g - Math

 b . ,'() * +, -(. ,
( 5
x0
f g : S ;! R
b ;< ! " # $ % b +, % / 01 2 3%4
.' 5 .-
- 67 0 )8 9:
=+, -() > . (1-15)
:3 @ (f + g ) (x0 ) = f (x0 ) + g (x0 )
0
0
.
(f g ) (x0 ) = f (x0 )g (x0 )+ f (x0 )g (x0 )
.
0
0
. S
0
0
b ;< 2
0
x0
f g
x0
- ( ) > .
0
x0
f
g
f +g
(BC
(9 DE ) (G
S = fx 2 S j g (x) 6= 0g
x0
: S ;! R
0
- = (H
+, 3 @
( fg ) (x0 ) = f (x0 )g (x0 ) ; f2(x0 )g (x0 )
g(x0 )
0
0
0
I J K - L ) 5 ' M L f g x f g x0 )
x x0
( + )( );( + )(
;
= f (xx) xf 0(x0 ) + g(xx) gx(0x0 )
;
;
;
;
(B C) . .N K =-L IJK
1
(G)
f g)(x) (f g)(x0 )
x x0
(
;
;
= f (x)g(x)x fx(0x0 )g(x0 )
;
;
= f (x)g(xx) xf0(x0 )g(x) + f (x0 )g(xx) xf0(x0 )g(x0 )
, = 1 x0
;
;
;
;
= ( f (xx) xf 0(x0 ) )g (x) + f (x0 )( g(xx) xg(0x0 ) )
g
;
;
;
;
, x0
g
. ,8-14 01 % S ) - ( ) ,
I J K G T$ @ L I J K - L M ( 5 ' M L .
x ;! x0
g(x) ;! g(x0 )
9
.N K -L GT$@L
(H)
x f x0 )
x x0
f
= f (x(x)g(xx00))g(fx)(gx(0x)0g)(x)
( g )( );( g )(
;
;
;
= ( f (xx) xf 0(x0 ) )g (x0 ) ; f (x0 ) g(xx) xg(0x0 ) ] g(x)1g(x0 )
2
;
;
;
;
x0
.N K GT$@L IJK -L g
U 5 'ML 1 K( J-(. +, 2 . (2-15)
p(x) = a0 + a1 x + + am xm
,
ak xk
(1)
J 2 , = ) ' - 2-5-14 1-5-14 =X Y 1-15 , 9 .' ) - / C + x
0
0
b ( q(x0 ) 6= 0
,
1
q
-() > XL
3 @ 0-N B *, +, ) N
, U ( 5 S
0
.
2
(2)
q (x) = b0 + b1 x + + bn xn
f (x) = pq((xx))
-() , .) %[ 1-15 2J) / b J= 0U/ ,
k = 0 1 : : : m
;
K (H) 1-15 5 .-N U J-(. 2
S = fx 2 R j q (x) 6= 0g
,
= 5
J-(. ,(BC)
p (x) = a1 + 2a2 x + + mamxm
.- U Z 7 b ( \< b J=
b ;< 2
q (x) 6= 0
S
' 1
0
5
x0
x0
b ;< = )
21 =
x
cos sin
,
,
,YY =+, ) '- 10 ^ 5 4-10 0
b 1 . (3-15)
=+, '= csc sec cot tan
() .-(= 7 B *, b ( 2 = ,
, .' / "/ =X -(= 1 3 @ U + , " . 5 N 4 _ ( ) _( ,
,
7 B *, b (
) 1-15 01
:' Z _( +, D,, - .-N B *, =+, J H7
x h
h
sin( + );sin
x
x
=
h
h
= (sin x) cos hh
1
;
.' ) 4 limh
0
!
1;cos h
h
x
h
(sin )(cos )+(cos )(sin );sin
= 0 limh
!
sin
0 h
h
=1
x
+ (cos x) sinh h
, Y Y - L 2-7-13 (
lim sin(x + hh) ; sin x = cos x
h 0
!
x
= 5
sin
+, ,U sin x = cos x
0
x
= 5
(3)
cos
+, ) N 4 D,, J= cos x = ; sin x
0
1 cot = cos tan = sin
csc = sin1 sec = cos
sin
cos
,
(
,
,
(4)
+, ) N 4 (H) 1-15 5 0# XL
# @ 0- ( ( ) B * , 3 % c H ) =
x
5
( * ) B * , b ( x
= 5
:-N K 5 =X -
tan x = sec2 x = 1 + tan2 x
0
cot x = ; csc2 x = ;(1 + cot2 x)
0
sec x = (tan x)(sec x)
0
3
(5)
(6)
(7)
csc x = ;(cot x)(csc x)
0
/ 0
b 5- ec ) ' 5 (8)
67 5 ^ 0
b -J .- U
(4-15)
,
f (x0 ) > 0
0
C L .- N S x0
5
b ; < .'() 2 = ) 0 < e < f (x0 )
0
%S ,
) ' ( ) 7 S
:' e
f : S ;! R
f (x0 ) = 0 f (x0 ) < 0
0
f (x0 ) > 0
0
- c .
0 < jx ; x0 j < - ( ) > 0
- ( ) > >0
) ,-L B *,
;e < f (xx) ;; fx(x0 ) ; f (x0 ) < e
0
0
f7E
f (x) ; f (x0 ) > f (x ) ; e > 0
0
x ; x0
0
(9)
:
) ) 'ML , -(N ec '= - h. J ) H 3 @ (
x0 < x < x0 + 0 U / ,
) >0
f (x)
x0
- < ,
), $ b M .-N
f (x0 )
X - ( ) < ,
'ML ) -= x0
5 M.)
2 1 f (x) x0
x f (x)
,
0U/ ,
x
5 1 2 1 =
,
x0
. *@ 0 XL 0
b 5 j= 4
0
5 M.) 21 =
f
f (x0 )
f (x) > f (x0 )
U x
f (x0 )
- < i < ' M L ) XL 2.) 0
b 5 2 f (0) > 0
MC (
(1-4-15)
,
f (x0 ) > 0
0 U / ,
- < *T 1 $MN .
0
x0 ; < x < x0
f (x) < f (x0 )
.
5 1 f (x0 ) > 0
0 U / ,
0
f
+, *@
) 1-4-15
b ( ) C 1 kk*@ b 7N \
"
-
" 0 M1 +, J $MN *@ C 1 =
7N 5 (= ' N 21 0 ) '= - = C N 2.) l -, "/
.) '=7 m *, (. B *, [ @ 3%c 0-(/ .- 0 nS 0 < e < ;f (x0 )
0
0 U / ,
) ' U S e
K ( . :' 0 < jx ; x0 j < f (x0 ) < 0
0
C L
) >0
;e < f (xx) ;; xf (x0 ) ; f (x0 ) < e
0
0
f7E
f (x) ; f (x0 ) < f (x) + e < 0
x ; x0
0
(10)
:N K / 5 ) -(N N B ec - h. J ) H 3 @ (
x0 < x < x0 + 0 U / ,
) >0
f (x) > f (x0 )
.
x0
5 M.) 21
x
f (x) < f (x0 )
,
f (x0 ) < 0
0 U / ,
0
x0 ; < x < x0
0U/ ,
' ,-N
x0
,
5 1 21
.
,
f : S ;! R
x
f (x) < f (x0 )
K( f (x) > f (x0 )
b b f (x) f (x0 )
. x0
M.) U nS D , , )
'N N
x0
f (x0 )
b jx ; x0 j < * T ( J) (
( ) 5 M J )
x2S
b ; < 5 1
x0
S x0
5
b ;<
= ) -N N -N N *T b (J) (
+ , - ( ) > XL .(
0
* T 5 2 j = 3 @ nS 2 - +, -< ,2-4-15 1-4-15
S x0
5
>0
f (x) f (x0 )
f
f (x0 ) < 0 f (x0 ) > 0
0
2 :!E D,, - .-N
f
,-N
+, 2 .N $@L ", DC K 2-4-15 1-4-15 5
' (
D,, )
(2-4-15)
b ;< f : S ;! R
f (x0 ) = 0
.
5
0
f (x0 )
5
+, (3-4-15)
0U/ ,-N x0
i 7 ,- N _ J i 7 + , ) * T ( J ) ( \ < D , , - f 1C - _J
.(2 $MN) -N < f 1C _J i7 -N < ) )8 !5E
X . *T b (J) (
i7 )
x4
C , * T b (
x2
, * T b (J)
x3 x1
b , 5 p @ =XY .*T (J) 0-N =q *T (
-
b
\ < ,2 $ M N
, < _J
.-(= D; x
= 5
+ , . 0- N B * ,
f (x) = x4 ; x3
3 @ :' .'() N #@ f : R ;! R
+ , .1 / ) S< . f (x) = 4x3 ; 3x2 = x2 (4x ; 3)
0
x4 ; x3 = x3 (x ; 1)
-,J
x=0
ec , .N #@
9 , #(
-N (J) -
0 1]
0<x<1
%Y
f (x) = x4 ; x3
x = 34 x = 0
x< 0
f
b ;< -< ) N ZLe
b +, . .-N *T b (J) (
b ;< 2 - (J) b ;< ( .-N #( - (J) -< , #(
,-N #@ - b (J) U S 5 .
. 0-N 0 ^J (BC) 3 $MN )
f (x) = 1 ; cos x
0
f
f (0) = f (1) = 0
]0 1
) 5 -N
f 1C
J . (J) 2 ' .' Z + , * T ( J ) ( 9
f (x) = x ; sin x
\ < 5 2 j = . N # @
2
+, -<
0 1]
x = 34
;< 9
+ , .2 p [ @ G r .((G) 3 $MN)
b L .' 5 05 2 ec #( %Y -* ! .N -=7 1 U = ) ) \
"
U r
) D(
.'() 4 n * X r ) r 5 @7 CL
e
6
\ < b J = ) * ,
a<c<b
) ,
c
, f : a b] ;! R
; < 3 @ .
- ( ) > f (a) = f (b) = 0
. e
! (5-15)
- N .
Z c
f (x) > 0
+, .
b ;< #@ ;< = / f (x) 6= 0
f Y ,
-() > e
) f
- < , % Y ;< 2 $-L 05 b ;< 2 - (
f (x) < 0
,
a b]
f (a) = f (b) = 0
b ;< 9 ,
f (c) = 0
D,, J= .
0
x
a b]
f
a b]
f (c) = 0
0
.
;< -() > XL .
) K / 5 ( a b]
f
b U S 5 .-N %Y - (
' 3-4-15 %S XL .
2
c
) -N #@
a<c<b
)
c
f Y ,-N
e
.'() - Z b ;< ,
(J) 5 0# 5
J _ J ) b a
c
; < ) E b r %* , 2
f < , 5 r . J " $@ i7 5
.) %*, b
\ < b J= ) * ,
0<c<b
) ,
f : a b] ;! R
, c
- ( ) > ."#$ b
;< 3 @ .-N ; f (a)
f (c) = f (bb) ;
a
0
: 5 b C*
! (6-15)
a b]
(11)
(b f (b)) (a f (a))
$@ i7 .
; f (a) (x ; a)
y = f (a) + f (bb) ;
a
B *, , S .' X b r *T '() ')
e
f (x)
5 J -< -()
; f (a) (x ; a)]
g(x) = f (x) ; f (a) + f (bb) ;
a
7
5 . = t + , I J K ) 5 ]a b
a b]
g
+ ,
:U S
g(a) = 0 g (c) = 0
:(* ,
0
g (b) = 0
a<c<b
) ,
c
;< X b r %S 9
; f (a ) = 0
f (c) ; f (bb) ;
a
0
2
. r 'ML )
y = f (x)
08 2 M % S . a b]
5 x
u .' ) 0 N r - ( = % * , 2 5 0
b 5 08 i c f (b) f (a)
b a
;
;
,' ( ) > x
5 v [ c 5 / 08 c ) )L 5 I N 5 c
5 ,
r
J , 3 @ 6-15 = ) J" + . X S i f * C 7 ' = 7 / f- * ) - = 7 + , 2 J $ J M , ) - ( . e
.' 5 ec
(7-15)
. 05 \< ) +, 2
. 4
I
f (a) = f (b)
f Y .
-() > e
'ML f (b) f (a)
b a
;
;
f
' = f (c) = 0
0
0U/ ,
I x
5
I b a
5
f : I ;! R
b ;< = 2
f (x) = 0
0
1J b ;< = '= ) b a
c
I
-() >
(1-7-15)
) .
; < U b r % S .
2
-N 05 2
a< b
.- 3%4
I
\< ) -(N +, f g : I ;! R
. 4 2
8
f ;g
-() > .'($ (2-7-15)
3 @ .- 0
(* , *@
I
f
I x
0U/ ,
5
b ;< = .
2
.-(= ec '=
f (b) ; f (a) b ; a
(* , C 1
I
f
I
0U/ ,
9 ,
x
f (b) f (a)
b a
a b]
\< K (11) 5 0 U / ,
]a b
f
x
= ' ,
0
;
b ;< = .
0
= f (c) > 0
;
2
f (a) < f (b)
f (x) > 0
f : a b] ;! R
jf (x)j M
0
a<b
)
I
b a
=
U b r %S .
f (x) < 0
0
f (a) > f (b)
(3-7-15)
' ,
(4-7-15)
a<b
)
I
b a
=
-() > . )$ ")* (8-15)
) - N N M 0
. :
) N
jf (b) ; f (a)j M jb ; aj
U (12) .N G[
. x
b ;< y = f (x)
J w(=/ Y)-L 05 XS GT$@L 5
9 .-(= ^J
y
f (b) ; f (a)
y
(* ,
(12)
J) u, w(=/
a b]
J 0
b 5 x
y
f (x)
0
) K/ 5
J ) * b;a
x
(* ,
J =
: J
j
yj M j
xj
(13)
' = -= j sin ; sin j j ; j
j cos xj 1 f (x) = cos x
.N K (12) 5 'ML ,
0
(14)
f (x) = sin x
' ,
:' %Y
j 1 1+ a ; 1 1+ b j ja ; bj
9
.1 b a
) K/ 5
= -= .2 (15)
, ,0-N B *, +, 05 .' Z ] ; 1 +1
f (x) = 1+1 x
+,
:' ;1
(1 + x)2
jf (x)j < 1
f (x) =
0
.N $@L (15) 0
9 , 1 1 5 H ,
10
x>0