lesson 17: the mean value theorem and the shape of curves
DESCRIPTION
The Mean Value Theorem is the Most Important Theorem in Calculus because it relates information about the derivative of a function to information about the function itself.TRANSCRIPT
. . . . . .
Section4.3TheMeanValueTheoremandtheshapeofcurves
Math1a
March14, 2008
Announcements
◮ Midtermisgraded.◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323
..Image: Flickruser Jimmywayne32
. . . . . .
Announcements
◮ Midtermisgraded◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323
. . . . . .
HappyPiDay!
3:14PM Digitrecitationcontest! Reciteallthedigitsyouknowof π(inorder, please). Pleaseletusknowinadvanceifyou’llrecite π inabaseotherthan10(theusualchoice), 2, or16.Onlypositiveintegerbasesallowed–nofairtomemorize πinbase π/(π − 2)...
4PM —Pi(e)eatingcontest! Cornbreadaresquare; pieareround.Youhave3minutesand14secondstostuffyourselfwithasmuchpieasyoucan. Theleftoverswillbeweighedtocalculatehowmuchpieyouhaveeaten.
ConteststakeplaceinthefourthfloorloungeoftheMathDepartment. .
.Image: FlickruserPaulAdamSmith
. . . . . .
Outline
Recall: Fermat’sTheoremandtheClosedIntervalMethod
TheMeanValueTheoremRolle’sTheorem
WhytheMVT istheMITC
TheIncreasing/DecreasingTestUsingthederivativetosketchthegraph
TestsforextemityTheFirstDerivativeTestTheSecondDerivativeTest
. . . . . .
Fermat’sTheorem
DefinitionLet f bedefinednear a. a isa localmaximum of f if
f(x) ≤ f(a)
forall x inanopenintervalcontaining a.
TheoremLet f havealocalmaximumat a. If f isdifferentiableat a, thenf′(a) = 0.
. . . . . .
TheClosedIntervalMethod
Let f beacontinuousfunctiondefinedonaclosedinterval [a,b].Weareinsearchofitsglobalmaximum, callit c. Then:
◮ Either themaximumoccursatanendpointoftheinterval, i.e., c = aor c = b,
◮ Or themaximumoccursinside (a,b). Inthiscase, c isalsoalocalmaximum.
◮ Either f isdifferentiableat c, inwhichcase f′(c) = 0byFermat’sTheorem.
◮ Or f isnotdifferentiableat c.
Thismeanstofindthemaximumvalueof f on [a,b],weneedtocheck:
◮ a and b◮ Points x where f′(x) = 0◮ Points x where f isnot
differentiable.
Thelattertwoarebothcalledcriticalpoints of f. ThistechniqueiscalledtheClosedIntervalMethod.
. . . . . .
TheClosedIntervalMethod
Let f beacontinuousfunctiondefinedonaclosedinterval [a,b].Weareinsearchofitsglobalmaximum, callit c. Then:
◮ Either themaximumoccursatanendpointoftheinterval, i.e., c = aor c = b,
◮ Or themaximumoccursinside (a,b). Inthiscase, c isalsoalocalmaximum.
◮ Either f isdifferentiableat c, inwhichcase f′(c) = 0byFermat’sTheorem.
◮ Or f isnotdifferentiableat c.
Thismeanstofindthemaximumvalueof f on [a,b],weneedtocheck:
◮ a and b◮ Points x where f′(x) = 0◮ Points x where f isnot
differentiable.
Thelattertwoarebothcalledcriticalpoints of f. ThistechniqueiscalledtheClosedIntervalMethod.
. . . . . .
MeettheMathematician: PierredeFermat
◮ 1601–1665◮ Lawyerandnumber
theorist◮ Provedmanytheorems,
didn’tquiteprovehislastone
. . . . . .
Outline
Recall: Fermat’sTheoremandtheClosedIntervalMethod
TheMeanValueTheoremRolle’sTheorem
WhytheMVT istheMITC
TheIncreasing/DecreasingTestUsingthederivativetosketchthegraph
TestsforextemityTheFirstDerivativeTestTheSecondDerivativeTest
. . . . . .
Rolle’sTheorem
Theorem(Rolle’sTheorem)Let f becontinuouson [a,b]anddifferentiableon (a,b).Suppose f(a) = f(b) = 0.Thenthereexistsapointc ∈ (a,b) suchthat f′(c) = 0. . .•
.a.•.b
.•.c
Proof.If f isnotconstant, ithasalocalmaximumorminimumin (a,b).Callthispoint c. ThenbyFermat’sTheorem f′(c) = 0.
. . . . . .
Rolle’sTheorem
Theorem(Rolle’sTheorem)Let f becontinuouson [a,b]anddifferentiableon (a,b).Suppose f(a) = f(b) = 0.Thenthereexistsapointc ∈ (a,b) suchthat f′(c) = 0. . .•
.a.•.b
.•.c
Proof.If f isnotconstant, ithasalocalmaximumorminimumin (a,b).Callthispoint c. ThenbyFermat’sTheorem f′(c) = 0.
. . . . . .
Rolle’sTheorem
Theorem(Rolle’sTheorem)Let f becontinuouson [a,b]anddifferentiableon (a,b).Suppose f(a) = f(b) = 0.Thenthereexistsapointc ∈ (a,b) suchthat f′(c) = 0. . .•
.a.•.b
.•.c
Proof.If f isnotconstant, ithasalocalmaximumorminimumin (a,b).Callthispoint c. ThenbyFermat’sTheorem f′(c) = 0.
. . . . . .
TheMeanValueTheorem
Theorem(TheMeanValueTheorem)Let f becontinuouson [a,b]anddifferentiableon (a,b).Thenthereexistsapoint c in(a,b) suchthat
f(b) − f(a)b− a
= f′(c). . .•.a
.•.b
.•.c
. . . . . .
TheMeanValueTheorem
Theorem(TheMeanValueTheorem)Let f becontinuouson [a,b]anddifferentiableon (a,b).Thenthereexistsapoint c in(a,b) suchthat
f(b) − f(a)b− a
= f′(c). . .•.a
.•.b
.•.c
. . . . . .
TheMeanValueTheorem
Theorem(TheMeanValueTheorem)Let f becontinuouson [a,b]anddifferentiableon (a,b).Thenthereexistsapoint c in(a,b) suchthat
f(b) − f(a)b− a
= f′(c). . .•.a
.•.b
.•.c
. . . . . .
ProofoftheMVT
Proof.Thelineconnecting (a, f(a)) and (b, f(b)) hasequation
y− f(a) =f(b) − f(a)
b− a(x− a).
ApplyRolle’sTheoremtothefunction
g(x) = f(x) − f(b) − f(a)b− a
(x− a).
Then g iscontinuouson [a,b] anddifferentiableon (a,b) since fis. Also g(a) = 0 and g(b) = 0 (checkboth). Sothereexistsapoint c ∈ (a,b) suchthat
0 = g′(c) = f′(c) − f(b) − f(a)b− a
.
. . . . . .
QuestionOnatollroadadrivertakesatimestampedtoll-cardfromthestartingboothanddrivesdirectlytotheendofthetollsection.Afterpayingtherequiredtoll, thedriverissurprisedtoreceiveaspeedingticketalongwiththetollreceipt. Whichofthefollowingbestdescribesthesituation?
(a) Theboothattendantdoesnothaveenoughinformationtoprovethatthedriverwasspeeding.
(b) Theboothattendantcanprovethatthedriverwasspeedingduringhistrip.
(c) Thedriverwillgetaticketforalowerspeedthanhisactualmaximumspeed.
(d) Both(b)and(c).
Bepreparedtojustifyyouranswer.
Answer(b)and(c).
. . . . . .
QuestionOnatollroadadrivertakesatimestampedtoll-cardfromthestartingboothanddrivesdirectlytotheendofthetollsection.Afterpayingtherequiredtoll, thedriverissurprisedtoreceiveaspeedingticketalongwiththetollreceipt. Whichofthefollowingbestdescribesthesituation?
(a) Theboothattendantdoesnothaveenoughinformationtoprovethatthedriverwasspeeding.
(b) Theboothattendantcanprovethatthedriverwasspeedingduringhistrip.
(c) Thedriverwillgetaticketforalowerspeedthanhisactualmaximumspeed.
(d) Both(b)and(c).
Bepreparedtojustifyyouranswer.
Answer(b)and(c).
. . . . . .
Outline
Recall: Fermat’sTheoremandtheClosedIntervalMethod
TheMeanValueTheoremRolle’sTheorem
WhytheMVT istheMITC
TheIncreasing/DecreasingTestUsingthederivativetosketchthegraph
TestsforextemityTheFirstDerivativeTestTheSecondDerivativeTest
. . . . . .
WhytheMVT istheMITC
TheoremLet f′ = 0 onaninterval (a,b).
Then f isconstanton (a,b).
Proof.Pickanypoints x and y in (a,b) with x < y. Then f iscontinuouson [x, y] anddifferentiableon (x, y). ByMVT thereexistsapointz ∈ (x, y) suchthat
f(y) − f(x)y− x
= f′(z) = 0.
So f(y) = f(x). Sincethisistrueforall x and y in (a,b), then f isconstant.
. . . . . .
WhytheMVT istheMITC
TheoremLet f′ = 0 onaninterval (a,b). Then f isconstanton (a,b).
Proof.Pickanypoints x and y in (a,b) with x < y. Then f iscontinuouson [x, y] anddifferentiableon (x, y). ByMVT thereexistsapointz ∈ (x, y) suchthat
f(y) − f(x)y− x
= f′(z) = 0.
So f(y) = f(x). Sincethisistrueforall x and y in (a,b), then f isconstant.
. . . . . .
WhytheMVT istheMITC
TheoremLet f′ = 0 onaninterval (a,b). Then f isconstanton (a,b).
Proof.Pickanypoints x and y in (a,b) with x < y. Then f iscontinuouson [x, y] anddifferentiableon (x, y). ByMVT thereexistsapointz ∈ (x, y) suchthat
f(y) − f(x)y− x
= f′(z) = 0.
So f(y) = f(x). Sincethisistrueforall x and y in (a,b), then f isconstant.
. . . . . .
Outline
Recall: Fermat’sTheoremandtheClosedIntervalMethod
TheMeanValueTheoremRolle’sTheorem
WhytheMVT istheMITC
TheIncreasing/DecreasingTestUsingthederivativetosketchthegraph
TestsforextemityTheFirstDerivativeTestTheSecondDerivativeTest
. . . . . .
TheIncreasing/DecreasingTest
Theorem(TheIncreasing/DecreasingTest)If f′ > 0 on (a,b), then f isincreasingon (a,b). If f′ < 0 on (a,b),then f isdecreasingon (a,b).
Proof.Itworksthesameasthelasttheorem. Picktwopoints x and y in(a,b) with x < y. Wemustshow f(x) < f(y). ByMVT thereexistsapoint c ∈ (x, y) suchthat
f(y) − f(x)y− x
= f′(c) > 0.
Sof(y) − f(x) = f′(c)(y− x) > 0.
. . . . . .
TheIncreasing/DecreasingTest
Theorem(TheIncreasing/DecreasingTest)If f′ > 0 on (a,b), then f isincreasingon (a,b). If f′ < 0 on (a,b),then f isdecreasingon (a,b).
Proof.Itworksthesameasthelasttheorem. Picktwopoints x and y in(a,b) with x < y. Wemustshow f(x) < f(y). ByMVT thereexistsapoint c ∈ (x, y) suchthat
f(y) − f(x)y− x
= f′(c) > 0.
Sof(y) − f(x) = f′(c)(y− x) > 0.
. . . . . .
ExampleFindtheintervalsofmonotonicityof f(x) = 2/3x− 5.
Solutionf′(x) = 2/3 isalwayspositive, so f isincreasingon (−∞,∞).
. . . . . .
ExampleFindtheintervalsofmonotonicityof f(x) = 2/3x− 5.
Solutionf′(x) = 2/3 isalwayspositive, so f isincreasingon (−∞,∞).
. . . . . .
ExampleFindtheintervalsofmonotonicityof f(x) = x2 − 1.
Solutionf′(x) = 2x, whichispositivewhen x > 0 andnegativewhen x is.Wecandrawanumberline:
. .f′.− ..0.0 .+
. . . . . .
ExampleFindtheintervalsofmonotonicityof f(x) = x2 − 1.
Solutionf′(x) = 2x, whichispositivewhen x > 0 andnegativewhen x is.
Wecandrawanumberline:
. .f′.− ..0.0 .+
. . . . . .
ExampleFindtheintervalsofmonotonicityof f(x) = x2 − 1.
Solutionf′(x) = 2x, whichispositivewhen x > 0 andnegativewhen x is.Wecandrawanumberline:
. .f′.− ..0.0 .+
. . . . . .
ExampleFindtheintervalsofmonotonicityof f(x) = x2 − 1.
Solutionf′(x) = 2x, whichispositivewhen x > 0 andnegativewhen x is.Wecandrawanumberline:
. .f′
.f
.−.↘
..0.0 .+
.↗
. . . . . .
ExampleFindtheintervalsofmonotonicityof f(x) = x2/3(x + 2).
SolutionWrite f(x) = x5/3 + 2x2/3. Then
f′(x) = 53x
2/3 + 43x
−1/3
= 13x
−1/3 (5x + 4)
Thecriticalpointsare 0 andand −4/5.
. .x−1/3..0.×.− .+
.5x + 4..−4/5
.0.− .+
.f′(x)
.f(x).
.−4/5
.0 ..0.×.+
.↗.−.↘
.+
.↗
. . . . . .
ExampleFindtheintervalsofmonotonicityof f(x) = x2/3(x + 2).
SolutionWrite f(x) = x5/3 + 2x2/3. Then
f′(x) = 53x
2/3 + 43x
−1/3
= 13x
−1/3 (5x + 4)
Thecriticalpointsare 0 andand −4/5.
. .x−1/3..0.×.− .+
.5x + 4..−4/5
.0.− .+
.f′(x)
.f(x).
.−4/5
.0 ..0.×.+
.↗.−.↘
.+
.↗
. . . . . .
ExampleFindtheintervalsofmonotonicityof f(x) = x2/3(x + 2).
SolutionWrite f(x) = x5/3 + 2x2/3. Then
f′(x) = 53x
2/3 + 43x
−1/3
= 13x
−1/3 (5x + 4)
Thecriticalpointsare 0 andand −4/5.
. .x−1/3..0.×.− .+
.5x + 4..−4/5
.0.− .+
.f′(x)
.f(x).
.−4/5
.0 ..0.×.+
.↗.−.↘
.+
.↗
. . . . . .
Outline
Recall: Fermat’sTheoremandtheClosedIntervalMethod
TheMeanValueTheoremRolle’sTheorem
WhytheMVT istheMITC
TheIncreasing/DecreasingTestUsingthederivativetosketchthegraph
TestsforextemityTheFirstDerivativeTestTheSecondDerivativeTest
. . . . . .
TheFirstDerivativeTest
Let f becontinuouson [a,b] and c in (a,b) acriticalpointof f.
Theorem
◮ If f′(x) > 0 on (a, c) and f′(x) < 0 on (c,b), then f(c) isalocalmaximum.
◮ If f′(x) < 0 on (a, c) and f′(x) > 0 on (c,b), then f(c) isalocalminimum.
◮ If f′(x) hasthesamesignon (a, c) and (c,b), then (c) isnotalocalextremum.
. . . . . .
TheSecondDerivativeTest
Let f, f′, and f′′ becontinuouson [a,b] and c in (a,b) acriticalpointof f.
Theorem
◮ If f′′(c) < 0, then f(c) isalocalmaximum.◮ If f′′(c) > 0, then f(c) isalocalminimum.◮ If f′′(c) = 0, thesecondderivativeisinconclusive(thisdoes
notmean c isneither; wejustdon’tknowyet).
. . . . . .
ExampleFindthelocalextremaof f(x) = x3 − x.
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Nexttime: graphingfunctions