Литвин І.І., Конопчук О.М., Желізняк Г.О. Вища математика
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111
?CA 6. '?"OA '"OP
6.1. ! /!+%* /*!+%,.
C ( ’ –
(.
& ( & =f(), ( ! . + !<, 7 <
&
&
x
c
'
'
o' 0
lim
.
", ( (9:
,
D
c
'
'
y
x
y
(
D
o0 'o0.
C(: '&=&
c
'+
D
'.
R (, (< ( ((. ((
D
',
(' ! !, < ! '9 (-
, '. C!, ' ( (( 9 , 9,
9 (( ( ! (
!).
/!+% /*!+%,
& = f() ! ) ! -
'& !) dy. C ! d& = f
c
(x)'.
/ f() = , < d = 1', ' d = ', ' (
. A(
d&=&c'x=&cd,
' ( & =f() ! ()< (' ( !
( .
C(,
dx
dy
&
c
, 9 ! ( ,
( (', (9) ( ( (-
.
6.2. -!< ;% /!+%*.
I & =f() (.1).
/( =
0
(! (
! ')
0
, ':
x
y
y
x
'
'
c
o' 0
0
limtg)(
D
.
112
C . 1 (, 7 (! '< KN ( (: ,
((< (( yc' PN, ((
D
'. R7 -
< ( , ( NP 9< ! 9(9, (
PK. ", ( (! < ) !)
& =f(). C (:
=tgM.
A, 7 MN='; tgD =y c, < PK=y c'=dy.
", ( & =f() ! '< -
( (!, ( ! (
0
, &
0
) ( !
0
'.
1. C ( &=(2
3
–4)
5
.
?;’;!!: C( ( ( :
&c=5(2
3
–4)
4
(2
3
–4)c=5(2
3
–4)
4
6
2
=30
2
(2
3
–4)
4
.
/ ( ( , < (
:
d&=30
2
(2
3
–4)
4
dx.
%&%>: d&=30
2
(2
3
–4)
4
dx.
2. C (
t
S
2
ln
1
.
?;’;!!: H! ( ( (
ttt
t
t
tt
t
t
S
33422
2
ln
2
ln
1
2
ln
)(lnln2
)(ln
)(ln
c
c
c
.
/ ( ( , < (
:
'y
'x
y N
P
y
0
M dy
K
0
0
.1.
D
6. [6* 6&!*/
113
dx
tt
dt
dS
3
ln
2
.
%&%>:
dx
tt
dt
dS
3
ln
2
.
3. "'! ! ( v=cos
3
3M M S/18,
'M=0,01.
?;’;!!: '! ( dv=v
c
(M)dM.
/9 ) , ( ( !
M=S/18:
01,0
;375,3
8
27
2
1
4
3
9
2
1
2
3
9
6
sin
6
cos9
18
;3sin3cos933sin3cos3
)3)(3sin(3cos3)3(cos3cos3
2
2
22
22
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
c
c
c
c
M
SSS
MMMM
MMMMM
d
v
v
C(, dv = –3,3750,01= –0,03375.
%&%>: dv = –0,03375.
6.3. *!! /!+%* !#$!: #!>. /
( () ( ( ( ! a'. R7
<) ) !), < '
'& | dy,
'
'
(.
5)!, 7 '& =f(+') – f() , < f(+') – f()
|
dy, (
f(+')
|
f()+dy.
+ ' ) ( ' '!, ' '!-
( ! 9, '! .
4. "'! ' ! &=
2
+2+5
( =2 ( =2,001.
?;’;!!: C( ( d =' =2,001 – 2=0,001. /
, '& ' ()< ( d&.
'!)< ): dy=yc(x)dx. H! (-
( ! = 2.
yc= 2x+2,
6. [6* 6&!*/
114
yc(2) = 22+2 = 6.
A(:
dy = 60,001 = 0,006;
'& | 0,006.
A! ! ( ):
'& = f(2,001) – f(2),
(
f(2) = 2
2
+ 22 + 5 = 13, f(2,001) = (2,001)
2
+ 22,001 + 5 = 13,006001.
A(
'& = 13,006001 – 13 = 0,006001.
/))! ( dy, '!, 7 ')
' ()< 0,000 001. / ') ' (< ( -
! (, '! ( ':
%02,0
006001,0
000001,0
Z
.
A ! ( ( ( ( ,
( (.
%&%>: '& | 0,006.
5. "'! ' ! & =
3
+
2
– 2 = 2,01.
?;’;!!: C( ( dx='=2,01–2=0,01. /
, ( '! ' !
< :
f(+')
|
f(x) + d&, ' f(2,01) | f(2) + dy.
H! ( ! = 2: f(2) = 2
3
+ 2
2
– 22 = 8.
( ): dy = &
c
()dx, ( ( (
! = 2:
&c= 3
2
+ 2 – 2, &c(2) = 32
2
– 2 = 12 + 4 – 2 = 14;
A( dy = 140,01 = 0,14;
C(: f(2,01) | 8 + 0,14 = 8,014.
%&%>: f(2,01) | 8,014.
6: C ' !
06,16
.
?;’;!!: & ' ' !
xy
=16,06.
C( ( : d =' = 16,06 – 16 = 0,06.
/ ,
f(16,06) | f(16) + d&, f(16) =
16
= 4 .
6. [6* 6&!*/
115
( ): d& = &
c
()dx, ( ! (-
( ! = 16.
0125,0
8
1
162
1
)16(;
2
1
)()(
c
c
c
f
x
xxf
.
A( d& = 0,1250,06 = 0,0075.
C(,
0075,0406,16 |
= 4,0075.
A! !
06,16
= 4,0074929.
%&%>:
06,16
= 4,0075.
7. C ' !
3
988,0
.
?;’;!!: R ( (, < f(0,988) | f(1) + dy,
;
3
xy
dx =' = 0,988 – 1= –0,012;
.11)1(;004,0)012.0(
3
1
;
3
1
)1(;
3
1
3
1
)(
3
32
3/23/1
c
c
c
fdy
y
x
xxy
A(:
.996,0004,01988,0
3
|
%&%>:
996,0988,0
3
|
.
8. "'’< ', ' ()< 4 , '9<-
0,96
3
. R '9< ' '?
?;’;!!: "'’< ' ' '!)< ): V =
3
. "-
'V | dV, dV | 0,96.
'!)< ) dV=V
c
(x)dx, (
.
)(xV
dV
dx
c
/9 <) ) ( ( V
! = 4: Vc= 3
2
, Vc(4) = 34
2
= 48.
A ( dx | 0,96/48 = 0,02, ' | 0,02.
%&%>: I' ' '9 ' 0,02 .
6. [6* 6&!*/
116
6.4. &.
1. ! ( .
2. B ()< ( ()?
3. C ( ( ?
4. 5 ! < ! ( ?
5. R ) ( ' '!?
6. C( ( ) & = (2
2
–1)(3–5
2
); ') V = ln sin
3
2t.
7. "'! ! (
9
2
tS
t ( 4 (
4,025.
8. "'! ' ! &=(1+ –
2
)
2
( 3 ( 2,998.
9. C( ' ! )
005,1
; ')
3
9843,0
.
10. "'’< ', ' ()< 40 '9< 0,05
! !. C( ( ' '.
11. H ! ()< 12,
30. & (') 9< '’<, 7 ))! , -
9 0,01.
"'! ' !
y = f(x)
( x
1
( x
2
.
12.
,5)(
3
xxxf
,2
1
x 01,2
2
x .
13.
,354)(
23
xxxxf
,1
1
x 03,1
2
x .
14.
,)(
23
xxxxf
,2
1
x 01,2
2
x .
15.
,132)(
3
xxxf
,2
1
x 001,2
2
x .
16.
,42)(
23
xxxxf
,2
1
x 04,2
2
x .
17.
,455)(
23
xxxxf
1
1
x ,06,1
2
x .
18.
,23)(
3
xxxf
,4
1
x 003,4
2
x .
19.
,266)(
23
xxxxf
,1
1
x 005,1
2
x .
"'! ' !
)(xfy
( !
20.
,2
2
1
4
1
)(
24
xxxxf
01,1
x
.
6. [6* 6&!*/
117
21.
,2
2
1
4
1
)(
24
xxxxf
02,4
x
.
22.
,2542)(
23
xxxxf
04,2
x
.
23.
,2542)(
23
xxxxf
04,1
x
.
24.
,1)(
3
xxxf
04,2
x
.
25.
43
3
1
)(
23
xxxxf
02,1
x
.
26.
,266)(
23
xxxxf
01,3
x
.
27.
,132)(
23
xxxxf
02,2
x
.
28.
,15)(
23
xxxxf
03,2
x
.
29.
,432)(
23
xxxxf
02,1
x
.
"'! ' !
30.
08,4
. 31.
7
)01,1(
.
32.
3
04,1
. 33.
3
98,26
.
34.
6
)03,2(
. 35.
D
31sin
.
36.
D
32cos
. 37.
D
47tg
.
6. [6* 6&!*/
118
/!+% /*!+%,
*!! !#$!: #!>
dxyxyxxy
dy&
c
|'
|'
)()(
:
1. C (
2
&
dxxedxedxydy
xx
22
2)(
c
c
.
2. C ' ! & = cosx
! = 45
0
30c
– ! .
=#,!< ;%
-!< ;%
&&
y
x
y
y
x
y
x
''
c
'
o'
o
c
'
'
c
'
'
o'
)(
0
0,
lim
0
D
DD
!
Y
y=f(x)
y
+
ǻ
y
D
(x)
x
(!
B
ǻ
y
D
dy
y A C
D
x x+
ǻ
x
X
)(xytg
c
D
3
'
ABCBC=ACtgD=
'
xtgD=yc
'
x
dxydy
c
dyy |'
dx
dy
y
dxydy
c
c
'
–
'
& –
'
& = &
2
– &
1
= f(x
2
) – f(x
1
)=
= f(x
1
+
'
x) – f(x
1
)
Y
y
2
y
y
x
x
1
x
2
X
xxyxyxxy '
c
|' )()()(
.69,00087,0
2
2
2
2
0087,0
45
)sin(
2
2
0087,0
45
)(cos45cos)5,045cos()0345cos(
|
c
|
c
D
D
DDDD
x
x
x
x
119
?CA 7. ""D"R "-?A
7.1. ! !;!!= %!=*.
) – (, () () <) (
( ! ( ( .
C( ( < ! !. A, (
S=S(t) () ( 9( v(t)=S
c
(t), 9
a(t)=S
cc
(t); 7 ( y = f(x), !
< (!, ( ( ( : k = f
c
(x).
5 < ' (!, (:
) ( 9( , .
') ( < (! ( , < .
9 !, () () ' ), ( (
(, ' () ' ()). +) () )
%!=*!!. C () () () ' (-
( ), ) ).
F(x) <
&%!F ( f(x)
<x<b, 7 F
c
(x)=f(x) ( <x<b.
A, ( f(x)=cosx ) < F(x)=sinx, (sinx)c=cosx.
C, 7 ( < <( . &(, F
1
(x)=sinx+1,
F
2
(x) = sinx + 3, F
3
(x) = sinx – 4, () 9 , (-
) F
i
c
(x)=cosx.
(
*: 7 F(x) – ( f(x) ( ,
F(x)+C, ( – '(- , < ) ( f(x)
.
(:
(F(x)+C)
c
=F
c
(x)+C
c
=f(x).
", ( ( f(x) ( ) F(x), 7'
, ' () ( ( (
! .
H F(x)+C f(x) <x<b )
!;!! %!= ( f(x) !)
³
dxxf )(
.
A f(x)dx – ( , f(x) – ( , –
, – ( .
"&:
³
c
x
CtgxCtgx
x
dx
22
cos
1
)(',
cos
.
%! F(x)+C ' ,
'(- ( OY (. 1) .
120
R7 f(x) < ( ! ( , )
) .
7.2. % !;!!= %!=*.
1. /( ! ( ; (
! ( :
(
dxxf
³
)
c
=f(x); d
dxxf
³
=f(x)dx.
2. &! ( ( :
³
xdF
=F(x)+C.
3. / :
³³
dxxfAdxxAf )()(
.
4. &! ( '!
'! ! ( :
.
2121
dxxfdxxfdxxfxf
³³³
r r
5. R7 F(x) < ) ( f(x), ( k b ( ! (k z 0),
C
k
bkxF
dxbkxf
³
.
(( 1-5 ( ( ' ! .
&(, (( 4:
xfxfdxxfxf
2121
r
c
r
³
,
Y
0
X
. 1
7. "^ _