Download - Lesson 7: The Derivative
![Page 1: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/1.jpg)
. . . . . .
Section2.1TheDerivativeandRatesofChange
V63.0121.034, CalculusI
September23, 2009
Announcements
I WebAssignmentsdueMonday. EmailmeifyouneedanextensionforYomKippur.
![Page 2: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/2.jpg)
. . . . . .
RegardingWebAssignWefeelyourpain
![Page 3: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/3.jpg)
. . . . . .
Explanations
Fromthesyllabus:
Graderswillbeexpectingyoutoexpressyourideasclearly, legibly, andcompletely, oftenrequiringcompleteEnglishsentencesratherthanmerelyjustalongstringofequationsorunconnectedmathematicalexpressions. Thismeansyoucouldlosepointsforunexplainedanswers.
![Page 4: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/4.jpg)
. . . . . .
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.
![Page 5: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/5.jpg)
. . . . . .
Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
![Page 6: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/6.jpg)
. . . . . .
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
![Page 7: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/7.jpg)
. . . . . .
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
![Page 8: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/8.jpg)
. . . . . .
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
![Page 9: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/9.jpg)
. . . . . .
Graphicallyandnumerically
. .x
.y
..2
..4 .
.
..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
![Page 10: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/10.jpg)
. . . . . .
Graphicallyandnumerically
. .x
.y
..2
..4 .
.
..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
![Page 11: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/11.jpg)
. . . . . .
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
![Page 12: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/12.jpg)
. . . . . .
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
![Page 13: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/13.jpg)
. . . . . .
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
![Page 14: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/14.jpg)
. . . . . .
Graphicallyandnumerically
. .x
.y
..2
..4 .
.
..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
![Page 15: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/15.jpg)
. . . . . .
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
![Page 16: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/16.jpg)
. . . . . .
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
![Page 17: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/17.jpg)
. . . . . .
Graphicallyandnumerically
. .x
.y
..2
..4 .
.
..3
..9
.
..2.5
..6.25
.
..2.1
..4.41 .
..2.01
..4.0401
.
..1
..1
.
..1.5
..2.25
.
..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
![Page 18: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/18.jpg)
. . . . . .
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
![Page 19: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/19.jpg)
. . . . . .
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
![Page 20: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/20.jpg)
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 15
1.5 − 12.51.1 − 10.51.01 − 10.051.001 − 10.005
![Page 21: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/21.jpg)
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5
− 12.51.1 − 10.51.01 − 10.051.001 − 10.005
![Page 22: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/22.jpg)
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.5
1.1 − 10.51.01 − 10.051.001 − 10.005
![Page 23: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/23.jpg)
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1
− 10.51.01 − 10.051.001 − 10.005
![Page 24: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/24.jpg)
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.5
1.01 − 10.051.001 − 10.005
![Page 25: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/25.jpg)
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01
− 10.051.001 − 10.005
![Page 26: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/26.jpg)
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01 − 10.05
1.001 − 10.005
![Page 27: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/27.jpg)
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01 − 10.051.001
− 10.005
![Page 28: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/28.jpg)
. . . . . .
Numericalevidence
t vave =h(t) − h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01 − 10.051.001 − 10.005
![Page 29: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/29.jpg)
. . . . . .
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
![Page 30: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/30.jpg)
. . . . . .
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
![Page 31: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/31.jpg)
. . . . . .
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.
![Page 32: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/32.jpg)
. . . . . .
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.
![Page 33: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/33.jpg)
. . . . . .
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.
![Page 34: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/34.jpg)
. . . . . .
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.
![Page 35: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/35.jpg)
. . . . . .
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
![Page 36: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/36.jpg)
. . . . . .
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
![Page 37: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/37.jpg)
. . . . . .
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
![Page 38: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/38.jpg)
. . . . . .
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
![Page 39: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/39.jpg)
. . . . . .
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
![Page 40: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/40.jpg)
. . . . . .
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
![Page 41: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/41.jpg)
. . . . . .
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.
![Page 42: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/42.jpg)
. . . . . .
UpshotTheinstantaneouspopulationgrowthisgivenby
lim∆t→0
P(t + ∆t) − P(t)∆t
![Page 43: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/43.jpg)
. . . . . .
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.
![Page 44: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/44.jpg)
. . . . . .
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.
![Page 45: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/45.jpg)
. . . . . .
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
![Page 46: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/46.jpg)
. . . . . .
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
![Page 47: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/47.jpg)
. . . . . .
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
![Page 48: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/48.jpg)
. . . . . .
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
![Page 49: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/49.jpg)
. . . . . .
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
![Page 50: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/50.jpg)
. . . . . .
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
![Page 51: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/51.jpg)
. . . . . .
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
![Page 52: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/52.jpg)
. . . . . .
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
![Page 53: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/53.jpg)
. . . . . .
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
![Page 54: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/54.jpg)
. . . . . .
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
![Page 55: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/55.jpg)
. . . . . .
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
![Page 56: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/56.jpg)
. . . . . .
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
![Page 57: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/57.jpg)
. . . . . .
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.
![Page 58: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/58.jpg)
. . . . . .
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.
![Page 59: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/59.jpg)
. . . . . .
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.
![Page 60: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/60.jpg)
. . . . . .
Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
![Page 61: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/61.jpg)
. . . . . .
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.
![Page 62: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/62.jpg)
. . . . . .
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.
![Page 63: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/63.jpg)
. . . . . .
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.
![Page 64: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/64.jpg)
. . . . . .
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.
![Page 65: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/65.jpg)
. . . . . .
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
.
![Page 66: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/66.jpg)
. . . . . .
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
.
![Page 67: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/67.jpg)
. . . . . .
TheSure-FireSallyRule(SFSR) foraddingFractionsInanticipationofthequestion, “Howdidyougetthat?”
ab± c
d=
ad± bcbd
So
1x− 1
2x− 2
=
2− x2x
x− 2
=2− x
2x(x− 2)
![Page 68: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/68.jpg)
. . . . . .
TheSure-FireSallyRule(SFSR) foraddingFractionsInanticipationofthequestion, “Howdidyougetthat?”
ab± c
d=
ad± bcbd
So
1x− 1
2x− 2
=
2− x2x
x− 2
=2− x
2x(x− 2)
![Page 69: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/69.jpg)
. . . . . .
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
![Page 70: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/70.jpg)
. . . . . .
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
![Page 71: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/71.jpg)
. . . . . .
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
.
![Page 72: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/72.jpg)
. . . . . .
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
![Page 73: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/73.jpg)
. . . . . .
Graphicallyandnumerically
. .x
.y
..2
..4 .
.
..3
..9
.
..2.5
..6.25
.
..2.1
..4.41 .
..2.01
..4.0401
.
..1
..1
.
..1.5
..2.25
.
..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
![Page 74: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/74.jpg)
. . . . . .
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
![Page 75: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/75.jpg)
. . . . . .
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
![Page 76: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/76.jpg)
. . . . . .
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
![Page 77: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/77.jpg)
. . . . . .
Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
![Page 78: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/78.jpg)
. . . . . .
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.
![Page 79: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/79.jpg)
. . . . . .
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.
![Page 80: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/80.jpg)
. . . . . .
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.
![Page 81: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/81.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?Kinks
. .x
.f(x)
. .x
.f′(x)
.
.
![Page 82: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/82.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?Kinks
. .x
.f(x)
. .x
.f′(x)
.
.
![Page 83: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/83.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?Kinks
. .x
.f(x)
. .x
.f′(x)
.
.
![Page 84: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/84.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?Cusps
. .x
.f(x)
. .x
.f′(x)
![Page 85: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/85.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?Cusps
. .x
.f(x)
. .x
.f′(x)
![Page 86: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/86.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?Cusps
. .x
.f(x)
. .x
.f′(x)
![Page 87: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/87.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?VerticalTangents
. .x
.f(x)
. .x
.f′(x)
![Page 88: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/88.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?VerticalTangents
. .x
.f(x)
. .x
.f′(x)
![Page 89: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/89.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?VerticalTangents
. .x
.f(x)
. .x
.f′(x)
![Page 90: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/90.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?Weird, Wild, Stuff
. .x
.f(x)
Thisfunctionisdifferentiableat 0.
. .x
.f′(x)
Butthederivativeisnotcontinuousat 0!
![Page 91: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/91.jpg)
. . . . . .
Howcanafunctionfailtobedifferentiable?Weird, Wild, Stuff
. .x
.f(x)
Thisfunctionisdifferentiableat 0.
. .x
.f′(x)
Butthederivativeisnotcontinuousat 0!
![Page 92: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/92.jpg)
. . . . . .
Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
![Page 93: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/93.jpg)
. . . . . .
Notation
I Newtoniannotation
f′(x) y′(x) y′
I Leibniziannotation
dydx
ddx
f(x)dfdx
Theseallmeanthesamething.
![Page 94: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/94.jpg)
. . . . . .
MeettheMathematician: IsaacNewton
I English, 1643–1727I ProfessoratCambridge(England)
I PhilosophiaeNaturalisPrincipiaMathematicapublished1687
![Page 95: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/95.jpg)
. . . . . .
MeettheMathematician: GottfriedLeibniz
I German, 1646–1716I Eminentphilosopheraswellasmathematician
I Contemporarilydisgracedbythecalculusprioritydispute
![Page 96: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/96.jpg)
. . . . . .
Outline
RatesofChangeTangentLinesVelocityPopulationgrowthMarginalcosts
Thederivative, definedDerivativesof(some)powerfunctionsWhatdoes f tellyouabout f′?
Howcanafunctionfailtobedifferentiable?
Othernotations
Thesecondderivative
![Page 97: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/97.jpg)
. . . . . .
Thesecondderivative
If f isafunction, sois f′, andwecanseekitsderivative.
f′′ = (f′)′
Itmeasurestherateofchangeoftherateofchange!
Leibniziannotation:
d2ydx2
d2
dx2f(x)
d2fdx2
![Page 98: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/98.jpg)
. . . . . .
Thesecondderivative
If f isafunction, sois f′, andwecanseekitsderivative.
f′′ = (f′)′
Itmeasurestherateofchangeoftherateofchange! Leibniziannotation:
d2ydx2
d2
dx2f(x)
d2fdx2
![Page 99: Lesson 7: The Derivative](https://reader035.vdocuments.us/reader035/viewer/2022070302/548f65abb47959302a8b4603/html5/thumbnails/99.jpg)
. . . . . .
function, derivative, secondderivative
. .x
.y
.f(x) = x2
.f′(x) = 2x
.f′′(x) = 2