Диференціальне числення ФОЗ

M .

$" " y = f (x )

* * -

M (x, f ( x)) ( .2). . ' x

x -

N (x + x, f ( x + x)) . 7 " y = f (x ) ,

y = f ( x +

x ) − f (x ) . % M N MN . NMK ,

NKM ,

K , ( x + x, f ( x)) . 6 -

3 * " ,:

tg ϕ =

=

y

.

NK

MK

x

. (

x → 0 . # N , ' * + , "-

, M , MN , ' * M , *

MT . ϕ * " , " + " α .

# ! + :

α = lim ϕ .

x→0

#, * "

y = tg x

x [0, π 2), ,,

( , ',:

tg α = tg (lim ϕ) = lim tg ϕ = lim

y

= lim

f ( x +

x ) − f ( x )

.

x→0

x→0

x→0

x

x→0

x

+ t (3 *

x = x (t + t ) − x (t ) . # "

3 * " +

t ',:

v =

x

.

t

v (t ) = lim v

= lim

x

= lim

x (t +

t ) − x (t )

.

t →0

t →0

t

t →0

t

% m ( x )

+ ", 3 +

' ' x .

$" '

x +

x . # + " + x x +

x :

m = m ( x + x ) − m ( x ) .

ρc + " [ x, x +

x] ' * -

3 ":

ρc

=

m

.

x

+ " x ' * '

ρ = lim ρc

= lim

m

= lim

m ( x +

x ) − m ( x )

,

x→ 0

x→ 0

x

x→ 0

x

! (3 , + .

4. 6 3 * .

. ( N = N (t )

– * * , , ' -

t . 6 + + '

t " * * " -

',

N (t +

t ).

+

[t,t +

t ] , * " 3 ":

ν c

=

N

=

N (t + t ) − N (t )

.

t

t

t :

ν (t ) = limν

= lim

N

= lim

N (t + t ) − N (t )

.

c

t → 0

t → 0

t

t → 0

t

t . 6 +

t ( ! " ', * " * V (t + t ) . %-

! " ! ',

V = V (t +

t ) − V (t ).

! , * " 3 ":

Ic =

V

.

t

! " t

( * " " " -

*

t → 0 , !:

I (t ) = lim I

= lim

V

= lim

V (t +

t ) − V (t )

.

c

t

t

t→ 0

t → 0

t → 0

y′; y& ;

dy

;

f ′( x );

f& (x );

df

.

dx

dx

# , ":

f ′(x ) = lim

y

= lim

f ( x + x ) − f ( x )

.

x

x→ 0

x→ 0

x

sin

x

x

2

( 3 +

= lim

lim cos x +

= 1 cos x = cos x

x

x→ 0

x→ 0

2

ln ( x + x ) − ln x

ln

x +

x

(ln x )′ = lim

= lim

x

=

x

x→ 0

x

x→ 0

+

x

+

x

ln 1

ln 1

= lim

x

1

=

1

lim

x

=

1

1 =

1

x

x

x→ 0

x x x→ 0

x

x

x

x

lim

y

= lim

f (x0 + x) − f ( x0 )

,

x

x→ − 0

x→ −

0

x

y

= lim

f (x0

+ x) − f (x0 )

lim

,

x

x→ + 0

x→ +

0

x

, * " "

f ′

(x ) .

+

0

# , ":

f ′(x ) =

lim

y

, f

′(x ) =

lim

y

.

− 0

x→ − 0

x

+

0

x→ +

0

x

9

%

f ′( x0 )

y = f ( x)

x0

, * ,

' * !

f ′( x ) ,

f ′( x )

' *. % *:

−

0

+

0

f

′( x ) = f

′( x ) = f ′( x ) .

−

0

+

0

0

# 1. 6 ( y =| x | x0 = 0 .

/ ,:

f

′(0) = lim

| 0 + x |−| 0 |

= lim

|

x |

= lim

−

x

= −1 ,

−

x→ −

0

x

x→ − 0

x

x→ −

0

x

f

′(0) = lim

| 0 + x |−| 0 |

= lim

|

x |

= lim

x

= 1.

+

x→ +

0

x

x→ + 0

x

x→ −

0

x

6 +, *

f ′( x ) ≠

f

′( x ) , y =| x |

-

−

0

+

0

lim

y

= lim

f ( x0

+ x) − f ( x0 )

= ∞ .

x

x

x→ 0

x→ +

0

# + *, " y = f ( x) , x0 .

% "

x = x0 * , * " y =

f ( x)

8 lim

y

= + ∞ , + *, " y = f ( x)

, x

-

x→

0

x

0

,

"

',

+ ∞ .

*

!

lim

y

x→ − 0

x

lim

y

' * ' ' ' y = f ( x)

x

x→ +

0

x

' *

+

"

f ′( x )

f

′( x ) .

# ,

"

0

−

0

+

0

f ′( x ) = + ∞ ,

f ′( x ) = + ∞ ,

f ′( x ) = + ∞

. &,

"

0

− 0

+

0

f ′( x ) = lim

y

= − ∞ ,

f ′( x ) = − ∞ , f

′( x ) = − ∞ .

0

x→ 0

x

−

0

+

0

3

0 +

x − 3 0

3

x

1

f ′(0) = lim

= lim

= lim

= + ∞ .

x

x

x2

x→ 0

x→ 0

x→ 0 3

$ . 5.

,

f ′( x0 ) = + ∞ , ! f ′( x0 ) = − ∞ , + *, "

y = f ( x) , x0 $ .

$" , lim

y

= ∞ , +

x→ 0

x

f ′( x ) = − ∞ ,

f

′( x ) = + ∞ . # + *, lim

y

, '

0

0

x→ 0

x

. . , " , , " f

′( x ) = − ∞ ,

−

0

′( x ) = + ∞ . # * ,, , " y =

f

+

| x |

0

x0 = 0 ( . 6). (:

f ′(0) = lim

| x |

= + ∞ ,

f ′(0) = lim

| x |

= − ∞ .

+

x→ +

0

x

−

x→ −

0

x

.

7 "

y = f (x ) , * " x ,

"

y = f ( x +

x ) − f (x ) + ! ":

11