Дифференциальное исчисление. - 5 стр.

UptoLike

ВУЗ: 

МГТУ ГА | Москва

Рубрика: 

§1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ 5
òÅÛÅÎÉÅ. óÎÁÞÁÌÁ ÎÁÊÄ¾Í ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x):
f
0
(x) = (5x
3
− 3x
2
− 2x + 7)
0
= (5x
3
)
0
− (3x
2
)
0
− (2x)
0
+ 7
0
=
= 5(x
3
)
0
− 3(x
2
)
0
− 2x
0
+ 0 = 5 · 3x
2
− 3 · 2x − 2 · 1 = 15x
2
− 6x − 2.
éÔÁË, f
0
(x) = 15x
2
− 6x − 2. ôÅÐÅÒØ ÎÁÈÏÄÉÍ ÚÎÁÞÅÎÉÑ ÐÒÏÉÚ×ÏÄÎÙÈ ÐÒÉ
ËÏÎËÒÅÔÎÙÈ ÚÎÁÞÅÎÉÑÈ x:
f
0
(0) = 15 · 0
2
− 6 · 0 − 2 = 0 − 0 − 2 = −2,
f
0
(2) = 15 · 2
2
− 6 · 2 − 2 = 60 − 12 − 2 = 46,
f
0
(−1) = 15 · (−1)
2
− 6 · (−1) − 2 = 15 + 6 − 2 = 19.
ðÒÉÍÅÒ 8. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x) = (x
2
+ x) cos x.
òÅÛÅÎÉÅ.
f
0
(x) =
(x
2
+ x) cos x
0
= (x
2
+ x)
0
cos x + (x
2
+ x)(cos x)
0
=
= (2x + 1) cos x + (x
2
+ x)(−sin x).
ðÒÉÍÅÒ 9. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x) =
x
3
+2x
2
+5x+1
x
2
+2
x
.
òÅÛÅÎÉÅ.
f
0
(x) =
x
3
+ 2x
2
+ 5x + 1
x
2
+ 2
x
0
=
=
(x
3
+ 2x
2
+ 5x + 1)
0
(x
2
+ 2
x
) − (x
3
+ 2x
2
+ 5x + 1)(x
2
+ 2
x
)
0
(x
2
+ 2
x
)
2
=
=
(3x
2
+ 4x + 5)(x
2
+ 2
x
) − (x
3
+ 2x
2
+ 5x + 1)(2x + 2
x
ln 2)
(x
2
+ 2
x
)
2
.
ðÒÉÍÅÒ 10. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x) = arccos x ln x sh x.
òÅÛÅÎÉÅ.
f
0
(x) = (arccos x ln x sh x)
0
= ((arccos x ln x) sh x)
0
=
= (arccos x ln x)
0
sh x + (arccos x ln x)(sh x)
0
=
= ((arccos x)
0
ln x + arccos x(ln x)
0
) sh x + (arccos x ln x) sh x =
=
−
1
√
1 − x
2
ln x + arccos x
1
x
sh x + arccos x ln x ch x =
= −
1
√
1 − x
2
ln x sh x + arccos x
1
x
sh x + arccos x ln x ch x =
=
arccos x sh x
x
−
ln x sh x
√
1 − x
2
+ arccos x ln x ch x.
§1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ                                                               5

  òÅÛÅÎÉÅ. óÎÁÞÁÌÁ ÎÁÊÄ¾Í ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x):
 f 0(x) = (5x3 − 3x2 − 2x + 7)0 = (5x3)0 − (3x2)0 − (2x)0 + 70 =
         = 5(x3)0 − 3(x2)0 − 2x0 + 0 = 5 · 3x2 − 3 · 2x − 2 · 1 = 15x2 − 6x − 2.
éÔÁË, f 0(x) = 15x2 − 6x − 2. ôÅÐÅÒØ ÎÁÈÏÄÉÍ ÚÎÁÞÅÎÉÑ ÐÒÏÉÚ×ÏÄÎÙÈ ÐÒÉ
ËÏÎËÒÅÔÎÙÈ ÚÎÁÞÅÎÉÑÈ x:
                  f 0 (0) = 15 · 02 − 6 · 0 − 2 = 0 − 0 − 2 = −2,
                 f 0 (2) = 15 · 22 − 6 · 2 − 2 = 60 − 12 − 2 = 46,
             f 0(−1) = 15 · (−1)2 − 6 · (−1) − 2 = 15 + 6 − 2 = 19.
   ðÒÉÍÅÒ 8. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x) = (x2 + x) cos x.
   òÅÛÅÎÉÅ.
                        0
 f 0(x) = (x2 + x) cos x = (x2 + x)0 cos x + (x2 + x)(cos x)0 =
                                             = (2x + 1) cos x + (x2 + x)(− sin x).
                                                              3   2
   ðÒÉÍÅÒ 9. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x) = x +2x           +5x+1
                                                           x2 +2x    .
   òÅÛÅÎÉÅ.
           3       2
                               0
            x  + 2x   + 5x + 1
 f 0(x) =                         =
                 x2 + 2 x
          (x3 + 2x2 + 5x + 1)0(x2 + 2x ) − (x3 + 2x2 + 5x + 1)(x2 + 2x )0
       =                                                                  =
                                    (x2 + 2x )2
                  (3x2 + 4x + 5)(x2 + 2x ) − (x3 + 2x2 + 5x + 1)(2x + 2x ln 2)
                =                                                              .
                                            (x2 + 2x )2
  ðÒÉÍÅÒ 10. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ f (x) = arccos x ln x sh x.
  òÅÛÅÎÉÅ.
 f 0(x) = (arccos x ln x sh x)0 = ((arccos x ln x) sh x)0 =
                 = (arccos x ln x)0 sh x + (arccos x ln x)(sh x)0 =
        = ((arccos x)0 ln x + arccos x(ln x)0) sh x + (arccos x ln x) sh x =
                                            
                    1                      1
         = −√              ln x + arccos x     sh x + arccos x ln x ch x =
                  1 − x2                   x
                  1                          1
         = −√           ln x sh x + arccos x sh x + arccos x ln x ch x =
                1 − x2                       x
                                    arccos x sh x ln x sh x
                                 =                −√          + arccos x ln x ch x.
                                          x            1 − x2

Страницы