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Сборник задач по высшей математике. Тестовые методы контроля знаний. В 3 томах. Т.3

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. -   
3=a
7=b
2
7
3
2
+=+
b
3412
12
)52
(
<
<
X
P
)(x
f
)(x
f
)32(),(),
(
<
<
XPXDX
M
[
]
(5.28), 
)

,
 . )
, !
)( =
=
XM
XD
)(
−=b
)(
=
  
b
=<<
   
1
 
=
)(
xf
 
ab
[ ]
[ ]
bax
;;,
bax
;;,0
22
)37(
dxxfbXaP )()(
(5.28)
1
 
=
4
 
;
5
.
=
(5.22)
,
x
x
[ ]
[ ]
 
;7;3,
.7;3,0
-

5
3
5
1
0)()52(
2
2
dxdxdxxfXP
3
5
x
44
1
.
==+==<<
2
3
 4. *        
 
=
xf
)(
 
*
. $  '   
(5.24),
   
x
1
2
xe
;0,
2
<
x
.0,0
.
(5.22) –
(5.29)
 
   
-
λ−
x
)(
xf
=
 
xe
x
181
;;0,
λ
[ ]
.;0,0

2
1
λ=

2
(
)
β
α
(
)
=
(
)
)(
)(
8882
,03944
,04938
,0)25,1()5,2
(
=+=+=
0=m
=
σ
λ
=
, 
1
)(
= XDXM ;
1
2
)32(
2
1
2
1
)(;2
3
4
=
2
5,11
==<< eeeeXP
.
 5. .,  ,   
     
2
  #
16,0)( =xD
. .  ,     
5,3
. &      -
 #.
. 2     -
 ,    1  
,
-
  (5.31)
β
m
σ
 
 
=β<<α
XP )(
 
α σ
m
,
 
5,2)(=XM

 ,
XMm
 ,
  5 (5.32), 
xD=σ
– 
   #   1.
  !   ,  ( -
   
)5,32(
=<< )25,1()5,2(
 
16,0
5,25,3
)
 
 
 
5,22
16,0
XP
==
.
 ,    – 88,82 %,  #  11,18 %.
 6. " '# '  -
  
4
. *  , 
' #   '#,    #­  6 .
182
. .    , 
δ
)(
0=m
=
δ
=
σ
4332
,0)5,1(=
(
)
=⋅=
<
10=σ
,10,
1500
)
(
=σ=
=
XMm
30
1500
30<−<−
X
)
1530
;
1470
(∈X
)(x
f
#      ­  m '  
, 
 (5.33)
δ
,
σ
,
6
,

=δ< mXP 2)(
  5 (5.32).  

6
=<
2)6( XP =
( )
5,12
.
4
 #   1 
XP
8664,04332,026
.
, 
 7. .     
 
1500)(=XM
  -
4
-
 
. * ,    -
 « »     0,9973.
.  « »   (5.34)
.
,9973,0)1031500( =<XP
 
9973,0)3( =σ<mXP

&
.
301500 <X
-'  ,  ,

.
)  859 – 868    
 
. #   A.
183
859.
[
]
6
1
[
]
21
2
[
]
19
[
]
61
[
]
20
1
[
]
60
[
]
32
[
]
31
[
]
243
[
]
21
2
)(x
f
)(x
F
?)2
(=F
2
1
;4;2,
:
.4;2,0
x
xAx
[ ]
=
)(
xf
860.
861.
862.
863.
864.
865.
=
)(
xf
=
)(
xf
xf
xf
xf
xf
 
=
)(
 
=
)(
 
=
)(
 
=
)(
 
xAx
[ ]
x
2
xAx
[ ]
x
2
xAx
[ ]
x
3
xAx
[ ]
x
3
xAx
[ ]
x
4
xAx
[ ]
x
;5;2,
:
.5;2,0
;3;2,
:
.3;2,0
;5;4,
:
.5;4,0
;3;1,
:
.3;1,0
;4;2,
:
.4;2,0
;2;0,
:
.2;0,0
3
3
1
5
4
=
)(
866.
867.
868.
)  869 – 870    -
  

869.
xf
xf
xf
  .
xf
 
=
)(
x
 
=
)(
x
1
 
)(
=
2
 
x
5
5
x
x
xAx
[ ]
xAx
[ ]
xAx
[ ]
[ ]
[ ]
;2;1,
:
.2;1,0
;3;0,
:
.3;0,0
;2;1,
:
.2;1,0
. *    -
;3;1,
.3;1,0
:
5
2
184
870.
?)2
(=F
4
1
)(x
f
(
)
<
<
8
(
)
<
<
63
(
)
<
<
26
(
)
<
<
16
(
)
<<−
81
)(x
f
1
 
)(
xf
=
4
 
[ ]
x
x
[ ]
.5;1,0
;5;1,
:
)  871 – 875   
   X. #   ­ X   .
x
871.
872.
873.
 
xf
)(
=
4
 
2
x
xf
=
)(
21
 
3
x
=
)(
xf
26
 
[ ]
[ ]
[ ]
[ ]
[ ]
;3;1,
.3;1,0
.4;1,0
[ ]
.3;1,0
P
;4;1,
P =?
;3;1,
P =?
20
=?
30
42
:
:
:
3
26
19
x
x
x
x
2
x
x
-
3
x
=
)(
874.
875.
)  876 – 885    
  X    
876.
xf
4
 
3
x
4
xf
=
)(
81
 
=
)(
xf
 
x
x
x
x
xx
[ ]
[ ]
x
[ ]
;0,0
.2;0,0
[ ]
.3;0,0
;0,sin
π
.
2
;2;0,
P =?
;3;0,
P =?
π
;
2
31
21
: 1
:
15
16
:
.
185
877.
2
π
3
2
1
3
1
π
π
2
π
x
sin
xf
)(
=
2
 
x
[ ]
x
[ ]
π
π
;;0,
.;0,0
:
878.
879.
880.
881.
 
π
 
 
 
 
 
;
;0,sin2
3
.
π
 
.
 
.
π
;0,cos
2
.
;
;0,2sin2
4
π
;
;0,3sin3
6
;
:
:
:
:
3π−
12−
 
=
)(
xf
xf
xf
xf
 
x
 
 
=
)(
  
 
 
=
)(
  
 
 
=
)(
  
x
x
x
xx
π
;0,0
3
xx
π
;0,0
4
xx
π
;0,0
6
xx
π
;0,0
2
882.
883.
 
π
 
 
;
;0,cos2
6
.
π
 
.
;
;0,2cos2
4
186
:
:
(
236−+
2/)1
 
=
)(
xf
xf
 
x
 
 
)(
=
 
x
xx
π
;0,0
6
xx
π
;0,0
4
884.
2
π
2
π
)(x
f
[
]
4
3
80
4
9
80
27
5
3
4
15
16
2
9
20
27
)(x
F
 
=
)(
xf
  
x
xx
π
 
;0,0
6
π
 
.
;
;0,3cos3
6
:
(
3/)1
 
)(
=
885.
)  886 – 890     ­    X,   

886.
887.
xf
 
x
.
2
=
)(
xf
 
 
)(
=
xf
 
xx
x
2
x
x
9
x
xx
π
;0,0
8
[ ]
[ ]
[ ]
.1;0,0
 
.3;0,0
π
 
.
;3;0,
;1;0,3
;
;0,4cos4
8
: 4/)1
:
:
(
3
2
3
x
=
)(
888.
889.
890.
)  891 – 895     ­    ,    

xf
64
 
=
)(
xf
 
 
xf
=
)(
 
x
3
x
125
x
2
x
72
x
2
x
[ ]
x
[ ]
x
[ ]
.
[ ]
[ ]
[ ]
;4;0,
.4;0,0
;5;0,
.5;0,0
;6;0,
.6;0,0
: 3
:
:
15
187
891.
<
2
1
<
3
8
9
8
<
3
18
<
<
3
18
49
)(x
F
α
<
4
>
;0,0
x
: 2
;30,
.3,1
x
 
2
x
=
)(
xF
9
x
892.
893.
894.
x
 
2
x
xF
xF
xF
)(
=
16
x
x
 
2
x
)(
=
25
x
x
 
2
x
=
)(
36
x
>
>
>
;0,0
x
;40,
:
.4,1
;0,0
x
:
10
;
50,
25
.5,1
;0,0
x
: 4 2
;
60,
.0,1
895.
x
 
2
x
=
xF
)(
49
x
>
;0,0
10
;
70,
x
:
.7,1
)  896 – 900    
  

.
3
x
;2,0
)(
=
896.
xF
 
>
x
.3,1
xx
. *   
65
;
32,2
:
188
897.
<
<
<
4
27
<
4
),(ba
=
a
=
b
3
4
=
a
=
b
3
1
=
a
=
b
=
a
=
b
3
16
=
a
=
b
3
4
x
 
x
xF
)(
=
2
x
>
;0,0
;
20,
x
: 2
.2,1
898.
899.
900.
x
 
x
=
)(
xF
xF
xF
2
x
x
 
x
=
)(
3
x
x
 
x
=
)(
5
x
;1,0
1
x
;31,
: 10
2
>
.3,1
;0,0
;30,
x
>
:
.3,1
;0,0
125
x
>
;50,
:
.5,1
)  901 – 910     ­   ,  
 
901.
902.
903.
904.
905.
1,
1,
1,
1,
2,
.
5.
3.
7. : 4 3
9.
6.
: 3
: 2
: 5
: 4
189
906.
=
a
=
b
3
16
=
a
=
b
=
a
=
b
3
25
=
a
=
b
3
1
=
a
=
b
3
4
-
 #   #'  .
#-
)(x
f
2
1
4
1
4
1
16
2,
10.
: 6
907.
908.
909.
910.
3,
3,
4,
5,
9. : 6 3
13.
6.
9.
: 8
: 5
: 7
911. 6  '  #  0,5. 
*  ,    #  ' ,  ' 0,1.
: 0,4
912. 6  '  #  0,4. -
 #   #'  . *  ,    #  '#­,  ' 0,1.
: 0,5
)  913 – 918     ­   ,   -
   
2
x
xe
x
<
4
x
xe
<
x
;0,2
.0,0
;0,4
.0,0
913.
914.
=
)(
xf
 
=
)(
xf
.
:
1
:
190
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