Диференціальне числення ФОЗ
.pdf517.2
22.161.1
92
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( 1 1 24.10.2013).
1. .
– , " , * " -
+ " ( ' ( . «&
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" * " ! ’"
. * ! ! 0. .*' . 2 (! XVII . XIX . " ). 3, . & (, 3 . ! !4 " *.
5 * " " * " " ", * " " " -
. $" * , " " * * " ". 1. 6 .
$" " ' L ( . 1) * ( ( MN . % " MN , " * , ' * . #
, N , * + L , ! + ' *
M . & * MN " ".
$ . 1.
+ " MT MN , * " L
M . |
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$" " y = f (x ) |
* * - |
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M (x, f ( x)) ( .2). . ' x |
x - |
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N (x + x, f ( x + x)) . 7 " y = f (x ) , |
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y = f ( x + |
x ) − f (x ) . % M N MN . NMK , |
" ϕ – ( " -
3
NKM , |
K , ( x + x, f ( x)) . 6 - |
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3 * " ,: |
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tg ϕ = |
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MK |
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. ( |
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x → 0 . # N , ' * + , "- |
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, M , MN , ' * M , * |
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MT . ϕ * " , " + " α . |
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# ! + : |
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α = lim ϕ . |
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x→0 |
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#, * " |
y = tg x |
x [0, π 2), ,, |
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( , ',: |
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tg α = tg (lim ϕ) = lim tg ϕ = lim |
y |
= lim |
f ( x + |
x ) − f ( x ) |
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x→0 |
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$ . 2.
) ' ' * y = f (x ) x . .+-
!, * " * ! 3 .
2.6 , 3 * .
. ( * M , * " + " "
( . 3).
4
$ . 3.
% x (t ) – M t . .(
+ t (3 * |
x = x (t + t ) − x (t ) . # " |
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3 * " + |
t ',: |
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v = |
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. $ " t M + " : , , 3, *,
( " , 3 (!
! , * " * "). # "
3 * , * ' ' . # -
, + " 3 " " * ,! * " ( , ! . 8 +
(? 8 ' " * 3 v " - + t ", !:
v (t ) = lim v |
= lim |
x |
= lim |
x (t + |
t ) − x (t ) |
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t →0 |
t →0 |
t |
t →0 |
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8 !, ' * (3 ", (
, ! + . 3. 6 + ".
$" ( " ( ( + * + l ( . 4) -
( Ox , ! ( ( * ! " -
. + + * , ! ( , ',
', * " .
$ . 4.
5
% m ( x ) |
+ ", 3 + |
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' ' x . |
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x + |
x . # + " + x x + |
x : |
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m = m ( x + x ) − m ( x ) . |
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ρc + " [ x, x + |
x] ' * - |
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3 ": |
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ρc |
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m |
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x |
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+ " x ' * ' |
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ρ = lim ρc |
= lim |
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m |
= lim |
m ( x + |
x ) − m ( x ) |
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x→ 0 |
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! (3 , + . |
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4. 6 3 * . |
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. ( N = N (t ) |
– * * , , ' - |
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t . 6 + + ' |
t " * * " - |
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', |
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N (t + |
t ). |
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+ |
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[t,t + |
t ] , * " 3 ": |
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ν c |
= |
N |
= |
N (t + t ) − N (t ) |
. |
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t |
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" * 3 t → 0 ,
t : |
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ν (t ) = limν |
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= lim |
N |
= lim |
N (t + t ) − N (t ) |
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c |
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t → 0 |
t → 0 |
t |
t → 0 |
t |
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2 , , . 5. 6 * !.
. ( V (t ) – ! " ! " -
t . 6 + |
t ( ! " ', * " * V (t + t ) . %- |
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! " ! ', |
V = V (t + |
t ) − V (t ). |
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! , * " 3 ": |
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Ic = |
V |
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t |
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! " t |
( * " " " - |
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t → 0 , !: |
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I (t ) = lim I |
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= lim |
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V |
= lim |
V (t + |
t ) − V (t ) |
. |
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c |
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t→ 0 |
t → 0 |
t → 0 |
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0 + .
) + * – , , , . 0 ( " " – . 0 ,
, " " * + " ". 5 ! ' , *-
6
" * : ,' ' *
+ * * ' ( ' + ' '. *, , " , . 8 + ( *- ( %, « –
( + ( "».
2..
+ ( "
! .
. ( " + X ' y = f ( x ) . &- * * x X x x , !
x + x + + + X . # " y = f ( x ) ,
y = f ( x + x ) − f ( x ) .
. # y = f ( x) x , * " "
3 " y ( x , -
" , ". % , * " :
y′; y& ; |
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dy |
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f ′( x ); |
f& (x ); |
df |
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dx |
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# , ": |
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f ′(x ) = lim |
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= lim |
f ( x + x ) − f ( x ) |
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x→ 0 |
x→ 0 |
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' * ", ’" 1–5, " , + :
1. ( , y = f (x ) x ',
( f ′(x ) ( : tg ϕ = f ′(x ).
* " , ( .
2. / , 3 * t ', ( -
( : v (t ) = x′(t ) = x& (t ) .
* " , ( .
3 ' * ': x& (t ).
3.2 ( + " ' x ', (
m ( x ) + ", , + [0, x]:
7
ρ(x ) = m′(x ) = dm ( x ) .
dx
4. 9 * t ', ( *-
N (t ) ( :
ν (t ) = N ′(t ) = dN (t ) .
dt
5.0 * ! t ', ( ! "-
V (t ) ! ( :
I (t ) = V ′(t ) = dV (t ) . dt
$" .
1. 6 ( y = C ( ). / , " * x :
f ′(x ) = C′ = lim |
f (x + |
x ) − f (x ) |
= lim |
C − C |
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x→ 0 |
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x→ 0 |
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# ! . 2. 6 ( y = x .
/ , " * x :
= lim 0 = 0 .
x→ 0
f ′(x ) = x′ = lim |
f (x + |
x ) − f ( x ) |
= lim |
x + |
x − x |
= lim 1 = 1. |
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x→ 0 |
x→ 0 |
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3. 6 ( y = x2 . 6 ( " ,
x = 3 .
/ , " * x :
(x2 )′ = lim |
(x + x )2 − x2 |
= lim |
x2 + 2x x + x2 − x2 |
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x→ 0 |
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x→ 0 |
x |
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= lim |
(2x + |
x) x |
= lim (2x + |
x ) = 2x . |
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x→ 0 |
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x→ 0 |
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6, " x = 3 , |
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(x2 )′ |
= 2 3 = 6 . |
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x=3
4.6 ( y = sin x .
/ , " * x :
(sin x)′ = lim |
sin ( x + |
x) − sin x |
= (sin x)′ = lim |
sin ( x + |
x) − sin x |
= |
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x→ 0 |
x→ 0 |
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8
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sin |
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( 3 + |
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= lim |
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lim cos x + |
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= 1 cos x = cos x |
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x→ 0 |
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x→ 0 |
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2 |
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2
y = cos x ). 5. 6 ( y = ln x . / , " * x :
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ln ( x + x ) − ln x |
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ln |
x + |
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(ln x )′ = lim |
= lim |
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ln 1 |
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ln 1 |
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= lim |
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lim |
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1 = |
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x→ 0 |
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x x x→ 0 |
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( * * – . «& »).
8 ": ', y = ln 5 ? " + , ' * , -
' *, ', 1 . " + , ?
5
3. .
6 ,' ! " ! ' ' - " * " " " ! – .
. 8 " y = f (x) x0 , ,
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lim |
y |
= lim |
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f (x0 + x) − f ( x0 ) |
, |
x |
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x→ − 0 |
x→ − |
0 |
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" " , * " y = f (x) x0 , -
, * " " f−′(x0 ) .
8 " y = f (x) x0 , , "
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y |
= lim |
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f (x0 |
+ x) − f (x0 ) |
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lim |
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x→ + 0 |
x→ + |
0 |
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" " , * " y = f (x) x0 ,
, * " " |
f ′ |
(x ) . |
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+ |
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# , ": |
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f ′(x ) = |
lim |
y |
, f |
′(x ) = |
lim |
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− 0 |
x→ − 0 |
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+ |
0 |
x→ + |
0 |
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9 |
% |
f ′( x0 ) |
y = f ( x) |
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' * ! |
f ′( x ) , |
f ′( x ) |
' *. % *: |
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+ |
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f |
′( x ) = f |
′( x ) = f ′( x ) . |
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# 1. 6 ( y =| x | x0 = 0 . |
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/ ,: |
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f |
′(0) = lim |
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= lim |
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= lim |
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= −1 , |
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− |
x→ − |
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x→ − 0 |
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x→ − |
0 |
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f |
′(0) = lim |
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= lim |
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= lim |
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= 1. |
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+ |
x→ + |
0 |
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x→ + 0 |
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0 |
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6 +, * |
f ′( x ) ≠ |
f |
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x0 = 0 ,.
$" " " . .( " y = f ( x)
x0 , (:
lim |
y |
= lim |
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f ( x0 |
+ x) − f ( x0 ) |
= ∞ . |
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x→ + |
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# + *, " y = f ( x) , x0 . |
% " |
x = x0 * , * " y = |
f ( x) |
M ( x0 , f ( x0 )) .
8 lim |
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y |
= + ∞ , + *, " y = f ( x) |
, x |
- |
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x→ |
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, |
" |
', |
+ ∞ . |
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lim |
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x→ − 0 |
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lim |
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' * ' ' ' y = f ( x) |
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x→ + |
0 |
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x |
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' * |
+ |
" |
f ′( x ) |
f |
′( x ) . |
# , |
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0 |
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− |
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f ′( x ) = + ∞ , |
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f ′( x ) = + ∞ , |
f ′( x ) = + ∞ |
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. &, |
" |
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0 |
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− 0 |
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+ |
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f ′( x ) = lim |
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f ′( x ) = − ∞ , f |
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# 2. $" ' y = 3 x . 6 (:
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0 + |
x − 3 0 |
3 |
x |
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f ′(0) = lim |
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= lim |
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= lim |
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= + ∞ . |
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x→ 0 3 |
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& (0;0) ' , " " x = 0 ( . 5).
10
$ . 5. |
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f ′( x0 ) = + ∞ , ! f ′( x0 ) = − ∞ , + *, " |
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y = f ( x) , x0 $ . |
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$" , lim |
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f ′( x ) = − ∞ , |
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′( x ) = + ∞ . # + *, lim |
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. . , " , , " f |
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′( x ) = + ∞ . # * ,, , " y = |
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x0 = 0 ( . 6). (: |
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f ′(0) = lim |
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= + ∞ , |
f ′(0) = lim |
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= − ∞ . |
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$ . 6.
4. , ’
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& ".
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7 " |
y = f (x ) , * " x , |
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y = f ( x + |
x ) − f (x ) + ! ": |
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11 |