. Ifthelimit
f′(a) = limh→0
f(a + h) − f(a)h
exists, thefunctionissaidtobe differentiableat a and f′(a) isthederivativeof f at a.
. . . . . .
Thedefinition
Alloftheseratesofchangearefoundthesameway!
DefinitionLet f beafunctionand a apointinthedomainof f. Ifthelimit
f′(a) = limh→0
f(a + h) − f(a)h
exists, thefunctionissaidtobe differentiableat a and f′(a) isthederivativeof f at a.
. . . . . .
Derivativeofthesquaringfunction
ExampleSuppose f(x) = x2. Usethedefinitionofderivativetofind f′(a).
Solution
f′(a) = limh→0
f(a + h) − f(a)h
= limh→0
(a + h)2 − a2
h
= limh→0
(a2 + 2ah + h2) − a2
h= lim
h→0
2ah + h2
h= lim
h→0(2a + h) = 2a.
. . . . . .
Derivativeofthesquaringfunction
ExampleSuppose f(x) = x2. Usethedefinitionofderivativetofind f′(a).
Solution
f′(a) = limh→0
f(a + h) − f(a)h
= limh→0
(a + h)2 − a2
h
= limh→0
(a2 + 2ah + h2) − a2
h= lim
h→0
2ah + h2
h= lim
h→0(2a + h) = 2a.
. . . . . .
Derivativeofthereciprocalfunction
Example
Suppose f(x) =1x. Usethe
definitionofthederivativetofind f′(2).
Solution
f′(2) = limx→2
1/x− 1/2x− 2
= limx→2
2− x2x(x− 2)
= limx→2
−12x
= −14
. .x
.x
.
. . . . . .
Derivativeofthereciprocalfunction
Example
Suppose f(x) =1x. Usethe
definitionofthederivativetofind f′(2).
Solution
f′(2) = limx→2
1/x− 1/2x− 2
= limx→2
2− x2x(x− 2)
= limx→2
−12x
= −14
. .x
.x
.
. . . . . .
TheSure-FireSallyRule(SFSR) foraddingFractionsInanticipationofthequestion, “Howdidyougetthat?”
ab± c
d=
ad± bcbd
So
1x− 1
2x− 2
=
2− x2x
x− 2
=2− x
2x(x− 2)
. . . . . .
TheSure-FireSallyRule(SFSR) foraddingFractionsInanticipationofthequestion, “Howdidyougetthat?”
ab± c
d=
ad± bcbd
So
1x− 1
2x− 2
=
2− x2x
x− 2
=2− x
2x(x− 2)
. . . . . .
Whatdoes f tellyouabout f′?
I If f isafunction, wecancomputethederivative f′(x) ateachpoint x where f isdifferentiable, andcomeupwithanotherfunction, thederivativefunction.
I Whatcanwesayaboutthisfunction f′?
I If f isdecreasingonaninterval, f′ isnegative(well,nonpositive)onthatinterval
I If f isincreasingonaninterval, f′ ispositive(well,nonnegative)onthatinterval
. . . . . .
Whatdoes f tellyouabout f′?
I If f isafunction, wecancomputethederivative f′(x) ateachpoint x where f isdifferentiable, andcomeupwithanotherfunction, thederivativefunction.
I Whatcanwesayaboutthisfunction f′?I If f isdecreasingonaninterval, f′ isnegative(well,
nonpositive)onthatinterval
I If f isincreasingonaninterval, f′ ispositive(well,nonnegative)onthatinterval
. . . . . .
Derivativeofthereciprocalfunction
Example
Suppose f(x) =1x. Usethe
definitionofthederivativetofind f′(2).
Solution
f′(2) = limx→2
1/x− 1/2x− 2
= limx→2
2− x2x(x− 2)
= limx→2
−12x
= −14
. .x
.x
.
. . . . . .
Whatdoes f tellyouabout f′?
I If f isafunction, wecancomputethederivative f′(x) ateachpoint x where f isdifferentiable, andcomeupwithanotherfunction, thederivativefunction.
I Whatcanwesayaboutthisfunction f′?I If f isdecreasingonaninterval, f′ isnegative(well,
nonpositive)onthatintervalI If f isincreasingonaninterval, f′ ispositive(well,
nonnegative)onthatinterval
. . . . . .
Graphicallyandnumerically
. .x
.y
..2
..4 .
.
..3
..9
.
..2.5
..6.25
.
..2.1
..4.41 .
..2.01
..4.0401
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..1
..1
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..1.5
..2.25
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..1.9
..3.61.
..1.99
..3.9601
x m3 52.5 4.252.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
. . . . . .
Whatdoes f tellyouabout f′?FactIf f isdecreasingon (a,b), then f′ ≤ 0 on (a,b).
Proof.If f isdecreasingon (a,b), and ∆x > 0, then
f(x + ∆x) < f(x) =⇒ f(x + ∆x) − f(x)∆x
< 0
Butif ∆x < 0, then x + ∆x < x, and
f(x + ∆x) > f(x) =⇒ f(x + ∆x) − f(x)∆x
< 0
still! Eitherway,f(x + ∆x) − f(x)
∆x< 0, so
f′(x) = lim∆x→0
f(x + ∆x) − f(x)∆x
≤ 0
. . . . . .
Whatdoes f tellyouabout f′?FactIf f isdecreasingon (a,b), then f′ ≤ 0 on (a,b).
Proof.If f isdecreasingon (a,b), and ∆x > 0, then
f(x + ∆x) < f(x) =⇒ f(x + ∆x) − f(x)∆x
< 0
Butif ∆x < 0, then x + ∆x < x, and
f(x + ∆x) > f(x) =⇒ f(x + ∆x) − f(x)∆x
< 0
still!
Eitherway,f(x + ∆x) − f(x)
∆x< 0, so
f′(x) = lim∆x→0
f(x + ∆x) − f(x)∆x
≤ 0
. . . . . .
Whatdoes f tellyouabout f′?FactIf f isdecreasingon (a,b), then f′ ≤ 0 on (a,b).
Proof.If f isdecreasingon (a,b), and ∆x > 0, then
f(x + ∆x) < f(x) =⇒ f(x + ∆x) − f(x)∆x
< 0
Butif ∆x < 0, then x + ∆x < x, and
f(x + ∆x) > f(x) =⇒ f(x + ∆x) − f(x)∆x
< 0
still! Eitherway,f(x + ∆x) − f(x)
∆x< 0, so
f′(x) = lim∆x→0
f(x + ∆x) − f(x)∆x
≤ 0
. . . . . .
Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
. . . . . .
Differentiabilityissuper-continuity
TheoremIf f isdifferentiableat a, then f iscontinuousat a.
Proof.Wehave
limx→a
(f(x) − f(a)) = limx→a
f(x) − f(a)x− a
· (x− a)
= limx→a
f(x) − f(a)x− a
· limx→a
(x− a)
= f′(a) · 0 = 0
Notetheproperuseofthelimitlaw: if thefactorseachhavealimitat a, thelimitoftheproductistheproductofthelimits.
. . . . . .
Differentiabilityissuper-continuity
TheoremIf f isdifferentiableat a, then f iscontinuousat a.
Proof.Wehave
limx→a
(f(x) − f(a)) = limx→a
f(x) − f(a)x− a
· (x− a)
= limx→a
f(x) − f(a)x− a
· limx→a
(x− a)
= f′(a) · 0 = 0
Notetheproperuseofthelimitlaw: if thefactorseachhavealimitat a, thelimitoftheproductistheproductofthelimits.
. . . . . .
Differentiabilityissuper-continuity
TheoremIf f isdifferentiableat a, then f iscontinuousat a.
Proof.Wehave
limx→a
(f(x) − f(a)) = limx→a
f(x) − f(a)x− a
· (x− a)
= limx→a
f(x) − f(a)x− a
· limx→a
(x− a)
= f′(a) · 0 = 0
Notetheproperuseofthelimitlaw: if thefactorseachhavealimitat a, thelimitoftheproductistheproductofthelimits.
. . . . . .
Howcanafunctionfailtobedifferentiable?Kinks
. .x
.f(x)
. .x
.f′(x)
.
.
. . . . . .
Howcanafunctionfailtobedifferentiable?Kinks
. .x
.f(x)
. .x
.f′(x)
.
.
. . . . . .
Howcanafunctionfailtobedifferentiable?Kinks
. .x
.f(x)
. .x
.f′(x)
.
.
. . . . . .
Howcanafunctionfailtobedifferentiable?Cusps
. .x
.f(x)
. .x
.f′(x)
. . . . . .
Howcanafunctionfailtobedifferentiable?Cusps
. .x
.f(x)
. .x
.f′(x)
. . . . . .
Howcanafunctionfailtobedifferentiable?Cusps
. .x
.f(x)
. .x
.f′(x)
. . . . . .
Howcanafunctionfailtobedifferentiable?VerticalTangents
. .x
.f(x)
. .x
.f′(x)
. . . . . .
Howcanafunctionfailtobedifferentiable?VerticalTangents
. .x
.f(x)
. .x
.f′(x)
. . . . . .
Howcanafunctionfailtobedifferentiable?VerticalTangents
. .x
.f(x)
. .x
.f′(x)
. . . . . .
Howcanafunctionfailtobedifferentiable?Weird, Wild, Stuff
. .x
.f(x)
Thisfunctionisdifferentiableat 0.
. .x
.f′(x)
Butthederivativeisnotcontinuousat 0!
. . . . . .
Howcanafunctionfailtobedifferentiable?Weird, Wild, Stuff
. .x
.f(x)
Thisfunctionisdifferentiableat 0.
. .x
.f′(x)
Butthederivativeisnotcontinuousat 0!
. . . . . .
Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
. . . . . .
Notation
I Newtoniannotation
f′(x) y′(x) y′
I Leibniziannotation
dydx
ddx
f(x)dfdx
Theseallmeanthesamething.
. . . . . .
MeettheMathematician: IsaacNewton
I English, 1643–1727I ProfessoratCambridge(England)
I PhilosophiaeNaturalisPrincipiaMathematicapublished1687
. . . . . .
MeettheMathematician: GottfriedLeibniz
I German, 1646–1716I Eminentphilosopheraswellasmathematician
I Contemporarilydisgracedbythecalculusprioritydispute
. . . . . .
Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
. . . . . .
Thesecondderivative
If f isafunction, sois f′, andwecanseekitsderivative.
f′′ = (f′)′
Itmeasurestherateofchangeoftherateofchange!
Leibniziannotation:
d2ydx2
d2
dx2f(x)
d2fdx2
. . . . . .
Thesecondderivative
If f isafunction, sois f′, andwecanseekitsderivative.
f′′ = (f′)′
Itmeasurestherateofchangeoftherateofchange! Leibniziannotation:
d2ydx2
d2
dx2f(x)
d2fdx2
. . . . . .
function, derivative, secondderivative
. .x
.y
.f(x) = x2
.f′(x) = 2x
.f′′(x) = 2