attachments_22-12-2011_21-07-09 / semestr1_2
.pdf-
:
•;
•;
•;
•.
-
.
! " , " [3], ' 1, . 15-25 ' – +.
, .
, -
. -
-
.
. + X Y , , x y -
. ! ' +-
x X |
y Y , '- |
|||
, + X f - |
||||
|
|
|
|
y = f ( x), x X . |
. + X "# ( ) / 0 |
||||
f ( x) " |
D( f ) ; x ' / 0. .+ |
|||
f ( X ) |
y Y , , x X , |
|||
|
|
& / 0. |
||
! + |
f ( X ) + - , C ( . . |
|||
x X |
+ C ), / 0 - |
|||
, , - |
||||
|
|
|
|
y = f ( x) = C = const . |
' |
|
y = f (x) ' - |
||
Oxy |
+ Oxy |
(x, f ( x)) , x X .
( :
116
1. : ": / 0
/ . , / 0
F (x, y) = 0, ', / 0 y(x) .
! " " -, ', + '-, / .
2.< / ": ' / / 0.
3.= " ": / 0 " 0, -
' / 0.
* .
1. > " / 0 |
y = |
|
2 |
|
. |
||
|
|
|
|
||||
x2 |
4 |
||||||
|
|
|
|
|
+ ,. + + " 0,
+ . = ", "
: x2 4 > 0 . @ x ( ; 2) (2;+ ) .
: D( f ) = {( ; 2) (2;+ )} .
2. > " / 0 y = 3x + log2 (x + 1) .
+ ,. ' -
, ' / + . C, " -
x + 1 > 0, x ( 1;+ ) .
: D( f ) = ( 1;+ ).
-
1. D / 0
. E 0 y = f (x) " X
&., " ' x X f ( x) = f (x) .
E 0 y = f (x) " X &., -
" ' x X f ( x) = f (x) .
E 0, , ,
/ 0 / 0 . F , :
1.@" G G / 0 -
.
2.< / G / 0 .
3.< / G / 0 -
.
* . |
f (x) = x4 2x2 , . . ( x)4 = x4 ,( x)2 = x2 , |
|||
1. E 0 |
||||
f ( x) = f (x) . |
|
|||
2. E 0 |
f (x) = |
x3 2x |
. H, |
|
x2 |
||||
|
|
|
117
f ( x) = |
( x)3 2( x) |
= |
x3 + 2x |
= |
x3 2x |
= f (x). |
|
( x)2 |
x2 |
x2 |
|||||
|
|
|
|
3.E 0 f (x) = 5x + 3 / 0 " ' , -
x x ', , "
, , / 0.
2./ 0
. E 0 y = f (x) " |
X |
||
&, T > 0 , : |
|
|
|
1) |
x X , x T X x + T X ; |
|
|
2) |
" ' x X f (x) = f (x + T ) ; |
|
|
3) |
T - . J - T - |
||
/ 0 y = f (x) . |
|
|
|
* . |
|
|
|
E 0 f (x) = 3sin 2x + 1 T = , |
|
||
' , / 0 y = sin x |
2 , |
- |
sin 2x .
3.@' / 0
. E 0 y = f (x) " X
0 & + X0 , ( X0 X ), -
M , " ' x X0 |
f (x) M . E 0 |
y = f (x) 0 & + |
X0 , |
m , " ' x X0 f (x) m .
E 0 y = f (x) 0 & + X0 , -
L > 0 , " ' x X0 | f (x) | L
.
@, / 0 ' ' ', ' '-
' .
@ +, ' / 0 +
' / 0.
* .
E 0 |
f (x) = |
1 |
|
' |
. = |
|||||
x2 +1 |
||||||||||
|
|
1 |
|
|
|
|
|
|||
x2 0, 0 |
|
|
1, |
, '. |
||||||
x2 |
+ 1 |
|||||||||
|
|
|
|
|
|
|
||||
4. . / 0 |
|
|||||||||
. |
E 0 y = f (x) #/ - |
|||||||||
+ |
X , |
" x1, x2 X |
x1 < x 2 , |
f (x1 ) < f (x 2 ) , "- ' "- -
/ 0.
118
E 0 |
y = f (x) |
#/ |
X , " |
|||
x1, x2 X x1 < x2 , |
f (x1 ) > f (x2 ) , |
"- - |
||||
' - / 0. |
||||||
! / 0 y = f (x) X " X , - |
||||||
0 |
X . |
|
|
|
|
X , |
E 0 #/ + |
||||||
x1 < x2 , |
f (x1 ) f (x2 ) , " x1, x2 X , |
#/ |
||||
+ X , |
|
x1 < x2 , |
f (x1 ) f (x2 ) , " |
|||
x1, x2 X . |
|
|
|
X |
||
, ", " |
||||||
/ 0 / 0, |
X . |
|
|
|||
* . |
|
|
|
|
|
|
E 0 y = x3 . H, -x1 < x2 , x13 < x23 .
119
, , 1 |
. |
|
||
|
|
|
|
|
/ 0 y = f ( x) , X - + , |
||||
Y0 = f ( X ) - + . , / 0 " |
||||
X ( . 4.1 / 0). = ' + - |
||||
y Y0 " - |
x X , . . |
|
||
+ Y0 |
/- |
y |
|
|
0 x = ( y) . |
J / 0 - |
|
|
|
|
|
|
||
/ 0 y = f ( x) , |
Y0 y |
|
|
|
/ 0 - "- |
|
|
||
. H " / 0 Y0 - " - |
|
|
|
|
+ , X - + - |
|
x |
|
|
. ! / 0 , |
O |
x |
|
|
/ x = ( y) |
|
X |
|
|
- - y = f ( x) - |
M. 4.1 |
|
||
x . |
|
|
|
|
" " ' / 0, "
- " / 0 y = ( x) . F -'
, ' / " / 0 y = f ( x) y = ( x)
" ' ' .
! / 0 y = f ( x) " + X ,
' y Y0 , " '
x X . , " / 0 / 0 y = f ( x) . > ' " + X0 +-
X , y = f ( x) + ". M /-
0 y = f ( x) , |
x X0 , / 0 x = ( y) , " . |
|
* . |
y = x3 , |
|
E 0 |
( . . X = ), - |
|
+ Y0 |
= . C, , / 0 " / 0 |
y = 3x , X = .
(
M /:
y = f (u), u U , u = ( x), x X .
120
/ / 0 y = f (u) + U ,
– / 0 u = ( x) + X . @" , /, -
, # #, #
.
. + X * + / 0
x X , u = ( x) - + + U / 0 f (u) .
|
u & 0 + /- |
||||||||||
0, |
x , + X * , |
||||||||||
|
. |
|
|
|
|
|
|
|
|
||
|
* |
. |
/ 0: / 0 y = f (u) = lg u, |
||||||||
|
M |
||||||||||
+ U = (0, + ) , |
/ 0 u = ( x) = 1 x2 , +- |
||||||||||
X = |
|
. , |
|
( |
X |
) |
= |
( |
,1 |
||
|
|
|
|
] . |
|||||||
. + |
( X ) |
U " , ( |
+), ' / +
/ 0 + X * x , u = 1 x2 > 0 . M- -
' " 1 < x < 1. F, / -
+ / 0
y= lg (1 x2 ), ' x X * = ( 1,1)
@ , / 0, , -, " ' .
O , ' " : +, -
, +, 0 . 3 0 & / 0 / 0,
' ' "
' .
P ' " / 0:
1)0 0 / 0 (' " '); -
/ 0
y = a0 xn + a1xn 1 + ... + an 1x + an ,
' ak - , n - 0 0 ;
2)" 0 / 0 – 0 - 0 / 0;
3)0 / 0; ' " /- 0, ' -
. / 0:
121
|
|
|
|
|
|
|
2x + 1 |
|
|
y = |
3x |
2 |
+ 2, |
y = |
. |
||||
|
5x + 3 |
Q " / 0, ' ",
/ 0. - 0 / 0 -
:
1)/ 0 y = xa , ' a - 0 x > 0 ;
2)/ 0 y = ax (a > 0,a 1) ;
3)' / / 0 y = loga x(a > 0,a 1);
4) |
' / 0 y = sin x , y = cos x , y = tgx '; |
5) |
" ' / 0 y = arcsin x , y = arccos x |
'.
O + , / 0 ' "
- 0 / 0, + / 0,
/ (+, ,
+ ) 0 / 0 / 0,
. >, ,
/ 0
|
( |
|
|
) |
1 x2 |
|
|
|
|
|
|
y = lg |
|
x + |
1 + x2 |
|
, y = arctg |
2x4 |
, y = tg |
|
|
|
. |
|
|
|
x |
x |
. " , / 0 / 0, -
+ " /,
+ /,
/ . =, , / 0
x + 2, x 0, |
|
|
|
|
|
y = |
|
|
|
|
|
x2 , x > 0 |
|
|
|
|
|
, . |
|
|
|
|
|
+ , & |
|
|
|
|
|
* 1. > " / 0 |
y = |
|
3 |
. |
|
2 |
x |
||||
|
|
|
+ ,: + + , . . x 2 , , , " / 0 +
x x = 2 .
* 2. > " / 0 y = |
x 2 |
+ |
5 x |
. |
|||
+ ,: O + + + " 0- |
|||||||
, . . |
+ |
x 2 0 |
|||||
5 x 0 , |
, , + " x 2 x 5 . , |
||||||
, |
x |
2;5 . |
|
|
|
|
|
|
[ ] |
|
|
|
|
122
* 3. , " / 0 y = x3 + sin x -
.
+ ,: / 0 x x ,-
0. F ' , y = sin x - / 0 -
, ,
y( x) = ( x)3 + sin( x) = x3 sin x = (x3 + sin x) = y(x),
, , / 0 .
* 4. , " / 0 y = x2 + cos 2x -
.
+ ,: / 0 x x ,-
0. F ' , / 0 y = cos x -
, ,
y( x) = ( x)2 + cos 2( x) = x2 + cos 2x = y(x) ,
, , / 0 . |
|
3x |
|
|
||
* 5. , " / 0 y = |
|
|
. |
|||
x + |
5 |
|||||
+ ,: E 0 y |
|
|
||||
|
x = 5 . C, - |
|||||
+ y( x) |
x = 5 . @ |
x = 5 . , |
/ 0 + " , , . . / 0 – / 0 " ' .
* 6. , " / 0 y = x2 + 3x + 7 -
.
+ ,: / x x : y( x) = ( x)2 + 3( x) + 7 = x2 3x + 7 .
=. . y( x) = y(x) y( x) = y(x) .
= ", / 0 , , . . , / 0 " ' .
* 7. , / 0 y = 3x2 + 7 '
, ' .
+ ,: 3x2 + 7 7 , , ' . = x2 + ' "- , / 0 '-
.
* 8. , / 0 y = 3x3 + 2x2 |
+ |
||||
[0;+ ). |
|
|
|
|
|
+ ,: 0 x1 < x2 . = ' x13 < x23 |
x12 < x22 . C, |
||||
y(x ) = 3x2 |
+ 2x2 |
< 3x3 + 2x2 = y(x ), |
|||
1 |
1 |
1 |
2 |
2 |
2 |
. . / 0 +.
123
* 9. , / 0 y = x2 3x " +
( ;0] .
+ ,: x1 < x2 0 . H 0 x x1 < x2 -
, " , ' "
x12 > x22 . T + " x1 < x2 0 ,
3x1 > 3x2 . P , "
x12 3x1 > x22 3x2 , , , y(x) " - +.
; 4.1
1.H / 0 " / 0.
2.H ' / / 0.
3.H G / 0, G
/ 0, / 0 " ' . T +, "-
' /.
4.H / 0. .
5.P/ / 0, ' - +. .
6.P/, / 0 .
7.H " / 0. T + G -
.
8.O " ' / 0 ? T + " + .
9., / 0 ,.
10.H " / 0.
11.H / 0.
4.2.* ". * .
-
:
•* " .
•".
•; , .
• < & & ", .
.
! " , " [3], ' 2, . 26-42 ' – .
* "
124
/ 0
, ' ' + . .
-' ' – -
.
. E 0, + ,
& " "#.
F, f (n),(n = 1, 2,...) . C /- 0 " G + " " :
f(1) = x1, f (2) = x2 , f (3) = x3,L, f (n) = xn ,L.
+- 0 x1, x2 ,L, xn ,L + -
& " {xn}.
, |
|
xn , + |
||||||
xn = f (n) |
- / |
& |
" {xn }. |
|||||
D |
' |
/, |
||||||
|
|
|
1 |
|
|
n |
||
" xn . >, {xn} = |
|
, |
{xn} = ncos |
|
. |
|||
|
|
|||||||
|
|
|
n |
|
2 |
|||
. {xn} 0 &, |
||||||||
|
M > 0, |
, n |
xn M.
: ', {xn} 0 &
( ), ' M , 0 &-
( ), - ' m . @, ' ' ,
. ! + ' ,
+ " '. @, '
( 0 ') " :
. {xn} 0 &, -
" ' M > 0, n0 , -
xn0 > M .
* .
1. {xn} = {1n} ', n -
xn 1.
2. {xn} = {n 5} ' , xn 4
n .
3.{xn} = { 1 n} ' ; xn 1.
4.{n 5} { 1 n} '.
. {xn} #/,
125