Week 6
MVT Taylor's Thin
31.7 Binomial Series Them
V1 new Ctx E k xk xGR
V2 EIR Hx I
21 1 K
1 1 1
Smooth function VS Taylor's expansion
an
fix II anx
Ex fix e 70
0 O
Ext Prove 1 cos 1 4
for 10 I
Pf
Method 1 MVT
Partlifex Cosx 1 on 0 I
flo 0 and f is cont on 20
and differentiable on 10 E
By MVT
f f 5 sin
E 0 E
For XE 10 suffice to show that sinXtX 70
for X ECO
For flat 0 It 0
To check prove xe o six CO
g Sinx on 0 Is
Then 910 0 and by MUT
sinx 911 6,15 120
E 0
Sinx 4 610
Part 2 fix 1 cost for ECOTE
10 o V f E 70
Let 10 fat f's sin
We suffice to show that 4 210 E f x 70
f 10 D by Part I
f 12 8 7 f 3 1 2 05 00
flex 70 xt 0
fix 70 0
Overall 1 Eccosx 1 1
24
XE 0 I
ID
Method 2 Taylor's Theorem
Cos X 1 1 R Ix
Cost 1 Es R 5 10
44 0
cosx 1 Ro 3
0
f x cost
f x Sinx
f x cost
f x sin
f x Cosy
451
x six
f cost
EX fir IR diff suppose fix fix
on R 8 fixo 0 for some do
Prove fix o x7Xo
Pf f x fix 70
g e fix
91 7 5 fix e flex
e flex fix 70
And 91 07 0
gex 70 for x Xo
fix 70 for Xo
BY
Ex3 Show that ae txt for a
has exactly one root
int For fix aet Hx
Show f fix