matan-1_2

% )$ &

C f 7 <

[x0 − δ, x0 + δ] L , 7 α = f (x0 − δ): β =

f (x0 + δ) C 7 [α, β] <

f −1: 7 :

, 7 y0 = f (x0) (α, β) ! 7 7 ∆y 7 7: , y0 + ∆y (α, β) L , 7 ∆x = f −1(y0 + ∆) − f −1(y0) 7

lim ∆x.

∆y→0 ∆y

) 7 7: , ∆y = 0: ∆x = 0: 7 <

, f −1(y0 + ∆y) = f −1(y0): , ,

7 f −1 C M 7 ∆y = 0 <

∆x 1 ∆y = ∆y .

∆x

C ∆y → 0: ∆x → 0: x = f −1(y) <

, x0 C M 7

(f −1)

= lim

∆x

=

lim

1

=

1

.

∆y

∆y

f

y0

∆y

→

0

∆x

→

0

∆x

x0

C$" 3 3 C 7 3 2 y = arcsin x y = arccos x

7 −1 < x < 1: y = edJVXx y = edJJVXx 7 , 7 ' 7

M D C y = arcsin x: x = sin y L

1

1

1

1

(arcsin x)x =

=

=

=

√

.

(sin y)

cos y

2

y

1 x2

) 7

cos y > 0

y

1 − sin

−

|y| < π/2

, 7 : ,

, , 7

(arccos x)x =

1

=

1

= −

1

= −√

1

;

(cos y)

sin y

1

2

1 x2

(edJVXx) =

1

= cos2 y =

=

1

;

−

y

−

1 − cos y

x

(VXy)y

1 + VX2y

1 + x2

(edJJVXx)

=

1

=

sin2 y =

1

=

−

1

;

(JVXy)y

−

−1 + JVX2y

1 + x2

x

% )$ &

(edWgx)

=

1

=

1

=

1

=

1

;

x

Wgy

Jgy

Wg

2

√

2

(edJgx)

=

(

1 )y

=

1

=

1y + 1

=

x

1+ 1

;

x

(

Jgy

)

Wgy

Jg

2

√

2

−

1

2

1− 1

x

1

1

y

y

(edVgx)

=

= Jg y =

=

;

(Vgy)

Vg2y

1

x2

x

1

−

−

1

y

1

1

(edJVgx)x =

= −Wg2y = −

=

.

(JVgy)

JVg2y

−

1

1

−

x2

y

% 7 7 D

, 7 <

C y = y(t) x = x(t) [t0 − δ, t0 + δ]: δ > 0: , 7 <

x = x(t) 7 #

7 α = x(t0 −δ) β = x(t0 + δ)

t = t(x): 7

C 7 : , yt					0
x	x	= 0 # y(x) = y(t(x))
t0: , 7	t0
7			y
		y =	t0	, x0 = x(t0).
			x
		x0
			t0

5 : 7 7

D 7 7

		yx0 = yt0 ·	tx		yt
				0 =	0	,
					x
					t0
tx	0 = 1/xt	0 7

5 7: 7 7 D 7

8

#"L " 3 4 !

' #

5 M 7 , <

7 M 7 D 7

% ! , && ( &$(

+ 5 0 9 >> ; 9 ? > ; ?

LC "5"("'$" 2- # , x0 X R

N O: ,

N O f : X → R:

Ox0 R : , f (x) ≤ f (x0) Nf (x) ≥ f (x0)O x Ox0 ∩X # , 7 7 7 7 < 7 7 , 7 &: ,

D Q &

C$" 2-

f (x) =

x2, −1 ≤ x ≤ 2, 4, x > 2

5 M x = −1, x = 2 Q ,<

7 7 7 ; x = 0 Q ,

7 7 7 ; x > 2 Q , M <

7 7 : 7 ,< 7 7 7 7 <

7 7 7 :

LC "5"("'$" 2 3 # , x0 X M 7 7 <

f : X → R 7 &:

, x0: 7 X

! 7 7 E 7 , M 7 7 < , 7 , x = −1 , 7 D M 7 7

#"L " 2- N 7 7 7 M 7 7 O

f : X → R

& x0# ) fx0 = 0

C	, x0 7 7
f (x)	−	f (x	) = (f	+ α(x))(x	−	x	),
		0	x0			0

α(x) Q , 7 x → x0 C <

x0 Q , 7 7 7 : ,

x Ox0 X

C fx0 = 0 # , 7 <

α(x) x → x0 Ox0 : , D

x Ox0 fx0 + α(x) 7 : , fx0

% ! , && ( &$(

) , : D x Ox0 ∩Ox0 \{x0} (fx0 + α(x))(x −x0)

7 7 x > x0 x < x0:

f (x) − f (x0) ≥ 0 C ,

) " '$" 2- # 7 7 D 7 <

M 7 7 7 : <

, 7 , 5 D M 7 7 N , x = −1 7 7 E 7 O

7: , fx0 = 0: :

) " '$" 2 3 7 , 7 , :

: , , M 7 7 <

7

#"L " 2 3 N 7 O f C[a, b]

(a, b) f (a) = f (b)# ) ξ (a, b)

fξ = 0

C f C[a, b]: 7 ! E <

, xm, xM [a, b]: D 7 7 7 <

7 7 , " f (xm) = f (xM ): f

[a, b] : : fξ = 0 ξ (a, b) " f (xm) < f (xM ): f (a) = f (b): ,

(a, b) " 7 , 7 , ξ C 7

7 fξ = 0

#"L " 2 2 N 7 , D ( O

f C[a, b] (a, b)# )

ξ (a, b) f (b) − f (a) = fξ (b − a).

7 7 7

F (x) = f (x)

−

f (b) − f (a)

(x

−

a).

b

−

a

F C[a, b] F 7 (a, b): , 7

F (b) = f (b)

−

f (b) − f (a)

(b

−

a) = f (a) = F (a).

b

−

a

C M 7 7 ξ (a, b) : ,

F

= f

−

f (b) − f (a)

= 0.

ξ

ξ

b

−

a

: ! 1 > 1 3 . 4

&

% ! , && ( &$(

5 7 7 7 , 7 8 <

,

f (b) − f (a)

b − a

M 7 AB Ox C fξ y = f (x) , (ξ, f (ξ)): 7

: AB

("5 #!$" 2- *

(a, b) y = f (x)

fx x (a, b) !' E

fx ≥ 0 f (x) 7 ;

f

= 0

f (x) ;

x

fx ≤ 0 f (x) 7 .

C x , x

(a, b): , 7 x < x C 7 (

x

(x , x ) f

=

f (x ) − f (x )

.

x

x

−

x

C fx

0

x

(a, b): f (x )

f (x ) x , x

(a, b):

:

M 7

≥

x < x

≥

f 7 , <

, fx = 0 fx ≤ 0

C 7 (a, b) f 7

#

x	(a, b) f	= lim			f (t) − f (x)			= lim ϕ(t).
	x	t	→	x	t	−	x	t	→	x

! 7 f ϕ(t) ≥ 0 t (a, b)

C M 7 lim ϕ(t) ≥ 0 7 <

t→x

D , ,

#"L " 2 4 N 7 % E , D O

x = x(t) y = y(t) 0 [α, β]

(α, β) x								= 0	t	(α, β)#
)							t
		y(β)	−	y(α)		y
τ	(α, β)				=	τ	.
		x(β)		x(α)
			−			x
						τ

% % = >

C 7 , : , x(α) = x(β) " M :

7 , t (α, β)

x(t) 7 7 7

F (t) = y(t) − y(α) − y(β) − y(α) (x(t) − x(a)). x(β) − x(α)

[α, β] 7 (α, β) % 7 : F (α) = F (β) = 0 C 7 , τ (α, β) : ,

Fτ = yτ − y(α) − y(β) xτ = 0. x(α) − x(β)

+, : , xτ = 0: , 7 7

% E

) " '$" 2 2 7 ( , D

, , 7 % E x(t) = t

+ + 9

LC "5"("'$" 4- C y = f (x) 7 , x0

n , C )

f

	f		f	(n)		n
f (x) = f (x0) +	x0	(x − x0) + · · · +		x0	(x − x0)		+ rn(x0; x),
	1!		n!

rn(x0; x) Q ) C x0 = 0 <

7 # , *

#"L " 4- . x x0 f

n

& ! n + 1 ! ϕ &

!' ! !

ξ ' x x0

rn(x0; x) = ϕ(x) − ϕ(x0) f (n+1)(x − ξ)n. ϕξ n! ξ

; 9 0 3 3 $ 4

&

8 : 2 ! $ 4 $

&

% % = >

' I 7 x, x0 7 7 7 <

F (t) = f (x) − f (t) +	f	f (n)
	1! (x − t) + · · · +	n! (x − t)n .
	t	t

F I 7 <

D , D: , 7

	Ft = − ft +	f			f (n+1)	(x − t)n +
		1! (x − t) + · · · +			n!
		t			t
ft +	f			f (n)		= −	f (n+1)	(x − t)n.
	1! (x − t) + · · · +			(n − 1)!(x − t)n−1			n!
	t			t			t

C 7 F (t) ϕ(t) I 7 % E

, D : , 7

F (x) − F (x0)				F
			=	ξ	,
				ϕ
ϕ(x)	−	ϕ(x0)
				ξ

ξ Q , 7 x x0 C

Fξ 7 , : ,

F (x) − F (x0) = 0 − F (x0) = −rn(x0; x),

, 7 7

("5 #!$" 4- C ϕ(t) = x − t: , 7

" %

rn(x0; x) = n1!fξ(n+1)(x − ξ)n(x − x0).

("5 #!$" 4 3 C 7 ϕ(t) = (x − t)n+1 , 7

1

1		fξ(n+1)(x − x0)n+1.
rn(x0; x) =
	(n + 1)!
(" 4- f (x) = Pn(x0; x) + o((x					x0)n) x		x0
Pn(x0; x) = c0 +c1(x−x0)+· · ·+cn(x−x0)n#			) &
				−		→

Pn ! #

x − x0

% % = >

! 7 7

c0 = lim f (x);

c1 = lim

f (x) − c0

; . . . ;

x→x0

x→x0

x − x0

cn = lim

f (x) − (c0 + c1(x − x0) + · · · + cn−1(x − x0)n−1)

.

x→x0

(x − x0)n

(" 4 3 ϕ : I → R 2I R 0

x0

x0 R

(n)

3

n ! ϕ(x

) = ϕ

= ϕ

=

· · ·

= ϕx =

0

x0

x0

0

0# ) ϕ(x) = o((x − x0)n) x → x0

5 7 7 7 7 7 ,

C n = 1 # 7 ϕ , x0 7 7

ϕ(x) − ϕ(x0) = ϕx0 (x − x0) + α(x)(x − x0),

α(x) Q , 7 x → x0 C ϕ(x0) = ϕx0 = 0:ϕ(x) = o(x − x0)

C 7: , n = k ≥ 1 5 < 7: , n = k + 1 C

	(k+1)			(k)		ϕx(k)
ϕ	x0		ϕ	x0	x0	= x→x0
		=				lim

(k)

− ϕx0 ,

ϕ(xk0+1) : , ϕ(xk)

I D , x0 +7 E : : I: 7 , : , ϕ(x), ϕx, . . . , ϕ(xk): k ≥ 1:

7 I 7 x0 C k ≥ 1: ϕx

7

(ϕx)x0 = · · · = ϕx(k) x0

= 0.

C ϕ	= o((x	−	x	)k) x	→	x	#
x			0			0
7 (

ϕ(x) = ϕ(x) − ϕ(x0) = ϕξ (x − x0) = α(ξ)(ξ − x0)k(x − x0),

ξ Q , : 7 x x0 N |ξ − x0| < |x − x0|O

x: , 7 α(ξ(x)) Q , 7 x → x0 C M 7

|ϕ(x)| ≤ |α(ξ)| · |x − x0|k · |x − x0|

% % = >

, 7: , ϕ(x) = o((x − x0)k+1)

#"L " 4 3 I 0 x0 R# .

f : I → R x0 n

!

	f		fx(n)		n	n
	x0	(x − x0) + · · · +	0	(x − x0)
f (x) = f (x0) +						+ o((x − x0)	)
	1!		n!

x → x0

C 7 , Pn(x0; x) 77 4-

7 7: 7 7 7

ϕ(x) = f (x) − f (x0) + fx0 (x − x0

1!

L, : ϕ(x0) = ϕx0 = · · · = ϕ(xn0) o((x − x0)n)

) + · · · + fx0 (x − x0)n . n!

= 0 C 77 4 3 ϕ(x) =

LC "5"("'$" 4 3 7 : 7 4 3: <

7 #

$ 7 , E 7 7 7 , <

8

f

f

(n)

f

(n+1)

n+1

x0

x0

n

ξ

f (x) = f (x0)+

(x−x0)+· · ·+

(x−x0)

+

(x−x0)

−

1!

n!

(n + 1)!

7 # , 7 , 7 7 ( :

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

x → x0 Q 7 # , 7 , 7 7 C

C , D , : <

7 , , M D 7

# L N 7 C O:

N 7 ( % E O C 7 <

E x → x0 ! 7 7 <

E D D

L D n E 7 7

# x0 = 0 D M 7 D

+ 44 ' 0 1$ 1

&

f0(n)

= cos πn2

% % = >

C$" 4- C f (x) = ex: f0(n) = 1 C M 7

	1		1		1			eξ
ex = 1 +		x +		x2	+ · · · +		xn +		xn+1.
	1!		2!			n!		(n + 1)!

C$" 4 3 C f (x) = sin x:

f0	= sin 2		=	(−1)m, n = 2m + 1,	m {0} N.
(n)		πn		0, n = 2m,
C M 7

	sin x = x	−	1	x3	+	1	x5	−· · ·	+	(−1)m	x2m+1
			3!			5!				(2m + 1)!

n = 2m + 1 n = 2m + 2.

C$" 4 2 C f (x) = cos x:

0, n = 2m + 1, = (−1)m, n = 2m,

+	sin(ξ + π2 (n + 1))		x	n+1	,
	(n + 1)!

m {0} N.

C M 7

cos x = 1

−

1

x2

+

1

x4

− · · ·

+

(−1)m

x2m +

cos(ξ + π2 (n + 1))

xn+1,

2!

4!

(2m)!

(n + 1)!

n = 2m n = 2m + 1.

C$" 4 4 C f (x) = ln(1+x): f0(n) = (−1)n−1(n−1)!. C M 7

ln(1 + x) = x

1

x +

1

x2

+

(−1)n−1

xn +

(−1)n

xn+1

.

−

2

3

− · · ·

n + 1 ·

(1 + ξ)n+1

n

C$" 4 6 C f (x) = (1 + x)α: α R # f0(n) = α(α − 1) . . . (α − n + 1) C M 7

(1 + x)α = 1 +

α

x +

α(α − 1)

x2

+

· · ·

+

α(α − 1) . . . (α − n + 1)

xn+

1!

2!

n!

α(α − 1) . . . (α − n + 1)(α − n)

·

xn+1

.

(1 + ξ)n+1−α

(n + 1)!

! D 7 D , , 7 ( <

C 7 , , 7 D C % E