R f - math.ucdavis.edushu/teaching/Summer_2018/MAT125B_8... · snap Ifl ingff in.at ifI swap f case...
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Transcript of R f - math.ucdavis.edushu/teaching/Summer_2018/MAT125B_8... · snap Ifl ingff in.at ifI swap f case...
OH i
M 12 30 3 00 PM
R f i oo 3 30 pm
Theorem Absolute property If feaca bthen Ifl ERca.br and
ISabfixsdx.es IfualdxRemark Ixtyle 1 1 191
III xile naI f
Isifixidxle fablfixildxproof Let Po fa xocx c cXw b be a
partition of Taib then on each Exia Xi
HI a P H I
II fautt iIt
Case 1 denote A Ixia Xi
suppose snap f z int f 70
so snap Ifl snap f
int Ift int fcase 2 appose ozszpfzina.tt
sosnap Ifl ingffin.at if I swap f
case 3 suppose SIP f z o z inaff
then Sfp Ifl maxf snap f inaff
guff o
mathI
inaf Ifl E min I snap f inafffft.Y.it eeoxstoEUglIfiLg Ift E Volf CflSince ft Ila b then as 8 I O
Urift Lrcf o
so by the squeeze theorem
Usc Ifl Ifl o as Tso
So Ifl e Ila bNotice that
Ifk l E fix E lfcxstv xe a.LI
by monotonicity of integrals
fabifcxsldxefabfcxsdxefablfcxsldx
elfabfcx.de ESabIfcxsldxA
Remark The converse is not true
e.gfix f
it to.sn
if KE Ion iQ
then Ifix I I C Iconbut f g Rca b
file fne kca.bl.tnb
An Safia dx Tff So if
th An Life fab fax d x
I f feefuxidxwhat is the condition for this to be true
Theorem Interchanging limit and integral
Let Effi be a sequence of integrablefunctions defined on Each Suppose that
f a f uniformly then f c Rca band
thy fabfax d fablesfax dx
fab fan dx
proof HW
Recall 9ft f tabs 77K
then we say f f uniformly ifHE o 7 NOTE s.t.vn
fncxi f ex I E for xo Eaib
we say f f point ifb Taib I 0 I 0 Sit
b n N N cx.es
fncxs fcx
fix1xEENxNkati
Example let Erny be an enumeration
of all the rational numbers in Coil
define Anti as it n l
fuk e Il it tri r mgo if xefo.dllri.rs e rn
fu C Reais t uz l
pointwise limit
fix if xo co.in Q
o i f XE Io 1 l IQ
f if Rlah
free fabfacade lez o o
7 Fundamental theorem of calculus
Relation between differentiation and
integration
Def n Antiderivative For f table 51kthe antiderivative of f is a
continuous function 112 sitF is differentiable on ca b and
data F let fix for all xeca b
Example i f tou Tk fix X
Text 31 3C
Remark If anti derivative exists theremust be infinitely many
Example f I i i Ph
if o X E I
f it.ieon o l x
on l l o C x I
Fix a 1 1 is not not differentiableat x D
f does not have an anti derivative
Hw Problem 17