matan-1_2

% 0 : & ' '

) " '$" 4- $ 7 # , 7 , 7

7 ( 7 , 7

	f		f	(n)		n		n+1
f (x) = f (x0) +	x0	(x − x0) + · · · +		x0	(x − x0)		+ O((x − x0)		)
	1!		n!

x → x0

+ / * & E 9 9 > ; ' #

# > ;

# 7 7 : 7 , : E D 7

M 7 7 C M 7

#"L " 6- f : Ox0 → R

Ox0	x0 x0
	•	•

Ox0 # Ox0=

l O• x0 r O• x0 2 3# )

NTO ( x

•

(f

< 0))

( x

•

(f > 0))

Ox0

x

Ox0

x

f x0 & ;

NTTO ( x

l •

(f

< 0))

( x

r

•

(f

> 0))

Ox0

x

Ox0

x

f x0 ;

NTTTO ( x

l •

(f

> 0))

( x

r

•

(f

< 0))

Ox0

x

Ox0

x

f x0

•

NTO ! 7 (

x

(f < 0))

Ox0

x

L ( x

l •

(f

< 0))

( x

r

•

(f

< 0)) '

Ox0

x

Ox0

x

l O• x0 r O• x0 7 ( f 7

C f , x0: x

l

•

Ox0 x

r O• x0

f (x ) >

lim

f (x ) = f (x0) =

lim

f (x ) > f (x ).

x →x0−

x →x0+

C M 7 , x0 7 , M 7 7

, 7

•

NTTO ! 7 ( l Ox0 f 7 <

•

: r Ox0 7 C

% 0 : & ' '
, x0 f : 7 7 x			l	•	x	r	•
				Ox0			Ox0
f (x ) > lim f (x ) = f (x0) =	lim	f (x ) < f (x ),
x →x0−	x →x0	+

x0 Q , 7 7 7

, NTTTO 7 , NTTO

5 7 D 7 7 7 7 Q

, 7 M 7 7

7 M , " 7 7 7 f ,

x0: 7 7 7

#"L " 6 3 y = f (x)

Ox0 x0 R x0

n N ! # . fx0 = · · · = fx(n0 −1) = 0 fx(n0 ) = 0

n x0 & n

fx(n0 ) > 0 0 x0 f

fx(n0 ) < 0 0 #

C 7 7 7 # , 7 , 7 7 C <

	fx(n0 )
f (x) = f (x0) +		(x − x0)n + α(x)(x − x0)n,
	n!

α(x) Q , 7 x → x0 ) E 7 M 7

f (x) − f (x0) = n1!(fx(n0 ) + β(x))(x − x0)n,

β(x) Q , 7 x → x0: 7 :

7 7 " fx(n0 ) = 0: x <

, D x0 fx(n0 ) + β(x)

7 fx(n0 ) ) , : n Q , : D , x0

(fx(n0 ) + β(x))(x − x0)n 7 : , 7 n Q

: M 7

fx(n0 )

LC "5"("'$" 6- f : (a, b) → R

: x1, x2 (a, b) α1, α2 R+ (α1 + α2 = 1) 7 7

f (α1x1 + α2x2) ≤ α1f (x1) + α2f (x2)

% 0 : & ' '

7 ,

f : (a, b) → R , : ,

,

D : M

LC "5"("'$" 6 3 f : (a, b) → R

: x1, x2 (a, b) α1, α2 R+ (α1 + α2 = 1) 7 7

f (α1x1 + α2x2) ≥ α1f (x1) + α2f (x2)

' 7 : , ,

E D : M

LC "5"("'$" 6 2 ! N O (a, b)

f : (a, b) → R N

O: x1 = x2 6- N6 3O 7 7

:

f (α1x1 + α2x2) < α1f (x1) + α2f (x2) (f (α1x1 + α2x2) > α1f (x1) + α2f (x2)).

#"L " 6 2 $ (a, b)

y = f (x) fx

(a, b)# &

! fx #

C f : (a, b) → R 7 (a, b) # E x = α1x1 + α2x2: α1 + α2 = 1: α1, α2 > 0 7 7

α1	=	x2 − x	, α2	=	x − x1	.
		x2 − x1			x2 − x1

# 7

	f (x)	x2 − x	f (x	) +	x − x1	f (x	).
		≤ x2 − x1			x2 − x1
			1			2

L , 7

(x2 − x)f (x1) + (x1 − x2)f (x) + (x − x1)f (x2) ≥ 0.

# 7 7: , (x2 −x1) = (x2 −x) + (x −x1): M 7 , 7

f (x) − f (x1)		≤	f (x2) − f (x)
x − x1			x2 − x

% 0 : & ' '

x1 < x < x2 C , 7

+ 7 7 x , x1: 7

x2 C , 7

	f	≤	f (x2) − f (x1)	≤	f	,
			x2 − x1
	x1				x2

7 fx

+, M : 7 ( <

	f	≤	f	=	f (x) − f (x1)	<	f (x2) − f (x)	= f	≤	f	,
					x − x1		x2 − x
	x1		ξ1					ξ2		x2

x1 < ξ1 < x < ξ2 < x2: ,

7

# fx (a, b) 7 N 7 O <

# a < x1 < x < x2 < b 7 (

	f	=	f (x) − f (x1)	≤	(<)	f (x2) − f (x)	= f	,
			x − x1			x2 − x
	ξ1						ξ2

x1 < ξ1 < x < ξ2 < x2

("5 #!$" 6- C f : (a, b) → R !'

(a, b) ! !

fx ≥ 0# D fx > 0 #

5 , 7 (

7 fx

, 7 <

#"L " 6 4 $ (a, b)

y = f (x) fx

(a, b)# &

! fx #

("5 #!$" 6 3 C f : (a, b) → R !'

(a, b) ! !

fx ≤ 0# D fx < 0 #

LC "5"("'$" 6 4 C f : Ox0 → R Q : <

•

Ox0 , x0 R " 7 l Ox0

% 1 ) ;$

•

f N O: 7 r Ox0 f <

N O: , (x0, f (x0))

# 7 7: , x0 f <

7 : 7 6 2: 6 4 7 7 D <

7 : , fx0 = 0 "
•	l •
Ox0 , x0	Ox0 7 :
r O• x0 : x0 ,

+ 7 1 ! '

! 7 ,

#"L " 0- N ( O f : (a, b) →

R g : (a, b) → R (a, b) (−∞ ≤ a <

b

≤

+

) g

= 0 x

(a, b)

∞

x

f

lim

x

= A

(

−∞ ≤

A

+

).

x

→

a+ g

≤

∞

x

) ! !'

243

lim f (x) = 0)

( lim g(x) = 0)

(x a+

x

→

a+

→

∞

2443 lim g(x) =

x a+

→

lim

f (x)

= A.

x→a+ g(x)

	, x → b−
	C 7 , : , A < +∞f							: 7 , p q
: , A < p < q C				lim		x		= A: ,
				x→a+ gx
c		(a, b) : , 7 x			(a, c) f				/g < p < q #
								x	x
g			= 0 (a, b): g(x) 7 (a, b): 7 : <
		x

, c1 (a, c) , , a: 7 , : , g(x) = 0 (a, c1) C M 7 D x, y (a, c1): y < x 7 % E D 7 7

f (x) − f (y)				f
			=	ξ	,
				g
g(x)	−	g(y)
				ξ

( A < 4 1 -2 3 .

% 1 ) ;$

ξ (y, x) (a, c1) L

	f (x) − f (y)	< p,	(6.1)
	g(x) − g(y)

x, y (a, c1)

# 7 7 7 8

Q , NTO: D 7

y → a+: , 7

f (x)	≤ p < q, x (a, c1);
g(x)

Q , NTTO: y x , 7 a:

7 7

g(x) − g(y)

g(y)

= 1

> 0.

(6.2)

g(x)

− g(x)

5 7 N0-O , N0 3O: , 7

g(x)

− g(x)

f (x) − f (y)

< p 1

g(y)

,

f (x)

g(y)

f (y)

< p − p

+

.

g(x)

g(x)

g(x)

! c2 (a, c1) , 7 a:

x (a, c2)

7 7

f (x)

g(y)

f (y)

< p − p

+

< q.

g(x)

g(x)

g(x)

$: D , D NTO NTTO (a, cq):

, f (x)/g(x) < q

" A = −∞: q > A 7

M 7 ,

" A = +∞: q˜ p˜ : , q˜ < p˜ < A: ,

7 7 (a, cq˜): , q˜ <

f (x)/g(x): lim f (x)/g(x) = +∞

x→a+

" A R: , q˜ < A < q 7

(a, cq˜) ∩ (a, cq): , 7 7

q˜ <	f (x)		< q,		lim	f (x)		= A.
	g(x)
					x→a+ g(x)

% 3	) . $
+ D	' 1 B ' 9 A

LC "5"("'$" 1- C y = F (x) y = f (x) <

7 7 (a, b): , 7 F <

7 M 7 7 "

x	(a, b) F	= f (x),	(7.1)
	x

F (x) f (x)

7 (a, b)

) " '$" 1- ' 7 : , F (x)

f (x) , 5 : F (x) Q f (x): Φ(x) = F (x) + C: C

: :

(Φ(x))x = (F (x) + C)x = Fx + Cx = Fx = f (x).

LC "5"("'$" 1 3 D D <

f (x) 7 7 (a, b)

f (x) M 7 7 : ,				7<
7			f (x)dx
E		f (x)dx = F (x) + C.		(7.2)

) F (x) Q < f (x) 7 <

: f (x) Q	F
	: <
(a, b): C Q ) F

f (x)dx Q

C 7 <

Fxdx = dF (x):

f (x)dx = dF (x).

(7.3)

L D <

: :

C M 7 7 <

: 7 N1-O: 7 N1 3O

% 3 ) . $

$ 7 D 7

8

xαdx =

xα+1

dx

+ C, α = −1;

= ln |x| + C;

α + 1

x

axdx =

ax

exdx = ex + C;

+ C, a > 0, a = 1;

ln a

sin xdx = − cos x + C;

cos xdx = sin x + C;

dx

dx

= −JVXx + C;

= VXx + C;

cos2 x

sin2 x

Wgxdx = Jgx + C;

Jgxdx = Wgx + C;

dx

= edJVXx + C = −edJJVXx + C;

1 + x2

√

dx

= edJWTYx + C = −edJJkWx + C;

1 x2

−

dx

dx

−JVgx + C.

dx = Vgx + C;

dx =

Jg2x

Wg2x

# 7 7 <

!L& #!L -

df (x)dx = f (x)dx.

$ N1 3O N1 2O : ,

df (x)dx = d(F (x) + C) = dF (x) = f (x)dx,

dC = 0

!L& #!L 3

dF (x) = F (x) + C.

N1 3O N1 2O

#

% 3 ) . $

!L& #!L 2 . f (x) g(x) !

(a, b) ! α β

αf (x) + βg(x) ! (a, b)

(αf (x) + βg(x))dx = α f (x)dx + β g(x)dx.

C F (x) G(x) Q f g < αF (x) + βG(x) Q

αf (x) + βg(x):

(αF (x) + βG(x))x = αFx + βGx = αf (x) + βg(x).

# 7 7: 7 8

7

7 7 7 D

C 7 E 7 7 7 ' , 7

#"L " 1- y = f (x)

˜

∆ ∆ 0

f (x) ∆# . g(y) ! G(y)

˜

∆ ' g(f (x))f (x)

∆ !' &

	g(f (x))f (x)dx = G(f (x)) + C,	(7.4)
g(f (x))df (x) = G(f (x)) + C.		(7.5)

5 : D 7 7 ∆ <

7 F (x) = G(f (x))

Fx = Gy(f (x)) · fx = g(y)fx.

L : , G(y) Q g(y):

G(f (x)) Q g(f (x)) · fx

7 N1 4O N 7 N1 6OO

L 7 7 , ,

M 7

% 3 ) . $

("5 #!$" 1- NTO .

f (x)dx = F (x) + C,

f (ax + b)dx =

f (ax + b) ad(ax + b) = aF (ax + b) + C;

1

1

NTTO

f (x) dx =

f (x)

= ln |f (x)| + C;

f (x)

df (x)

f (x))α+1

NTTTO

(f (x))αf (x)dx = (f (x))αdf (x) =

(

+ C, α = −1.

α + 1

' , 7 7 7 7

F FN					f (x)O 7 7 <
7 : 7 7
(x + a)α		(x + a)α		(x 1−α−	, α = 1
dx	=	d(x + a)	=	ln |x 1+αa| + C, α = 1;
				+a)

x2 + a2 = a

1 + (x/a)2 =

dx

1

d(x/a)

√a2

x2

=

1 ( (x/a)2

dx

d x/a)

−

−

a1 edJVXxa + C = −a1 edJJVXxa + C. = edJWTYxa + C = −edJJkWxa + C.

C D D , 7

: <

7 7 7 N : 7

O C 7 D 7

f (x)dx.

7 7 7 7 8 x = g(t) #

	f (x)dx = x = g(t),	dx = g (t)dt	=	f (g(t))g (t)dt.

$ , : , 7 D

7 7 7

		dx = x = a sin t,	dx = a cos tdt	=	a2 cos2 tdt =
	a2 − x2