Дифференциальное исчисление. - 21 стр.

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§3. ðÒÏÉÚ×ÏÄÎÙÅ É ÄÉÆÆÅÒÅÎÃÉÁÌÙ ×ÙÓÛÉÈ ÐÏÒÑÄËÏ× 21
òÅÛÅÎÉÅ.
Á) C
3
5
=
5!
3! · (5 − 3)!
=
3! · 4 · 5
3! · 2!
=
4 · 5
1 · 2
= 10;
Â) C
2
6
=
6!
2! · (6 − 2)!
=
4! · 5 · 6
2! · 4!
=
5 · 6
1 · 2
= 15;
×) C
4
10
=
10!
4! · (10 − 4)!
=
6! · 7 · 8 · 9 · 10
4! · 6!
=
7 · 8 · 9 · 10
1 · 2 · 3 · 4
= 210.
ðÕÓÔØ y = u·v, ÇÄÅ u É v ¡ ÎÅËÏÔÏÒÙÅ ÆÕÎËÃÉÉ ÏÔ ÐÅÒÅÍÅÎÎÏÊ x, ÉÍÅÀÝÉÅ
ÐÒÏÉÚ×ÏÄÎÙÅ ÌÀÂÏÇÏ ÐÏÒÑÄËÁ. ôÏÇÄÁ ÓÐÒÁ×ÅÄÌÉ×Á ÆÏÒÍÕÌÁ ìÅÊÂÎÉÃÁ:
y
(n)
= (uv)
(n)
= C
0
n
u
(n)
v + C
1
n
u
(n−1)
v
0
+ C
2
n
u
(n−2)
v
00
+ . . .
. . . + C
k
n
u
(n−k)
v
(k)
+ . . . + C
n−1
n
u
0
v
(n−1)
+ C
n
n
uv
(n)
=
= u
(n)
v + nu
(n−1)
v
0
+
n(n − 1)
2!
u
(n−2)
v
00
+ . . .
. . . +
n(n − 1) . . . (n −k + 1)
k!
u
(n−k)
v
(k)
+ . . . + nu
0
v
(n−1)
+ uv
(n)
.
æÏÒÍÕÌÕ ìÅÊÂÎÉÃÁ ÕÄÏÂÎÏ ÐÒÉÍÅÎÑÔØ × ÓÌÕÞÁÅ ¤ÐÒÏÓÔÙÈ¥ ÆÕÎËÃÉÊ u É v.
÷ ÓÌÕÞÁÑÈ n = 1, n = 2 É n = 3 ÆÏÒÍÕÌÁ ìÅÊÂÎÉÃÁ ÐÒÉÎÉÍÁÅÔ ×ÉÄ:
y
0
= (uv)
0
= u
0
v + uv
0
; y
00
= (uv)
00
= u
00
v + 2u
0
v
0
+ uv
00
;
u
000
= (uv)
000
= u
000
v + 3u
00
v
0
+ 3u
0
v
00
+ uv
000
.
úÁÍÅÔÉÍ, ÞÔÏ ÐÅÒ×ÁÑ ÉÚ ÜÔÉÈ ÆÏÒÍÕÌ Ñ×ÌÑÅÔÓÑ ÆÏÒÍÕÌÏÊ ÐÒÏÉÚ×ÏÄÎÏÊ ÐÒÏ-
ÉÚ×ÅÄÅÎÉÑ Ä×ÕÈ ÆÕÎËÃÉÊ.
éÓÐÏÌØÚÏ×ÁÎÉÅ ÓÌÅÄÕÀÝÅÊ ÆÏÒÍÕÌÙ ÞÁÓÔÏ ÐÏÚ×ÏÌÑÅÔ ÓÏËÒÁÔÉÔØ ×ÙÞÉÓÌÅ-
ÎÉÑ ËÏÜÆÆÉÃÉÅÎÔÏ×.
C
k
n
= C
n−k
n
.
ðÒÉÍÅÒ 8. ÷ÙÞÉÓÌÉÔØ ÐÑÔÕÀ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y = x
5
e
x
.
òÅÛÅÎÉÅ. ðÏÌÁÇÁÑ u = x
5
É v = e
x
, ÎÁÈÏÄÉÍ:
u
0
= 5x
4
, u
00
= 20x
3
, u
000
= 60x
2
, u
(4)
= 120x, u
(5)
= 120,
v
0
= v
00
= v
000
= v
(4)
= v
(5)
= e
x
.
ôÅÐÅÒØ ×ÙÞÉÓÌÑÅÍ ËÏÜÆÆÉÃÉÅÎÔÙ ÐÒÉ ÐÒÏÉÚ×ÏÄÎÙÈ:
C
0
5
=
5!
0! · (5 − 0)!
= 1, C
1
5
=
5!
1! · (5 − 1)!
= 5, C
2
5
=
5!
2! · (5 − 2)!
= 10,
C
3
5
= C
5−2
5
= C
2
5
= 10, C
4
5
= C
5−1
5
= C
1
5
= 5, C
5
5
= C
5−0
5
= C
0
5
= 1.
§3. ðÒÏÉÚ×ÏÄÎÙÅ É ÄÉÆÆÅÒÅÎÃÉÁÌÙ ×ÙÓÛÉÈ ÐÏÒÑÄËÏ×                                             21

   òÅÛÅÎÉÅ.
                 5!       3! · 4 · 5 4 · 5
 Á) C53 =               =             =         = 10;
          3! · (5 − 3)!     3! · 2!       1·2
                 6!       4! · 5 · 6 5 · 6
 Â) C62 =               =             =         = 15;
          2! · (6 − 2)!     2! · 4!       1·2
      4           10!        6! · 7 · 8 · 9 · 10 7 · 8 · 9 · 10
 ×) C10 =                 =                      =              = 210.
           4! · (10 − 4)!          4! · 6!          1·2·3·4
  ðÕÓÔØ y = u·v, ÇÄÅ u É v ¡ ÎÅËÏÔÏÒÙÅ ÆÕÎËÃÉÉ ÏÔ ÐÅÒÅÍÅÎÎÏÊ x, ÉÍÅÀÝÉÅ
ÐÒÏÉÚ×ÏÄÎÙÅ ÌÀÂÏÇÏ ÐÏÒÑÄËÁ. ôÏÇÄÁ ÓÐÒÁ×ÅÄÌÉ×Á ÆÏÒÍÕÌÁ ìÅÊÂÎÉÃÁ:

 y (n) = (uv)(n) = Cn0u(n) v + Cn1u(n−1)v 0 + Cn2u(n−2)v 00 + . . .
            . . . + Cnk u(n−k) v (k) + . . . + Cnn−1u0 v (n−1) + Cnnuv (n) =
                                               n(n − 1) (n−2) 00
                  = u(n) v + nu(n−1)v 0 +                 u     v + ...
                                                  2!
                   n(n − 1) . . . (n − k + 1) (n−k) (k)
           ...+                                   u       v + . . . + nu0v (n−1) + uv (n) .
                                  k!
æÏÒÍÕÌÕ ìÅÊÂÎÉÃÁ ÕÄÏÂÎÏ ÐÒÉÍÅÎÑÔØ × ÓÌÕÞÁÅ ¤ÐÒÏÓÔÙÈ¥ ÆÕÎËÃÉÊ u É v.
  ÷ ÓÌÕÞÁÑÈ n = 1, n = 2 É n = 3 ÆÏÒÍÕÌÁ ìÅÊÂÎÉÃÁ ÐÒÉÎÉÍÁÅÔ ×ÉÄ:
             y 0 = (uv)0 = u0 v + uv 0;    y 00 = (uv)00 = u00 v + 2u0v 0 + uv 00;
                     u000 = (uv)000 = u000v + 3u00v 0 + 3u0 v 00 + uv 000 .
úÁÍÅÔÉÍ, ÞÔÏ ÐÅÒ×ÁÑ ÉÚ ÜÔÉÈ ÆÏÒÍÕÌ Ñ×ÌÑÅÔÓÑ ÆÏÒÍÕÌÏÊ ÐÒÏÉÚ×ÏÄÎÏÊ ÐÒÏ-
ÉÚ×ÅÄÅÎÉÑ Ä×ÕÈ ÆÕÎËÃÉÊ.
   éÓÐÏÌØÚÏ×ÁÎÉÅ ÓÌÅÄÕÀÝÅÊ ÆÏÒÍÕÌÙ ÞÁÓÔÏ ÐÏÚ×ÏÌÑÅÔ ÓÏËÒÁÔÉÔØ ×ÙÞÉÓÌÅ-
ÎÉÑ ËÏÜÆÆÉÃÉÅÎÔÏ×.
                              Cnk = Cnn−k .
   ðÒÉÍÅÒ 8. ÷ÙÞÉÓÌÉÔØ ÐÑÔÕÀ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y = x5ex .
   òÅÛÅÎÉÅ. ðÏÌÁÇÁÑ u = x5 É v = ex , ÎÁÈÏÄÉÍ:
         u0 = 5x4,    u00 = 20x3,     u000 = 60x2,      u(4) = 120x,          u(5) = 120,
                           v 0 = v 00 = v 000 = v (4) = v (5) = ex .
ôÅÐÅÒØ ×ÙÞÉÓÌÑÅÍ ËÏÜÆÆÉÃÉÅÎÔÙ ÐÒÉ ÐÒÏÉÚ×ÏÄÎÙÈ:
                5!                       5!                       5!
   C50 =                = 1, C51 =               = 5, C52 =               = 10,
          0! · (5 − 0)!            1! · (5 − 1)!            2! · (5 − 2)!
     C53 = C55−2 = C52 = 10, C54 = C55−1 = C51 = 5, C55 = C55−0 = C50 = 1.

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