Week 8
Integrability
Extra exercises
Ext f x X EQ A a b
0 Q
What is Uct f
Pf f supp f P Sgp 03 0
LCF P My oxo 0.0 4 0
Ulf inf Ulf P 9 x x2 on a b
inpf 419 P
Ucg b a
Lef Ucf f not integrable on aid
EX Find the following limits
a Is x dx Hint fu f
PF
yn fix
0 ETO 1
Let fe 0,1 TBD
of x dx J xdx xdx
make that 0 on 20,8
methods S.si xdxefsxndx 8Sfdx
14
1
Also 0 xdx l dx 1 8
Now given 70 pick 8 1
Also because Sf xdx 0 N 0
sit n N dx c
Of xdx E
Even though for f
limf'fn f f 0
O O
EX3 f integrable on 9lb g on carb
s t glx fax for all but finitely many
c a b Show that 9 is integrable
and I'f 9
Pf Suppose IX x I ECaib
otherwise
Ix 0
To get U 2 0 define Partitions
Pn a t x.tt b
for a large enough
If xo a or b we denote
Pn a ath b
or Pn a b tib
O U Ix Pn 0
U Ix if U 1 0 P 0
Ix is integrable and
IX 4 dx 0
d
CIx is still integrable for CER
Now f g has finitely many difference
so f g x ELEK
where Xi Xn are the points f 9
and Ck f 9 Xk
f g is integrable
If g dx caffmandx o
fbfdx fbg.ch
BE
EX 4 fax sin C 1 1 1903
0 0
Is f integrable
Pf For Se 01 TBD
1 8 U E 8,8 V18.13 1 1
70 pick 870 48 1
S E
B c sink is continuous on 11 8 U 8,1
7 P on 11 8 and Pz on 8,1
Sit Ulf P1 f P
Ulf P3 Cf P3
Now Pa 5,8
Ucf Pr f Pa 3
Let D PIUP UP a partion 2411
Ulf P LHIP a 3 3 5 E
f is integrable on 1 1
Review
chain rule f diff at a 9 diff at fa
of a 9 flat F a
Intermediate value Thin for derivative
let f is differentiable on lab Ex XD
If c between f x and flex
7 between Xiix s t
flex C
Ex f diff on IR and 7107 0 f 1
f 2 1
slow f x for some 10,2
Pf By MVT 7 c 0 1 Sit flexi 1
7 2 1 2 s t f x2 0
By IVT for derivative 7 3 E X1 X2
sit f x 4
Taylor's Thm
fix on aib is n th derivative
fix n f c f x c it ftp.t
V1 Rncx fffCx c7
ye x C
V2 Additionally f is C
Rna L face dt
Corollary as assumptions as V2
Rnix Cx c fally
Cf smooth f locally equal to
Taylor's expansion