Local Linear Approximation of Nonlinear Functions

 ! " #$ % ! "& % % % % ' (% & ) ! #$
# * $ % %% !"
' #& $ !" + #$, ) !" ! & &% % ' !" " ! " " & %%" ! %%" -% " " %
./
2
y
1
-3
-2
-1
0
1
-1
-2
" x
2
3
0.2
x
1.3
1.2
1.4
0
0.1
-0.2
-0.1
0
0.1
x
0.2
-0.2
-0.1
-0.2
" " .
( " " " " % " " % . " +& " " % . " 0 ) %1 " " & 2% !& 34 "
% . 2 #5%6$3 " 7 " " & % " . " & & & " % " % . * ! ! '" " " 8 " " ! ! %" " " % .
2
1
-3
-2
-1
0
1
x
2
3
-1
-2
" 8
) #5 $ " " 9 # $ %% 5 ! '" " " !& * & ' ! '" " " . * "& & ) % % %% " * 2! *3 % "
:" % ! "
2 " ;3 ) " % :"
% " ;
" .
) " "
<
6
5
3
2
1
-2
-1
0
1
2
3x
4
5
6
-1
-2
" </ = ( " *
&" % % &" % ! !% ! &" % % 2 " %
%3 9 " " %& ) " &" % " % >
" ? %6& " * ! %"
&" % " " 8
-0.9
-0.92
-0.94
-0.96
-0.98
-1
-1.02
-1.04
-1.06
-1.08
-1.1
0.92
0.96
1 1.02
1.06
1.1
" 2
1
0
1
2
3
-1
-2
" ?
" " " ' ' " ) " &% ! " ! ! ' " # $ ! " " & % ! " :" %
- " !
" " @ " " % % % ;
-0.9
-0.92
-0.94
-0.96
-0.98
-1
-1.02
-1.04
-1.06
-1.08
-1.1
0.92
0.96
1 1.02
1.06
1.1
" @
) :" % & % " 9 !% % 7
%1 ! %1 ! " +& " 2 " 3 & % & % & " ' ) %" & ) :" % " " %" % <
-0.9
-0.92
-0.94
-0.96
-0.98
-1
-1.02
-1.04
-1.06
-1.08
-1.1
0.92
0.96
1 1.02
1.06
1.1
" % !% % >
& !% ( ) ! ! :" % & 2A
% % ' %% %3
:" % B% ! '% "
" ! ! % % ! !
" & 2 % 3 ) &" % "
" *
! % & ) % " " %6& " % " 5%6 " !
" " ! %" !
6
4
-2 -1
0
1
2
3
4
5
6
-2
-4
" -0.9
-0.92
-0.94
-0.96
-0.98
-1
-1.02
-1.04
-1.06
-1.08
-1.1
0.92
0.96
1 1.02
1.06
1.1
" =%% ! %" " C >
% " ) " & & % #%>&$ " & " & ?
B '" %6& " " &
" ' ! # $ &" %
% &5
. - & & 2! *3 "% "% # $ % % &" % %%
2 " ! # $3
8 - :" %
; D . % 8 " & %1 % ' ! 2' )! 3
< B " ! % " ! " " / #7 " $
* " " ) " *
-% & * " ! % &" % A % !& " &
@
2 E %" !>
3 7 " !! %%
" " ' & ' '0
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.5
1
x
1.5
2
! " # 1.2
1.1
1
0.9
0.8
0.9
1
x
1.1
1.2
$%& '
( ) * ' +
,
) - . / ( 0
( 1 0 ( 2 ( 1
1.2
1.1
1
0.9
0.8
0.9
1
1.1
1.2
3 3 4 1 2 +
1 !" !" !" !" !" ! " !" !" !" !)" !5" !" (1 6
$ (1 6