Сборник задач по высшей математике. Часть II. Пределы. Производные. Графики функций. Самохин А.В - 90 стр.

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90 çÌÁ×Á III. äÉÆÆÅÒÅÎÃÉÁÌØÎÏÅ ÉÓÞÉÓÌÅÎÉÅ
÷ ÔÏÞËÁÈ, × ËÏÔÏÒÙÈ x
0
(t) 6= 0, ÉÍÅÅÍ
y
0
x
(t) =
y
0
(t)
x
0
(t)
=
7 sin t
7(1 − cos t)
=
sin t
1 − cos t
=
=
2 sin
t
2
cos
t
2
cos
2
t
2
+ sin
2
t
2
−
cos
2
t
2
− sin
2
t
2
=
2 sin
t
2
cos
t
2
2 sin
2 t
2
=
cos
t
2
sin
t
2
= ctg
t
2
.
äÌÑ ÎÁÈÏÖÄÅÎÉÑ ÐÒÏÉÚ×ÏÄÎÏÊ y
00
xx
(t) ÐÏ ÐÅÒ×ÏÊ ÆÏÒÍÕÌÅ ×ÙÞÉÓÌÉÍ ÓÎÁÞÁÌÁ
(y
0
x
(t))
0
:
(y
0
x
(t))
0
=
ctg
t
2
0
= −
1
2 sin
2 t
2
.
ôÅÐÅÒØ ÎÁÈÏÄÉÍ y
00
xx
(t):
y
00
xx
(t) =
(y
0
x
(t))
0
x
0
(t)
=
−
1
2 sin
2
t
2
7(1 − cos t)
= −
1
28 sin
4
t
2
.
éÔÁË, y
0
x
(t) = ctg
t
2
, y
00
xx
(t) = −
1
28 sin
4
t
2
.
ðÒÉÍÅÒ 3. îÁÊÔÉ y
00
xx
(t) ÆÕÎËÃÉÉ, ÚÁÄÁÎÎÏÊ ÐÁÒÁÍÅÔÒÉÞÅÓËÉ:
x(t) = ln(1 − t), y(t) = (t + 1)
2
.
òÅÛÅÎÉÅ. ÷ÏÓÐÏÌØÚÕÅÍÓÑ ÆÏÒÍÕÌÏÊ:
y
00
xx
(t) =
y
00
(t)x
0
(t) − x
00
(t)y
0
(t)
(x
0
(t))
3
.
äÌÑ ÜÔÏÇÏ ×ÙÞÉÓÌÉÍ ÐÒÏÉÚ×ÏÄÎÙÅ x
0
(t), x
00
(t), y
0
(t), y
00
(t):
x
0
(t) = (ln(1 − t))
0
=
1
1 − t
(1 − t)
0
= −
1
1 − t
;
x
00
(t) = (x
0
(t))
0
=
−
1
1 − t
0
= −
(1 − t)
−1
0
= (1 −t
−2
)(1 −t)
0
= −
1
(1 − t)
2
;
y
0
(t) =
(t + 1)
2
0
= 2(t + 1);
y
00
(t) = (y
0
(t))
0
= (2(t + 1))
0
= 2.
90                                            çÌÁ×Á III. äÉÆÆÅÒÅÎÃÉÁÌØÎÏÅ ÉÓÞÉÓÌÅÎÉÅ

÷ ÔÏÞËÁÈ, × ËÏÔÏÒÙÈ x0(t) 6= 0, ÉÍÅÅÍ

              y 0 (t)      7 sin t            sin t
  yx0 (t)   = 0       =                 =           =
             x (t) 7(1 − cos t) 1 − cos t
                              2 sin 2t cos 2t              2 sin 2t cos 2t   cos 2t     t
            =         2 t      2 t
                                             2 t    2 t =
                                                                   2 t    =     t = ctg .
                   cos 2 + sin 2 − cos 2 − sin 2              2 sin 2        sin 2      2

                             00
äÌÑ ÎÁÈÏÖÄÅÎÉÑ ÐÒÏÉÚ×ÏÄÎÏÊ yxx  (t) ÐÏ ÐÅÒ×ÏÊ ÆÏÒÍÕÌÅ ×ÙÞÉÓÌÉÍ ÓÎÁÞÁÌÁ
  0     0
(yx (t)) :
                                      0
                     0     0         t           1
                   (yx (t)) = ctg         =−           .
                                    2        2 sin2 2t
                00
ôÅÐÅÒØ ÎÁÈÏÄÉÍ yxx (t):

                                  (y 0
                                       (t))0     − 2 sin1 2 t        1
                          00
                         yxx (t) = x0        =              2
                                                              =−            .
                                   x (t)       7(1 − cos t)      28 sin4 2t

éÔÁË, yx0 (t) = ctg 2t , yxx
                          00             1
                             (t) = − 28 sin4 t .
                                                  2
     ðÒÉÍÅÒ 3. îÁÊÔÉ             00
                                yxx (t)   ÆÕÎËÃÉÉ, ÚÁÄÁÎÎÏÊ ÐÁÒÁÍÅÔÒÉÞÅÓËÉ:

                               x(t) = ln(1 − t),        y(t) = (t + 1)2.

     òÅÛÅÎÉÅ. ÷ÏÓÐÏÌØÚÕÅÍÓÑ ÆÏÒÍÕÌÏÊ:

                                 00         y 00 (t)x0(t) − x00 (t)y 0(t)
                                yxx (t)   =                               .
                                                       (x0(t))3

äÌÑ ÜÔÏÇÏ ×ÙÞÉÓÌÉÍ ÐÒÏÉÚ×ÏÄÎÙÅ x0 (t), x00(t), y 0 (t), y 00 (t):

                          1                1
  x0(t) = (ln(1 − t))0 =     (1 − t)0 = −     ;
                         1−t              1−t
                            0
    00        0   0       1                   0                           1
  x (t) = (x (t)) = −            = − (1 − t)−1 = (1 − t−2 )(1 − t)0 = −          ;
                         1−t                                            (1 − t)2
                    0
  y 0 (t) = (t + 1)2 = 2(t + 1);
                     0
  y 00 (t) = (y 0 (t)) = (2(t + 1))0 = 2.

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