matan-1_2

! 5 - &

7 y = 1/x y = ln x: 7 7 7

, 0

C$" = 3 y = sin x2: ,

R: 7 R: , D xn = π2 (n + 1) xn = π2 n 7 7 |f (xn) − f (xn)| = 1: 7

lim |xn − xn| = 0

n→∞

C M 7

7 7 7 <

[a, b] C / < ( 7 ,

{Oxδ11/2, . . . , Oxδnn/2} 7 7 δ = 12 min{δ1, δ2, . . . , δn} C 7: ,x , x [a, b] D: , |x − x | < δ |f (x ) − f (x )| < ε

5 : 7 {Oxδ11/2, . . . , Oxδnn/2} [a, b]: M 7 xi : , |xi − x | < δ/2 L

|x − xi| ≤ |x − x | + |x − xi| < δ + δi/2 ≤ δi/2 + δi/2 = δi.

: x , x Oxδii : , :

|f (x ) − f (x )| ≤ |f (x ) − f (xi)| + |f (xi) − f (x )| < ε/2 + ε/2 = ε.

5 ) ' 9 > ; A

9 > ;

, 7 7 0 3 7 7 ,

D L,

("5 #!$" .- N 7 0 3O *

#

("5 #!$" . 3 N 7 0 3O *

#

C y = f (x) Q 7 <

C 7: , 7 7 D

! 5 - &

5 : 7 0 3 D

: f (x − 0) = f (x + 0): f (x) = f (x − 0) = f (x + 0) !

.-: 7 7 7

# X Q 7 , dom f :

D f (x) % , x X 7 <

, r(x) : , f (x − 0) < r(x) < f (x + 0) : ,

r : X → Q I : r(x ) = r(x ):

x < x : 7 0 3 f (x + 0) ≤ f (x − 0)

# 7 7: 7 ,

7 X 7 7 7 7 Q: :

7 7: ,

#"L " .- C y = f (x) [a, b]

f [[a, b]]

f (a) f (b)

C 7 f C[a, b] ! 7

, 7 f (a) f (b) !

=- 7 , 7 f (a) f (b) # 7

7: f [[a, b]] Q 7 f (a) f (b)

C f Q 7 [a, b] : c [a, b]

Q , # 7 0 3 7 <

(f (c − 0), f (c)) (f (c), f (c + 0))

, f ' 7 M <

7 f (a) f (b): M 7 7 [a, b] 7 D , :

7 f (a) f (b) 7 , im f

(" .- ( X R

y = f (x) f [X]

x = f −1(y) !'

y = f (x)

L f : X → f [X] I C 7: ,

I 5 : (f (x ) = f (x )) (x = x ) <

7 D x , x X C f :

X → f [X] :

f −1 : f [X] → X $7 D 7 x = f −1(y) C f 7 :

! 5 - &

x , x X (x < x f (x ) < f (x )).

C 7 f −1: 7 7

y , y f [X] (f −1(y ) < f −1(y ) y < y ).

, 7 7 <

,

#"L " . 3 (

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

x = f −1(y) !

' #

! 77 7 [a, b] y = f (x)

f [[a, b]] x = f −1(y): 7

D 7 : , f ! 7 <

y = f (x) 7 .- : , f [[a, b]] Q

7 f (a) f (b) C f [[a, b]] = [f (a), f (b)] C f −1[[f (a), f (b)]] = f −1 ◦ f [[a, b]] = [a, b]:

7 x = f −1(y) 7 .- M

7 f (a), f (b)

C$" .- y = sin x <

[−π/2, π/2] ) , : 7 . 3

x = arcsin y [sin(−π/2), sin(π/2)] = [−1, 1]

C$" . 3 , 7 y = cos x [0, π] 7 1 −1 C < M 7 [−1, 1]

7 π 0 x = arccos y

C$" . 2 C 7 7 7 : <

7 −π/2 π/2 x = edJVXy:

7 π 0 x = edJJVXy

C$" . 4 y = Wgx 7

, ! 7 . 3

7

x = edWgy

! 5 - &

C D y = Wgx , <

: , 7 : ,

x = edWgy , 7 5 :

y = Wgx = ex − e−x .

2

L , 7 e2x − 2yex − 1 = 0 C <

C$" . 6 , 7 y = Jgx , D (−∞, 0) (0, +∞): 7

! "

. ) -

# # /

+ ( 1 > ; & 9

LC "5"("'$" - - y = f (x)

x dom f : 7 dom f : ,

∆x = t − x: ∆f = f (x + ∆x) − f (x); ∆x N, F FO ' : ∆f Q ' : 7 , x: x; : <

7 x X: 7 7 X;

: 7 x dom f :

C$" - - y = ex Q 7 5 <

C$" -3 y = ln |x| Q 7 5 <

:

M D

C$" -2 y = sin x 7 5 <

:

7 ,

C$" -4 C y = c 7 8

lim c − c = 0.

t→x t − x

LC "5"("'$" -3 7 <

! !

! !

$

#"L " - - C f x

f x

' ! &

fx = fx− = fx+.

+ 7 7 7 7 <

#"L " -3 D

& #

7 , t < 0: <

7 t > 0 $ 7 - - : ,

y = |x| 7

, x = 0: D

: , <

, <

7

C$" -0 7 7 : <

, M

, C 7 , <

y: <

7 7 y = f (t) # f˙(t0) M

N , D , 7 <

tO , t0 7 7 7 7 7 7

t0

' E 7 AB8

y = f (x0 + ∆x) − f (x0)(x − x0) + f (x0). ∆x

7 , 7 <

: , 7 ∆x → 0 <

<

: , x0

LC "5"("'$" -2 C y = f (x) <

7 , x0: 7 y = fx0(x − x0) + f (x0) 7

y = f (x) , (x0, f (x0))

# 7 7: ,

7: , , 7

M , <

7 7 Ox ! , :

% )$ &

7 , 7 Ox <

7 x = x0 # 7 7

N 8 O f

, x0

+ . 1 >9 & ' ; 1 A

9' ; > ; 1 > A ;

#"L " 3- y = f (x) y = g(x)

x & # )

cf f +g f ·g f /g 2 g(x) = 03

x

lim g(t) = g(x) g ,

% )$ &

, g , x

(" 3- ' lim f (t) = y# ) f (t) = y +

t→x

α(t) α(t) 0 t → x

C 7 α(t) = f (t) − y #

! 7 X, Y Q 7 R

#"L " 3 3 f : X → Y

x X g : Y → R

y = f (x) Y# ) g ◦ f : X → R

x X

(g ◦ f )x = gy · fx.

$ 7 f , x X : , f (t) − f (x) = (t − x)(fx + α(t)).

C g 7 , y = f (x) Y:

g(s) − g(y) = (s − y)(gy + β(s)).

#

(g ◦ f )(t) − (g ◦ f )(x) = g(f (t)) − g(f (x)) =

(f (t) − f (x))(gy + β(f (t))) = (t − x)(fx + α(t))(gy + β(f (t))).

L

(g ◦ f )(t) − (g ◦ f )(x) t − x

gy · fx + α(t)gy + β(f (t))fx + α(t)β(f (t)).

! , 7

% )$ &

C lim f (t) = y:

t→x

#"L " 3 2 y = f (x)

[x0−δ, x0 +δ] fx0 = 0# )

x = f −1(y) y = f (x)

y0 = f (x0)

(f −1)y0 = f1 .

x0