Дифференциальное исчисление. - 16 стр.

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16 §2. äÉÆÆÅÒÅÎÃÉÁÌ ÆÕÎËÃÉÉ
òÅÛÅÎÉÅ. äÉÆÆÅÒÅÎÃÉÁÌ dy ÆÕÎËÃÉÉ y(x) ÎÁÈÏÄÉÔÓÑ ÐÏ ÆÏÒÍÕÌÅ dy =
= y
0
(x)dx, ÐÏÜÔÏÍÕ
dy = dx
2
= (x
2
)
0
dx = 2xdx.
ðÒÉÍÅÒ 2. îÁÊÔÉ ÄÉÆÆÅÒÅÎÃÉÁÌ ÆÕÎËÃÉÉ y(x) = 5 cos 3x × ÔÏÞËÅ x
0
=
π
2
.
òÅÛÅÎÉÅ. ðÏ ÐÒÁ×ÉÌÕ ÎÁÈÏÖÄÅÎÉÑ ÄÉÆÆÅÒÅÎÃÉÁÌÁ ÆÕÎËÃÉÉ × ÔÏÞËÅ,
ÉÍÅÅÍ
dy = y
0
π
2
dx.
îÁÈÏÄÉÍ,
y
0
(x) = (5 cos 3x)
0
= −15 sin 3x ⇒ y
0
π
2
= −15 sin
3π
2
= 15.
óÌÅÄÏ×ÁÔÅÌØÎÏ, ÄÉÆÆÅÒÅÎÃÉÁÌ dy × ÔÏÞËÅ x
0
=
π
2
ÒÁ×ÅÎ 15 dx:
dy = 15 dx.
ðÒÉÍÅÒ 3. îÁÊÔÉ ÄÉÆÆÅÒÅÎÃÉÁÌ ÆÕÎËÃÉÉ y = sin
2
x
3
.
òÅÛÅÎÉÅ.
dy = y
0
dx =
sin
2
x
3
0
= 2 sin x
3
(sin x
3
)
0
dx =
= 2 sin x
3
cos x
3
(x
3
)
0
dx = 2 sin x
3
cos x
3
· 3x
2
dx = 3x
2
sin 2x
3
dx.
ðÒÉÍÅÒ 4. îÁÊÔÉ ÄÉÆÆÅÒÅÎÃÉÁÌ ÆÕÎËÃÉÉ xe
x
2
.
òÅÛÅÎÉÅ.
1-Ê ÓÐÏÓÏÂ. ÷ÏÓÐÏÌØÚÕÅÍÓÑ ÆÏÒÍÕÌÏÊ df(x) = f
0
(x)dx.
d
xe
x
2
=
xe
x
2
0
dx =
x
0
e
x
2
+ x
e
x
2
0
dx =
=
e
x
2
+ x · e
x
2
2x
dx = e
x
2
2x
2
+ 1
dx.
2-Ê ÓÐÏÓÏÂ. ðÒÉÍÅÎÉÍ ÆÏÒÍÕÌÕ d(uv) = udv + vdu, ÉÍÅÅÍ
d
xe
x
2
= e
x
2
dx + xd
e
x
2
.
ðÏ ÆÏÒÍÕÌÅ df (u) = f
0
(u)du ÐÏÌÕÞÁÅÍ
d
e
x
2
= e
x
2
d
x
2
= e
x
2
2xdx.
ôÁËÉÍ ÏÂÒÁÚÏÍ,
d
xe
x
2
= e
x
2
dx + x(2xe
x
2
dx) = e
x
2
2x
2
+ 1
dx.
16                                                  §2. äÉÆÆÅÒÅÎÃÉÁÌ ÆÕÎËÃÉÉ

  òÅÛÅÎÉÅ. äÉÆÆÅÒÅÎÃÉÁÌ dy ÆÕÎËÃÉÉ y(x) ÎÁÈÏÄÉÔÓÑ ÐÏ ÆÏÒÍÕÌÅ dy =
= y 0 (x)dx, ÐÏÜÔÏÍÕ
                     dy = dx2 = (x2)0dx = 2xdx.
  ðÒÉÍÅÒ 2. îÁÊÔÉ ÄÉÆÆÅÒÅÎÃÉÁÌ ÆÕÎËÃÉÉ y(x) = 5 cos 3x × ÔÏÞËÅ x0 = π2 .
  òÅÛÅÎÉÅ. ðÏ ÐÒÁ×ÉÌÕ ÎÁÈÏÖÄÅÎÉÑ ÄÉÆÆÅÒÅÎÃÉÁÌÁ ÆÕÎËÃÉÉ × ÔÏÞËÅ,
ÉÍÅÅÍ
                                     0 π
                                       
                              dy = y      dx.
                                       2
îÁÈÏÄÉÍ,
                                            0 π             3π
                                               
       0                0
      y (x) = (5 cos 3x) = −15 sin 3x ⇒ y         = −15 sin    = 15.
                                               2             2
óÌÅÄÏ×ÁÔÅÌØÎÏ, ÄÉÆÆÅÒÅÎÃÉÁÌ dy × ÔÏÞËÅ x0 = π2 ÒÁ×ÅÎ 15 dx:
                                   dy = 15 dx.
  ðÒÉÍÅÒ 3. îÁÊÔÉ ÄÉÆÆÅÒÅÎÃÉÁÌ ÆÕÎËÃÉÉ y = sin2 x3.
  òÅÛÅÎÉÅ.
                      0
 dy = y 0 dx = sin2 x3 = 2 sin x3(sin x3)0 dx =
               = 2 sin x3 cos x3 (x3)0 dx = 2 sin x3 cos x3 · 3x2dx = 3x2 sin 2x3dx.
                                                        2
  ðÒÉÍÅÒ 4. îÁÊÔÉ ÄÉÆÆÅÒÅÎÃÉÁÌ ÆÕÎËÃÉÉ xex .
  òÅÛÅÎÉÅ.
  1-Ê ÓÐÏÓÏÂ. ÷ÏÓÐÏÌØÚÕÅÍÓÑ ÆÏÒÍÕÌÏÊ df (x) = f 0 (x)dx.
   2   2 0                   2 0 
      x        x          0 x2
 d xe    = xe      dx = x e + x ex        dx =
                                   2              
                                             x2          2
                                = e + x · e 2x dx = ex 2x2 + 1 dx.
                                    x
                                                              

     2-Ê ÓÐÏÓÏÂ. ðÒÉÍÅÎÉÍ ÆÏÒÍÕÌÕ d(uv) = udv + vdu, ÉÍÅÅÍ
                           2               2
                             x     x2
                        d xe    = e dx + xd ex .

ðÏ ÆÏÒÍÕÌÅ df (u) = f 0 (u)du ÐÏÌÕÞÁÅÍ
                           2     2        2
                         d ex = ex d x2 = ex 2xdx.
                                       

ôÁËÉÍ ÏÂÒÁÚÏÍ,
               2    2           2        2
            d xex = ex dx + x(2xex dx) = ex 2x2 + 1 dx.
                                                   

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