4.4 Mean Value Theorem - STEM Math & Calculus · 2019-12-10 · This was created by Keenan Xavier...
Transcript of 4.4 Mean Value Theorem - STEM Math & Calculus · 2019-12-10 · This was created by Keenan Xavier...
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4.4 Mean Value Theorem
Standards
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§ Extreme Value Theorem
tffk ) is Continuous on the closed interval [ a ,D,
thenfklattainsitsAbs max
-- . . .
.
tabsmax&*snwmm* '
¥€¥t[ Exampkffindabsmnxtmiwfrftxtzs - Mx on E3D .
absm.ms#p:ftx1=zt.80xSz4.so=OfKHSz4.soS(tt- 161=0
z' -16=02-4=16
z=2,
-2.
Step# criticality endpoints
f (2) : -128 fly =3
f H)=128 ffD= -3
Slept:The abs Maxis 128 occurring at X=2& Nbs minis -128 occurringat ×=2 .
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n@NkanValueThemempkasenolIThemainreswttofthistpicistheMeanValueTheaom.Topovelunderstandthiscmoept.weneedtoKnWtheRoHe1sThewemBasicldeaofRo11eisthenem-lftheendpointva1uesoffareequahtslopeofthethenfmusthareanexbemevalneat-tungentlheissomenumberstricHybetnuntheM0iendpoints.p
wvidedthatfiscmhnnms•... ... .f... .• Mtheintemallfis differentiable
,# then f 'kj=o .
Ca b
Hsohol=
÷A slope might be Otniw( or more ) between numbers
•[[¥¥• hehqehowtheralnesoffare±a b ¢f'÷
•
Rolle 's Theorem F)Isuppose that I I•
§i¥EhEstM¥Eewn¥asYeEnBan⇐Then
,there must exist atleastanumberc
In [ aidsuohthatf 'H=O .
Rolle's Theorem witlleadustothe Mean Value Theorem.
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Meanvaluetheorenx
suppose that1
. ftx) is continuous on [a, D
2. fk ) is differentiable on (a , b)
then,
there exists a number c in ( a,b) such
that§ (d = flbtftfs
- a .
noted The Mean Value Theorem is a more general case of Rolle's Theorem
which has additional hypothesis that fk#b ) and in conclusion states
that there is a number c such that f 'H= 0 :
Example 1) Find a point c Satisfying the conclusion of MVT for fatty in
[2,8] .
FITIs f continuous on GD ? yesSecond Is f differentiable on (2 , 8) ? yes
Now, ftc) : f (b) -f
b - aLet a=2& b=8
.
so, ftfifztt = ¥6
Also,
fly = te Erik C't
.f '
Cx ) =- Ice = ¥
.
So now set f' 4 = ¥6 =) ¥ = I
- a = - 16c
2 = 16
C = ± 4.But since c= - 4 isn't in [2,8] , C= -4 is not the answer .
Therefore I
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Example D Find a point c Satisfying the conclusion of MVT frftx ) =pon [9,25] .
First Is f continuous on [0,253? yesSecond : Is f is differentiable on (9,25) ? yes
Now,
f 'C D= flb)-fCa÷
a.
Let a- 9,
b : 25. So
, ftp.#=z5f=sIAlso fH=F .
Now, f ' H Izi B
= Ifso
, Frets8=2 R
4¥Html16 = c
Therefore ,c-- 16 meeting MVT 's requirements
.