Сборник задач и упражнений по математике
TRANSCRIPT
2 30 2017 2
« - »
: . . – . .- . , , . - -
. . − ..- . , , . ..
. . []: . .2 / . . , . . , . . , . . , . . , . ; . . . - . - – . : , 2017. – 71 . ISBN 978-5-528-00224-8 , , , - . , , , - . . - «64 » ( . . , . . , . . , . . ).
22.172
ISBN 978-5-528-00224-8 © . . , . . , . . , . . , . . , . , 2017 © , 2017
3
§ 3. 8
§ 4. 9
§ 5. , 10
§ 6. 11
11. 14
§ 1. 14
§ 2. 19
§3. 23
§4. - 24
12. 26
§2. 27
§3. . - 30
§4. 31
§5. . - 32
13. 33
§ 1. 33
4
§ 4. . 37
§ 5. . 39
§ 6. , -, 42
14. 44
§1. 44
§2. 46
§3. 47
15. 49
§1. . - 49
§2. - 50
§3. 51
54
10.1−−−−10.11 , -
( – - ).
10.1. . 10.2. .
10.3. xy cosln= xy tg=′ . 10.4. xeCy 4−= 04 =+′ yy .
10.5. 3xCy = yxy ′=3 . 10.6. .
10.7. xeCy 3= 03 =−′ yy . 10.8. .
10.9. . 10.10.
dy yx .
10.12. ( )xy φ= : ttex = , tey −= . - ,
( ) 01 2 =++ y dx
10.13−−−−10.18 - ( – ).
10.13. . 10.14. . 10.15. 022 =−+ Cxyx .
10.16. xCxy cossin += . 10.17. . 10.18. .
10.19. , , : ) , ) .
10.20. , - , a2 .
25xy = yyx 2=′ 2
( ) 022 =+−′⋅− yxyyx
21,,
6
10.21. , - .
10.22−−−−10.24 , - .
10.22. , . 10.23. , , .
10.25−−−−10.27 - . - .
10.25. . 10.26. . 10.27. .
10.28−−−−10.32 - , - .
10.28. , . 10.29. , . 10.30. ,
§ 2.
10.33−−−−10.55 ( ) - .
10.33. . 10.34. . 10.35. . 10.36. 0=+′ x
y y .
10.51. . 10.52. .
10.53. . 10.54. .
yx =− 22 ( ) 30 =y xexCCy 2 21 )( += ( ) 10 =y ( ) 00 =′y
xxx eCeCeCy 3 2
2xy =′ yxy +−=′ 1−=′ xy
( )1;0 −M yxy 2+=′ ( )0;3M xyy −=′ ( )2;4M
y
( ) ( ) 011 =−−+ dyxdxy xydydxy =⋅+12
0)( =−′+ yyxxy yyy ln′=
dxxyydyxydyxdx 22 3266 −=− ( ) xxyyy sin1 =+′
0222 22 =′−++ yxxyx 0 1
tg 2
1 2 =
10.56−−−−10.70 ( ) .
10.56. , . 10.57. , ( ) 4
π 1 =y .
0=+ ydx , . 10.64. , .
10.69. , . 10.70. ( ) ( ) 011 426 =+−+ dyxydxyx , .
10.71. , ( )2;2− , , - , .
10.72. , ( )1;1 −− , , Ox , ..
10.73. 0N . , - ( - k>0 ). ( )tN - . .
10.74. , . 5 , , 1 8 , 3 – 40 ?
10.75. m . , ,
VF α−= , 0>α - . -
, .
022 =− dxydyx 3
xdxyydyx sincossincos ⋅=⋅
02 =+ dxdye yx ( ) 00 =y ydxdyyx =ln ( ) 11 =y
( ) 012 =++ dxxdyex ( ) 00 =y 0=+ ′ ye
x
8
10.76. , - .
5. cV /200 = . 10 . -
.
§ 3.
10.77−−−−10.94 ( ) - .
10.77. . 10.78. . 10.79. .
10.83. . 10.84. . 10.85. .
10.86. . 10.87. . 10.88. .
10.92. .10.93. . 10.94. 2
10.95−−−−11.102 ( ).
10.95. , ( ) 01 =y . 10.96. x
y xyyx tg⋅=−′ , ( )
10.101. ( ) 023 22 =+− dxxydyxy , ( ) 10 =y . 10.102. ( ) yxyxyxy ′⋅+=+ 2222 ,
( ) 21 =y .
24 2
( ) 04 =y ( ) 10 =y
x
yx + .
10.104. ( )1;1− . . .
10.105. , ( )2;1A , - , , - .
§ 4.
10.106−−−−10.117 -
.
10.106. 2
2
ln2 .
10.118. I R, - L E
ERI dt
dI L =+ .
( )tII = , tAE ωsin= ( L , R , A - ).
10.119−−−−10.130 ( ).
10.119. 2x x
12 =
=−− −
π =
y .
10.131. I R, - L E
.
, ( L, R, k - ), k – .
§ 5. ,
10.132−−−−10.156 -
.
10.140. . 10.141. . 10.142. .
10.143. . 10.144. . 10.145. .
10.155. . 10.156. ( ) 0 1
x x
1 += +
x xyy
xy 3cos=′′ x
( ) 12 2 −′=′′′ yyyx
( )2)(1 yyy ′+′=′′ yyy 23 =′′′
§ 6.
10.174–10.186 -
, .
10.174. xx ee 2, − 10.175. xx ee −, 10.176. x,1 . 10.177. xx exe , .
10.178. xx 3cos,3sin . 10.179. xexx ,cos,sin . 10.180. xxx exee 22 ,, .
10.181. . 10.182. 1,2cos,2sin xx . 10.183. 2,,1 xx . 10.184. xe− , ,xe
xx 2cos,2sin . 10.185. xxexe xx cos,sin,, −− . 10.186. xxx ,1,3cos,3sin .
10.187–11.206 - .
xy 2tg=′′ ( ) 00 =y ( ) 00 =′y 3
6
y 2cos
( ) 00 =y ( ) 30 =′y 3xyyx =′−′′ ( ) 01 =y ( ) 01 =′y
( ) 01 =′++′′ yey x ( ) 30 =y ( ) 20 =′y yxxy ′=′′ ln
( ) 2=ey ( ) 4=′ ey x
yyx 1=′+′′ ( ) 41 =y ( ) 01 =′y
0 sin
1 tg =+′−′′⋅
x yyx
093 =+′′ yy ( ) 11 =y ( ) 31 =′y 1643 −=′′ yyy ( ) 220 =y
( ) 20 =′y ( ) ( )yyy ′′−=′ 12 2 ( ) 20 =y ( ) 10 =′y
( ) 00 =y ( ) 30 =′y ( )22tg yyy ′=′′ ( ) 2
0 π=y
10.196. . 10.197. . 10.198. +′′+′′′ yy 4
013 =′+ y . 10.199. 0=′′+′′′ yy . 10.200. 0=′+′′′ yy . 10.201. 0=+′′′ yy .
10.202. 02 =−′′+′′′ yyy . 10.203. 0=′′′+′′′′ yy . 10.204. 0=′′+′′′′ yy .
10.205. 0=′+′′′′ yy . 10.206. 0=+′′′′ yy .
10.207 – 10.215 , - .
10.207. , , . 10.208. ,
. 10.215. , , .
10.216−−−−10.235 - , - .
10.216. . 10.217. . 10.218. +′+′′ yy 4
05 =− y . 10.219. . 10.220. .
x3cos2= . 10.224. . 10.225. .
10.231. . 10.232. xeyy −=′′+′′′ . 10.233. xyy 6=′′+′′′ .
10.234. xxeyy −=−′′ 2 . 10.235. xyy cos=−′′′′ .
065 =+′−′′ yyy 056 =+′−′′ yyy 096 =+′−′′ yyy
06 =′−′′ yy 09 =−′′ yy 09 =+′′ yy
0106 =+′−′′ yyy 0=+′+′′ yyy 04 =+′′ yy
032 =′−′′−′′′ yyy 02 =′+′′+′′′ yyy
03 =′−′′ yy ( ) 30 =y ( ) 20 −=′y 09 =−′′ yy ( ) 30 =y
( ) 30 −=′y 025 =+′′ yy ( ) 00 =y ( ) 10 −=′y
( ) 40 =y ( ) 30 −=′y 0168 =+′−′′ yyy ( ) 00 =y
( ) 50 −=′y 042 =+′+′′ yyy ( ) 10 =y ( ) 00 =′y
xeyyy −=+′−′′ 1023 xyyy 222 =+′−′′
xxeyyy 244 =+′+′′ xyyy cos2 =+′+′′
xeyy 323 −=′+′′ xyy 3sin22 =′−′′
xyy cos=+′′ xyy 2sin=+′′
xeyyyy 2485 =−′+′′−′′′ xyy 2−=′−′′′
10.236–10.248 - , .
10.236. 1223 +=+′−′′ xyyy , , . 10.237. =+′−′′ yyy 34
x−=1 , , ( ) 20 =′y . 10.238. 265 2 +=+′−′′ xyyy , ( ) 00 =y , ( ) 40 =′y .
10.239. 26 +=−′−′′ xyyy , ( ) 00 =y , ( ) 30 =′y . 10.240. 33 +=′+′′ xyy ,
( ) 00 =y , ( ) 30 −=′y . 10.241. 12 2 −=′−′′ xyy , ( ) 00 =y , .
10.242. xxeyy 4=+′′ , ( ) 20 −=y , ( ) 00 =′y . 10.243. xyy sin4=+′′ ,
( ) 10 =y , ( ) 20 =′y . 10.244. xyy 2sin−=′+′′ , ( ) 1=πy , ( ) 1=′ πy .
10.245. xyy 3cos69 =+′′ , ( ) 10 =y , ( ) 30 =′y . 10.246. =−′+′′ yyy 32
= xex248 , ( ) 10 =y , ( ) 2
3 0 −=′y . 10.247. xyy 2−=′+′′′ , ( ) 00 =y , ( ) 10 =′y ,
( ) 20 =′′y . 10.248. xeyy 8=−′′′′ , ( ) 10 −=y , ( ) 00 =′y , ( ) 10 =′′y , ( ) 00 =′′′y .
10.249–10.260 - .
10.249. x
yy 2sin
10.252. x
yy 3cos
12 + =
x
ex
ln 96 =+′+′′ .
10.260. xx eeyy 22 1−=′−′′ . 10.261–10.270 , -
.
( ) 00 =y ( ) 10 =′y
10.265. xexyy −+=′′+′′′ 6 . 10.266. xxeyy x cos+=−′′′′ .
10.267. x
eyy x
+−=+′−′′ .
10.269. 22cos xxyy +=′+′′ . 10.270. xxyy 2sin4 =+′′ . 10.271 – 10.279 () -
.
10.271.
11
15
11.1−−−−11.17 () , -
.
11.1. D — .
11.2. D — : .
11.4. D — : .
11.6. D :
11.11. − .
11.18–11.25 ,
D − , , .
11.18. 11.19. y y
. . . . . . . . . . . x . . . . . x . .
11.20. 11.21. . y . y . . . . . . . . . x . . . . . x . . . . . 11.22. 11.23. . y . y . . . . . . . . . . x . . . . . . x . . 11.24. 11.25. y y
. . . . . . . . . . . x . . . . . x . . . .
11.26 – 11.35.
, D - , , , - .
11.26. y 11.27.
( )∫∫ D
dxdyyxf ,
( )∫∫ D
dxdyyxf ,
17
. . . . . . . . . . x . . . . . x . . . . 11.28. y 11.29 , . . . . . . . . . x . . . . . x . . 11.30. y 11.31. y . . . . . . . . . . x . . . . . x . . . . 11.32. y 11.33. . . . . . . . . . . x . . . . . x . . . . 11.34. 11.35. y . . . . . . . . . . x . . . . . x . . . .
11.36 – 11.43
( ), - D - .
( )∫∫ D
dxdyyxf ,
18
. . . . . . . . . . . . x . . . . . . x
11.44. ( )∫ ∫ −3
.
11.76. ( )∫ ∫ 1
0 0
11.78. ∫ ∫ 4
0 0
22 y
dxdyyxf ,
D , .
11.96. ∫∫ D
= =
.
dxdyyxf ,
11.116. ∫∫ + D
11.124. , D:
∫∫ +D yx
dxdy 22
11.130. ( )∫∫ + D
11.133. ( )∫∫ −
+ D
11.138. 05,4 =−+= yxxy . 11.139. 6,4 2 =+−= yxyyx .
11.140. ( ) ( )0,14, 2
11.142. 2,,1 === xyxxy . 11.143. 4,4, 22 === yxyxy .
11.144. 2,4,1 === yxxy . 11.145. .
11.150. ( )0,0,cos,sin >=== xxxyxy . 11.151. 0,02,2 ==−+= yyxxy .
11.152 – 11.158 , ( - ).
11.152. )0(,2, 2222 ≥=+=+ yxyxxyx . 11.153. xyx 322 ≤+ ,
yyx 322 ≤+ . 11.154. 0,3,322 ===+ xxyyyx . 11.155. xyx 422 =+ ,
)( xy ≥ . 11.156. ( )2,4,0 2 ≥−== yyyxx . 11.157. ( )ρ cos12 −= .
11.158. ( )ρ cos12 += , ρ cos2= .
11.159. , 2=ρ -
( )ρ sin12 += .
11.160. – 11.172. , - .
11.160. 42 =++ zyx , 0,0,0 === zyx .
11.161. ,1,3 22 =++= yxyxz 0,0,0 === zyx .
11.162. ,4 2xz −= 0,5,0 === zyy .
11.163. ,2,2 =+= yxyz 0,0,0 === zyx .
11.164. xyxyzxz 2,,6,0 ===+= .
1,1 2 +==+ xyyx
11.167. 3,222 ≥+++= yxyxz , 3,3,0,0,0 ===== yxzyx .
11.168. ,6,122 xyyxz −=++= xyyz 2,1,0 === .
11.169. 0,0,9, 3
11.170. 1622 =+ yx , 0,,0 === zyzy .
11.171. 4=++ zyx , 0,422 ==+ zyx .
11.172. 10=++ zyx , 0,82,42 ==+=+ zyxyx . §4. -
11.173. , : , ,
, , ( )yx,ρ
, .
11.174. )1(ρ = , -
: .
11.175. , xyxy == ,2 ,
ρ ( )yx, ( ) yxyx 2, +=ρ .
11.176. R, ( )yx,ρ -
.
: .
: )0(,0,4 ≥=−= xyxy .
: 0,2,2 ==+= yxyxy .
1−=x 2=x
: , , , .
: 4,4,2 22 === xxyxy .
11.182. -
)1(ρ = , : 4,2,1,4 ==== xxxyxy .
11.183. -
, ( )1=ρ .
11.184. - )1(ρ = , : , , .
11.185. )1(ρ =
OX, : , , , .
)1(ρ = , : , , , .
)1(ρ = , : 0,42 =−= yyx .
12
12.1. . 12.2. . 12.3. +++ !3
2yx = 24 yx = 4=x 0≥y
1 94
xy =2 xy 42 = 1=y 3=y
xy =2 xy 42 = 1=y 3=y
+++ 7
6
5
4
3
2 +++
27
6
9
4
3
2
⋅ ++
⋅ +++
12.21. . 12.22. . 12.23. .
12.24. . 12.25. . 12.26. .
12.27- 12.34 , . - .
12.27. . 12.28. + ⋅
.
12.35. . 12.36. . 12.37. .
12.38. . 12.39. . 12.40. .
12.46- 12.61 , (
:
.
12.72- 12.83 , -
.
12.72. . 12.73. . 12.74. .
12.75. . 12.76. . 12.77. .
12.78. . 12.79. . 12.80. .
12.81. . 12.82. . 12.83. .
12.84- 12.91 , -
.
12.84. . 12.85. . 12.86. ∑ ∞
12.90. . 12.91. .
12.92 — 12.106 .
+++ 32 2
12.107. . 12.108. ( )
12.113. ( ) ( )∑
12.121. . 12.122. .
12.123. . 12.124. .
12.125. . 12.126. .
12.133. . 12.134. . 1 2.135. .
( ) .
12.151. . 12.152. . 12.153. .
12.154. . 12.155. . 12.156. .
12.157. . 12.158. . 12.159. .
12.160. . 12.161. ( ) n
n
nxn∑ ∞
=1 .
§5. . -
12.163- 12.174 -
, xe , xsin , sx, ( )x+1ln , ( )αx+1 .
12.163. . 12.164. . 12.165. . 12.166. .
12.167. . 12.168. . 12.169. . 12.170. .
12.171. . 12.172. . 12.173. . 12.174. .
( ) ∑ ∞
=
−⋅ +
( )α−xcos
1
12.177. , . 12.178. , .
12.179. , . 12.180. , .
12.181. , . 12.182. , .
12.183. , . 12.184. , .
12.185. , . 12.186. , .
12.187. , . 12.188. , .
12.189- 12.198 δ .
12.189. , . 12.190. , . 12.191. ,
12.194. 3 70, . 12.195. 5 40, . 12.196. , .
12.197. , . 12.198. , .
.
12.203. , . 12.204. , .
1
80 π=x
( ) xexf 2= 40 =x ( ) 2
x
001,0=δ 1sin 0001,0=δ
001,0=δ 01,0=δ 3 500 001,0=δ
e
01,0=δ ∫ −4,0
x
§ 1.
13.1. - . , ?
13.2. 8 ?
13.3. ( 1, 2 3 ) - 10 ?
13.4. 8 ( ). ?
13.5. 8 3 ( ). ? ( - ).
13.6. , , . - ?
13.7. . , - ?
13.8. 12 - , 3 ?
13.9. « » , ?
13.10. . ?
13.11. 2, 3, 5, 7, ?
13.12. 0, 1, 2, 3, 4, ?
35
13.13. 7 , – 9. - ?
§ 2. . 13.14. 2 . , : 1) 7; 2) 8, 4; 3) 8, , 4; 4) 5, 4.
13.15. 2 . , .
13.16. 6 , . - . , .
13.17. 1, 2, 3, 4, 5, – 8, 7, 8, 9, 10. . , : 1) 7; 2) 11; 3) 11?
13.18. , , 1000 , - , . , : 1) ; 2) ; 3) ; ) .
13.19. . , : 1) ; 2) , .
13.20. 4 5 . . , .
13.21. 4 5 . . . - . , .
13.22. 4 5 . , , . . . , , , .
13.23. , , , – . - ,
13.24. 20- 3 - . , : 1) ;
36
2) ; 3) .
13.25. , 4- 1 1 ?
13.26. 3 . - , , - , .
13.27. 90 - ( ). , - .
13.28. 86 ( ). , , 90 ( : 90, 89, 88, …, 6, 5).
13.29. «6 49» ( 6 49 ) : 1) ; 2) .
13.30. 15 , 10 - . . , : 1) 5, .. ; 2) 4, .. - ; 3) 3, .. - ; 4) 2, .. - .
13.31. 20 25. . , , . . , : ) ; ) ?
13.32. 100 . . 10 . , .
13.33. 1000 : 500 500 . - . , : 1) - ; 2) , – .
13.34. 50 28 . 15 . , 7 ( ).
13.35. 5 , . - . , - : 1) ; 2) - ; 3) .
37
13.36. 8 . , 2 , : 1) 8; 2) 12.
13.37. 5 ( 52 ). , : 1) ; 2) – 5 ; 3) – 4 .
§ 3. 13.38. OA L Ox
)(xB . , OB BA - , , 3/L . , .
13.39. R r . , , , . , .
13.40. 2 ( 51 =R , 102 =R ). . , - , . , - .
13.41. R . R . - . . , - .
13.42. - , . . , . , - .
13.43. , a2 . - ar < . , .
38
13.44.
2
a r < . , -
. , - .
13.45 12 13 . , . , , ( 12 13 ).
13.46. , - . , xy
, .
§4. .
13.47. , - . - : 8,0,9,0 21 == pp . . .
13.48. R . 4 . , : ) 4 ; ) - ; ) - .
13.49. . , , 0,1. - , .
13.50. 20 25 . - , .
13.51. 10 , 4 . . , - .
13.52. . 0,7, – 0,8. , .
a
4/1
i
39
13.53. 0,9, – 0,6, – 0,3. . , , ?
13.54. , , 0,03. , 4 ?
13.55. . . 0,15, - 0,3. - , .
13.56. . , , , : 0,3, 0,4, 0,6, 0,7.
13.57. , , . , - , 0,1. 0,15 0,2. , .
13.58. 10 6 3 . , : 1) ; 2) .
13.59. . - 0,2, – 0,6. ?
13.60. - 0,8. , .
13.61. 15 5 . - . , .
13.62. . 0,5 0,4. , , .
13.63. 100 . . . - , ?
40
13.64. . - . , . ?
13.65. 40 50. . . - , : 1) ; 2) ; 3) ; 4) .
§ 5. .
13.66. , , 5 . - 0,4; 0,8; 0,3; 0,2; 0,1. ?
13.67. 6 4 - . , , 0,95, – 0,8. . , - .
13.68. 2 . , - . , ( ).
13.69. 5 , 3 . , – 0,95, – 0,7. , , .
13.70. 10 , 8 ; 20 , 4 . - , . , .
13.71. 10 , 1,01 =p , – 4,02 =p . , .
13.72. 10 , . - , - , 0,95; 0,8. . : - ?
41
13.73. 45% , , 1- , 15% - 2-, – 3- . , , , - , 0,96, 0,84, 0,90 . - , .
13.74. - − . , 0,7, ( ) 0,92, - 0,65 . .
13.75. . 0,6 , − 0,7 , − 0,8 0,2, – 0,6 , – . - .
13.76. - . - . - : 1- , 2- − 2- 4 – , 3- − 3- 4- , 4- − 4 – . - 33,7% , 37,5% − , 20,9% − 7,9% − . , .
13.77. 1-, 2-, 3- 4:3:2. - 4% (.. 0,04), − 7%, − 9%. ?
13.78. 70%- . 20% 40% . - . , 1) , 2) -
13.79. , . - . 60% - , – 84%. . , - .
13.80. , , , , , 3:2. , -
42
, 0,1, – 0,2. - . , .
13.81. - . , - , 0,05, – 0,1. . , (- – ).
13.82. 10 , . - , - , 0,95; 0,8. . : - ?
13.83. 5 . , - 2- («») 0,9, («») 0,5. . : «» «»?
13.84. , 5% 0,25% . . , , , : 1) , 2) .
13.85. 97% - . - (.. ) 6% . . . - - , - ?
§ 6. , -,
13.86. 5 . , «» : ) ; ) ; ) .
13.87. . : 2 4- 3 6- ( - ).
13.88. . : ) 2- 2 4-; ) 2- 4- 3- 5-?
.
43
13.89. , . 0.1. .
13.90. ) , A 3- 4- , A 0,4. ) B , A 4- . B , 5 - , A 0,8.
13.91. AB C 2:1. 4 . , 2 C , – ( - ).
13.92. 0,4. , 5 ?
13.93. , 600 250 , 0,4.
13.94. , 1400 2400 - , 0,6.
13.95. 0,8. - , 100 75 .
13.96. 100 - 0,8. , : ) 75 90 ; ) 75 ; ) 74 .
13.97. - 0,7. , : ) - 1470 ; ) 1470 1500 ; ) 1470 .
13.98. 10% . , 400 349?
13.99. 10000 . - 100 .
A
2100
44
13.100. , , 0,004. 50 . , .
13.101. 60 . , 2 ?
13.102. , , 0,001. , 5000 .
13.103. - 10. , 60 3 ?
13.104. - 0,01. , 500 .
13.105. 1000 . - , , 0,003. - , .
14
§1.
14.1. 0,7. - L – . .
14.2. 0,4, 0,7, 0,8 . L – .
14.3. 5 3 . 3 . - .
14.4. , 4 , - , . - 0,7, 0,1. , - .
45
14.5. 0,8 0,7 . 5 . - , .
14.6. ( )cbaX ;;= ( )ba ; -
, ( )
,
− ⋅ ≤ −= − ⋅ ≥ −
, - . .
14.7. ( )0;1;1 +−=X ( )1;1 +−
( ) ( ) ( )
≥−⋅ ≤+⋅
, - . .
14.8. ( )π;0
( ) xAxf sin= .
( )xF . ( )xf , ( )xF -
14.9. ( )2;0
( ) Axxf = .
( )xF . ( )xf , ( )xF
( )5,1;5,0 .
14.10. ( )∞;0
( ) xexf λλ −⋅= , 0λ > . -
( )xF ( )xf , ( )xF . 1λ = -
( )2;0 . -
0,1.
46
14.11. ( )10;0 . ,
15=Z ? Z ( )20;0 .
14.12. 5 . - . - , 3- ?
14.13. , ( )λEX = , -
15 . , 20 .
14.14. -
2Y X= , ( )1;0RnX = − , (0,1).
§2.
14.15. 5 3 . 4 . - . , - .
14.16. - p=0,2. L – - . , .
14.17. - p . 1- -
. L – - . , .
47
( )ba ; . -
.
14.19. ( )1;1 +−
( ) ( )21 xAxf −⋅= . -
, - .
14.20. ( )1;1 +−
( )
, - .
14.21. ( ; ; )X a b c=
( )ba ;
, ( )
,
− ⋅ ≤ −= − ⋅ ≥ −
, - .
14.22. ( )X E λ=
(0; )∞ ( ) xexf λλ −⋅= , 0>λ . -
- , .
14.23. (0;3;1)X = ( )3;0 -
( )
≥− ≤
14.24. ( )0;3;3 +−=X
( )3;3 +−
48
x x
A xf
, - 0.9?
14.25.
( ) 22 xRAxf −⋅= . ,
, .
§3.
14.26. -
( ) ( )
a σ .
14.27. - , 60 / 10 /. , 80 /. - 50/ 70 /.
14.28. , , 15 1,442. - , - 2.
14.29. - 20 . , 15 .
14.30. 6 9 . 8 ? - .
14.31. - 5 0,81 2. - 0,99.
49
14.32. 15. - 0,04 2. 0,99.
14.33. ( )..50.,.350NX = . - 250.. , «». , , - , .
14.34. - 2XY = , ( )1;0NX = .
14.35 :
xi -1 0 1 pi 0.3 0.2 0.5 . )()()( YMXMYXM ⋅=⋅
14.36. : xi -4 0 4 pi 0.25 0.5 0.25 .
)()()( YDXDYXD +=+
§1. . - .
15.1. n=120, - , . - x2=4 ?
yi 0 1 3 pi 0.1 0.3 0.6
yi 2 4 6 pi 0.2 0.3 0.5
50
15.2. fj n=20. 4 6.
15.3. 110=n :
n6?
jf
51
§2.
15.5. :
?
15.6. :
?
15.7. :
mx - 0,95?
15.8. :
σx 0,99?
15.9. , - 15. ,
52
5 0,9.
15.10. ( ) ( ): 31; 33; 35; 36; 37. …
15.11. (32,6;41,1) . ?
15.12. (22,15;23,65) - . - …
15.13. 16 - 64 2. 0,99 - .
15.14. 14
2. 4 . .
§3. .
15.15. - ( )..50.,.350NX = .
. , , , 0,05.
15.16. 16 - 5,7 64 2. 0,01 , 8.
15.17. 16 - 5,7 72 2. 0,01 , - 642.
15.18. 15 25 . 15 16,3 , - 20 18.
53
- 0,05.
15.19. 20=n
0,05 - - . 4.
15.20. , 16 - - 0,5. 0,05.
15.21. - , D=0,81; D=0,64. : y+0,28x=1. ρ ?
15.22. - 25 36 15 , 0,75. 0,01.
15.23. , 60 105 , 69 140, 63 125 105 160. 0,05 - .
54
xy ++= 1 ln .
10.41. C x
− −
10.44. Cxy ln12 =+ . 10.45. Cyx ++= 3arctg 2 . 10.46. ( ) Cyyx ln2 =− .
10.47. Cx y +=
2
( )21
2
−= x
10.61. ( ) xy cos12 =− . 10.62. xy coscos = . 10.63. xey arccos= .
10.64. xy= . 10.65. x y
e−−=− 1
( )32 +− − xe x . 10.68. ( ) 2
1 1
10.70. 12
π arctg
x y
−=
y
x
=+ Cx
y
x
y
x
1= . 10.101. 322 yxy =− .
10.105. xxxy ln2 −= . 10.106.
10.125. ( )1−= − xey x . 10.126.
−⋅=
10.133. 213cos 9
10.135. 43
3 1
21 2
ln 1
211 14 CxCyC +=− . 10.151. Cy = ;
212 CxCy += . 10.152. ( )1 2arcsin Cey Cx −= + . 10.153. ( )12 cos CxCy +−= .
10.154. ( )211 sec CxCCy += . 10.156. 211
1 CxC
π cosln +−−= xxy .
10.160. xey 2
1
48
24
57
2
y −
=− 1
1 1 .
10.172. xy 3tg = . 10.173. xy −=ctg . 10.187. xx eeCy 2 2
3 1 += .
3 1
=
21
xeC −+ 3 . 10.197. ( )xeCy x 321 ++= − . 10.198. ( ++= − xeCy x 3cos2
2 1
10.201.
( )xxe x 3sin3cos 32 ++ − . .10.203. xeCxCxCy −+++= 4 2
321 .
10.209. xey x 2sin 2
1 −= . 10.210. xey 3
10.212. xy 5sin 5
10.215.
3
1 ++= − xx eeCy .
10.219. ( ) ( )12 32
58
2 −− −+= . 10.222. ++= xeCCy 2 21
xx 3sin 13
8 3cos
8
3
4
x2sin 3
1− . 10.228. ( )
321 xeCxCy x +++= .
10.231. xxx xeeCeCxxCy 2sincos 4321 ++++= − . 10.232. ++= xCy 21
xx xeeC −− ++ 3 . 10.233. ( )32 321 −+++= − xxeCxCy x . 10.234. += xeCy 1
( ) xx ex x
2
1
3
1
9
20
9
45
34
36
11
6
9
8
6
1
27
35
27
8
17
8
17
10.244. xxey 2sin 5
10.246. ( )5,134 23 +−+= − xxxeey xx . 10.247. 2sincos44 xxxy −+−= .
10.248. xxx xexxeey 2sin2cos3 ++++−= − . 10.249. +
xexC
22 −+ x . 10.263. ( ) xxxxCxCey x sin 25
3 cos
1 sin
xxxexxC x sin 4
1 21
xxxx 2
1 2sin2cos 21
xxxx 2sin 16
32 . 11.81. 18. 11.82.
1 2 −− ee .
11.94. 2π
188− . 11.103. 20
7
11.127. 2 π . 11.128. 12π18 − . 11.129. 4. 11.130.
2
4 + 11.149. 3
2
33 . 11.155. 2π− . 11.156. π . 11.157. π6 . 11.158. π5 . 11.159. 8π+ .
11.160. 3
32 . 11.161.
3
128
81 . 11.174. 4. 11.175.
11.184. , 4
15 51 . 11.187.
( ) 23
12.20. ( )
14
1 −
+= n
n
42 =u , 03 =u . 12.23. 31
2
2
( ) !2
∞= ∞→
n n
Slim , . 12.35. . 12.36. . 12.37. . 12.38. .
12.39. . 12.40. . 12.41. . 12.42. . 12.43. . 12.44. .
12.45. . 12.46. . 12.47. . 12.48. .
12.49. . 12.50. . 12.51. . 12.52. .
12.53. . 12.54. . 12.55. . 12.56. .
12.57. . 12.58. . 12.59. . 12.60. .
12.61. . 12.62. . 12.63. . 12.64. .
12.65. . 12.66. . 12.67. . 12.68. .
12.69. . 12.70. . 12.71. . 12.72. .
12.73. . 12.74. . 12.75. . 12.76. .
12.77. . 12.78. . 12.79. . 12.80. .
12.81. . 12.82. . 12.83. . 12.84. .
65
12.105. . 12.106. . 12.121. .
12.122. . 12.123. . 12.124. .
12.125. . 12.126. . 12.127.
. 12.128. . 12.129. .
12.130. . 12.131. . 12.132. -
. 12.133. 11 <<− x . 12.134. 11 <<− x . 12.135. 11 <<− x .
12.136 11 <<− x . 12.137. −3 < < 3. 12.138. 3
1
3
12.142. −∞ < < ∞. 12.143. −√3 − 4 < < √3 − 4. 12.144. 1 3
5 <<− x .
3 <<− x . 12.146. 1 < < 2. 2.147.−1 < < 0. 12.148. − − 1 <
< − 1. 12.150. −1 < < 1. 12.151. 11 <<− x . 12.152. 11 <<− x .
12.153. 2 ≤ < 4. 12.154. −3 ≤ ≤ 1. 12.155. −1 < ≤ 1.
12.156. −3 ≤ ≤ −1. 12.157. −5 ≤ < 3. 12.158. 13 ≤<− x .
66
12.163.
12.176. ( ) ( ) ( ) ( ) +−−+−−+−− −+ 112 32
12.179. ( )
12.192. 0,0175. 12.193. 0,158. 12.194. 4,125. 12.195. 2,09. 12.196. 7,937.
12.198. 0,7788. 12.199. 0,125. 12.200. 0,452. 12.201. 0,192. 12.202. 0,097.
12.203. 0,118. 12.204. 0,498. 12.205. 0,098. 12.206. 0,499.
13
13.1. 120. 13.2. 40320. 13.3 720. 13.4. 336. 13.5. 56. 13.6. 120.
13.7. 83. 13.8. 3 6
3 9
3 12 CCC ⋅⋅ . 13.9. 120. 13.10. 24. 13.11. 6. 13.12 96.
13.13. 756. 13.14 1) 1/6; 2) 1/18; 3) 0,5; 4) 1/18. 13.15. 0,75.
13.16. 1/720. 13.17. 1) 1; 2) 0,2; 3) 0,6. 13.18. 1) 0,008; 2) 0,096;
3) 0,384; 4) 0,512. 13.19. 1) 1/90; 2) 1/81. 13.20. 4/9. 13.21. 0,375.
13.22. 1/6. 3.23. 0,1. 13.24. n
m p = , ( ) ,
28 1 4
1 4 CCCm ⋅⋅= .
!4 . 13.29. 1) 6
CC 01765,0≈ .
13.30. 1) 24/91; 2) 45/91; 3) 20/91; 4) 2/91. 13.31. ) 62,0 ; ) 0,38.
13.32. 0,1. 13.33. 1) 499/1998; 2) 500/999. 13.34. 15 50
8 22
7 28
C CC .
13.35. 1) 0,6; 2) 0,3; 3) 0,9. 13.36. 1) 0,25; 2) 1/6. 13.37. n
m p = ,
44 2 42 CCm ⋅= ; 2) 5
134 Cm ⋅= ; 3) 62413 1 48 =⋅= Cm .
13.38. 1/3. 13.39. 22 / Rr . 13.40. 0,75. 13.41. π2
1 . 13.42. 0.5.
−− . 13.49. 0,18.
13.50. 5,0≈ . 13.51. 5/6. 13.52. P=0,38. 13.53. 0,306. 13.54. 0,88.
13.55. 0,955. 13.56. 0,95. 13.57. 0.388. 13.58. 1) 3/14; 2) 1/4. 13.59. 0,2.
13.60. 0,384. 13.61. 0,009. 13.62. 0,7. 13.63. 0,002. 13.64. 6,01 = ;
4,02 =P . 13.65. 1) 0,2; 2) 0,327; 3) 0,393; 4) 0,43. 13.66. 0,36.
13.67. 0,89. 13.68. 2/3. 13.69. 0,85. 13.70. 0,5. 13.71. 1/3. 13.72.
. 13.73. 0,918. 13.74. 0,95877. 13.75. 0,6448.
13.76. 0,574. 13.77. % 9
13.80. 3/7. 13.81. 1/3. 13.82. . 13.83. ,
«» . 13.84.: 1) 20/21; 2) 1/21.
13.85. 0,942. 13.86. ) 5/16; ) 3/16; ) 13/16. 13.87. 2 4.
13.88. ) ; ) 2 4. 13.89. 93,0≈ . 13.90. ) 0,18; ) 0,74. 13.91 8/27. 13.92. 0,922. 13.93. 0,024. 13.94. 0.004.
≈
) 4236,0≈ ; ) 5,0≈ . 13.98. 952,0≈ . 13.99. 0,0905. 13.100. 0,819.
13.101. 0,814. 13.102. 0,96. 13.103. 0,0308. 13.104. : 0,96. 13.105. 0.5.
14
14.6. ab
= 2 . 14.7. 1=A . 14.8. 5,0=A ; ( ) ( )xxF cos15,0 −= .
14.10. 11,1ln1,0 =x . 14.11. 0,875. 14.12. 0,4. 14.14. ( ) yyF 2= ; ( ) 1/f y y= .
14.15. 5,2=xm ; 784,6=xD ; 731,0=xσ . 14.16. pmx 5= ; ( )ppDx −= 15 .
14.17. p
mx 1= ;
0=xm ; 133,0=xD . 14.20. 0,5A = ; 0xm = ; 0,133xD = . 14.21. ab
A −
= 2 ;
3
λ 1=xm ;
λ =xD ;
λ 2
ln5,0 =x . 14.23. 667,0=A ; 333,1=xm ; 389,0=xD ; 268,15,0 =x .
14.24. 333,0=A ; 051,29,0 =x . 14.25. 2π
2
2R Dx = .
14.26. ,4=a 5=σ . 14.27. 0,977; 0,683. 14.28. 0,905. 14.29. 0,547.
14.30. 0,62. 14.31. 7,097. 14.32. 0,516. 14.33. 0,023.
14.34. ( ) ( )yOyF ˆ21+= ; ( ) ( )2/expπ2/1 yyyf −⋅= .
15
15.1. 375,02 =ν . 15.2. 63 =n . 15.3. 106 =n . 15.4. w2=0,2. 15.5. 5,1.
15.6. 2,808. 15.7. −0,777< xm <10,977. 15.8. 1,97< xσ <4,8. 15.9.
27. 15.10. 34.4. 15.13. 5,59< xσ <14,92. 15.14. 8. 15.15. 267,76.
15.16. , t−; −2,471<1,114< 2,471. 15.17. , 2χ -
; 4,6<18< 32,82,471. 15.18. F-
0,359<1,143<2,468; . t- -2,02<0,87< 2,02.
15.20. F- 0,151<2<4,236. 15.21. 0,315. 15.22.
70
t- 2,995<3,01. 15.23. 2χ - -
9,87>7,82.
1. ..
[ ] . 2 . . 1,2 / .. , .. , .
. – 6- . −: 21 , 2003.– 720 .− :http: //vipbook/info/nayka-i-ucheba/matematika/190523-danko-pe- popov-ag-vysshaya-matematika-v-uprazhneniyah-i-zadachah-chasti-12-6-e- izdanie.html
2. .. / .. .– : , 2004. – 336 .
3. : . / . ... – 2- ., .−:-,2008.–575 .
4. .., .., .., .. . 2 / .. , .. , .. , ... - 3- .,. .-−: – , 2003.- 576.
5. .. / .. .−: . ., 2003.-304.
6. .. / .. . −−−− -: , 2003.− 432.
7. .., .., .., .. 64 , 2. ,,2012, -284.
8. .. .—: , 2001, -400.
9. .. . , 2003,- 41.
71
2
6090 1/16 . . . . . 4,1. . . . 4,4. 500 .
____________________________________________________________________________
« - » 603950, , . , 65.
, 603950, ., , 65 http://www. nngasu.ru, [email protected]
« - »
: . . – . .- . , , . - -
. . − ..- . , , . ..
. . []: . .2 / . . , . . , . . , . . , . . , . ; . . . - . - – . : , 2017. – 71 . ISBN 978-5-528-00224-8 , , , - . , , , - . . - «64 » ( . . , . . , . . , . . ).
22.172
ISBN 978-5-528-00224-8 © . . , . . , . . , . . , . . , . , 2017 © , 2017
3
§ 3. 8
§ 4. 9
§ 5. , 10
§ 6. 11
11. 14
§ 1. 14
§ 2. 19
§3. 23
§4. - 24
12. 26
§2. 27
§3. . - 30
§4. 31
§5. . - 32
13. 33
§ 1. 33
4
§ 4. . 37
§ 5. . 39
§ 6. , -, 42
14. 44
§1. 44
§2. 46
§3. 47
15. 49
§1. . - 49
§2. - 50
§3. 51
54
10.1−−−−10.11 , -
( – - ).
10.1. . 10.2. .
10.3. xy cosln= xy tg=′ . 10.4. xeCy 4−= 04 =+′ yy .
10.5. 3xCy = yxy ′=3 . 10.6. .
10.7. xeCy 3= 03 =−′ yy . 10.8. .
10.9. . 10.10.
dy yx .
10.12. ( )xy φ= : ttex = , tey −= . - ,
( ) 01 2 =++ y dx
10.13−−−−10.18 - ( – ).
10.13. . 10.14. . 10.15. 022 =−+ Cxyx .
10.16. xCxy cossin += . 10.17. . 10.18. .
10.19. , , : ) , ) .
10.20. , - , a2 .
25xy = yyx 2=′ 2
( ) 022 =+−′⋅− yxyyx
21,,
6
10.21. , - .
10.22−−−−10.24 , - .
10.22. , . 10.23. , , .
10.25−−−−10.27 - . - .
10.25. . 10.26. . 10.27. .
10.28−−−−10.32 - , - .
10.28. , . 10.29. , . 10.30. ,
§ 2.
10.33−−−−10.55 ( ) - .
10.33. . 10.34. . 10.35. . 10.36. 0=+′ x
y y .
10.51. . 10.52. .
10.53. . 10.54. .
yx =− 22 ( ) 30 =y xexCCy 2 21 )( += ( ) 10 =y ( ) 00 =′y
xxx eCeCeCy 3 2
2xy =′ yxy +−=′ 1−=′ xy
( )1;0 −M yxy 2+=′ ( )0;3M xyy −=′ ( )2;4M
y
( ) ( ) 011 =−−+ dyxdxy xydydxy =⋅+12
0)( =−′+ yyxxy yyy ln′=
dxxyydyxydyxdx 22 3266 −=− ( ) xxyyy sin1 =+′
0222 22 =′−++ yxxyx 0 1
tg 2
1 2 =
10.56−−−−10.70 ( ) .
10.56. , . 10.57. , ( ) 4
π 1 =y .
0=+ ydx , . 10.64. , .
10.69. , . 10.70. ( ) ( ) 011 426 =+−+ dyxydxyx , .
10.71. , ( )2;2− , , - , .
10.72. , ( )1;1 −− , , Ox , ..
10.73. 0N . , - ( - k>0 ). ( )tN - . .
10.74. , . 5 , , 1 8 , 3 – 40 ?
10.75. m . , ,
VF α−= , 0>α - . -
, .
022 =− dxydyx 3
xdxyydyx sincossincos ⋅=⋅
02 =+ dxdye yx ( ) 00 =y ydxdyyx =ln ( ) 11 =y
( ) 012 =++ dxxdyex ( ) 00 =y 0=+ ′ ye
x
8
10.76. , - .
5. cV /200 = . 10 . -
.
§ 3.
10.77−−−−10.94 ( ) - .
10.77. . 10.78. . 10.79. .
10.83. . 10.84. . 10.85. .
10.86. . 10.87. . 10.88. .
10.92. .10.93. . 10.94. 2
10.95−−−−11.102 ( ).
10.95. , ( ) 01 =y . 10.96. x
y xyyx tg⋅=−′ , ( )
10.101. ( ) 023 22 =+− dxxydyxy , ( ) 10 =y . 10.102. ( ) yxyxyxy ′⋅+=+ 2222 ,
( ) 21 =y .
24 2
( ) 04 =y ( ) 10 =y
x
yx + .
10.104. ( )1;1− . . .
10.105. , ( )2;1A , - , , - .
§ 4.
10.106−−−−10.117 -
.
10.106. 2
2
ln2 .
10.118. I R, - L E
ERI dt
dI L =+ .
( )tII = , tAE ωsin= ( L , R , A - ).
10.119−−−−10.130 ( ).
10.119. 2x x
12 =
=−− −
π =
y .
10.131. I R, - L E
.
, ( L, R, k - ), k – .
§ 5. ,
10.132−−−−10.156 -
.
10.140. . 10.141. . 10.142. .
10.143. . 10.144. . 10.145. .
10.155. . 10.156. ( ) 0 1
x x
1 += +
x xyy
xy 3cos=′′ x
( ) 12 2 −′=′′′ yyyx
( )2)(1 yyy ′+′=′′ yyy 23 =′′′
§ 6.
10.174–10.186 -
, .
10.174. xx ee 2, − 10.175. xx ee −, 10.176. x,1 . 10.177. xx exe , .
10.178. xx 3cos,3sin . 10.179. xexx ,cos,sin . 10.180. xxx exee 22 ,, .
10.181. . 10.182. 1,2cos,2sin xx . 10.183. 2,,1 xx . 10.184. xe− , ,xe
xx 2cos,2sin . 10.185. xxexe xx cos,sin,, −− . 10.186. xxx ,1,3cos,3sin .
10.187–11.206 - .
xy 2tg=′′ ( ) 00 =y ( ) 00 =′y 3
6
y 2cos
( ) 00 =y ( ) 30 =′y 3xyyx =′−′′ ( ) 01 =y ( ) 01 =′y
( ) 01 =′++′′ yey x ( ) 30 =y ( ) 20 =′y yxxy ′=′′ ln
( ) 2=ey ( ) 4=′ ey x
yyx 1=′+′′ ( ) 41 =y ( ) 01 =′y
0 sin
1 tg =+′−′′⋅
x yyx
093 =+′′ yy ( ) 11 =y ( ) 31 =′y 1643 −=′′ yyy ( ) 220 =y
( ) 20 =′y ( ) ( )yyy ′′−=′ 12 2 ( ) 20 =y ( ) 10 =′y
( ) 00 =y ( ) 30 =′y ( )22tg yyy ′=′′ ( ) 2
0 π=y
10.196. . 10.197. . 10.198. +′′+′′′ yy 4
013 =′+ y . 10.199. 0=′′+′′′ yy . 10.200. 0=′+′′′ yy . 10.201. 0=+′′′ yy .
10.202. 02 =−′′+′′′ yyy . 10.203. 0=′′′+′′′′ yy . 10.204. 0=′′+′′′′ yy .
10.205. 0=′+′′′′ yy . 10.206. 0=+′′′′ yy .
10.207 – 10.215 , - .
10.207. , , . 10.208. ,
. 10.215. , , .
10.216−−−−10.235 - , - .
10.216. . 10.217. . 10.218. +′+′′ yy 4
05 =− y . 10.219. . 10.220. .
x3cos2= . 10.224. . 10.225. .
10.231. . 10.232. xeyy −=′′+′′′ . 10.233. xyy 6=′′+′′′ .
10.234. xxeyy −=−′′ 2 . 10.235. xyy cos=−′′′′ .
065 =+′−′′ yyy 056 =+′−′′ yyy 096 =+′−′′ yyy
06 =′−′′ yy 09 =−′′ yy 09 =+′′ yy
0106 =+′−′′ yyy 0=+′+′′ yyy 04 =+′′ yy
032 =′−′′−′′′ yyy 02 =′+′′+′′′ yyy
03 =′−′′ yy ( ) 30 =y ( ) 20 −=′y 09 =−′′ yy ( ) 30 =y
( ) 30 −=′y 025 =+′′ yy ( ) 00 =y ( ) 10 −=′y
( ) 40 =y ( ) 30 −=′y 0168 =+′−′′ yyy ( ) 00 =y
( ) 50 −=′y 042 =+′+′′ yyy ( ) 10 =y ( ) 00 =′y
xeyyy −=+′−′′ 1023 xyyy 222 =+′−′′
xxeyyy 244 =+′+′′ xyyy cos2 =+′+′′
xeyy 323 −=′+′′ xyy 3sin22 =′−′′
xyy cos=+′′ xyy 2sin=+′′
xeyyyy 2485 =−′+′′−′′′ xyy 2−=′−′′′
10.236–10.248 - , .
10.236. 1223 +=+′−′′ xyyy , , . 10.237. =+′−′′ yyy 34
x−=1 , , ( ) 20 =′y . 10.238. 265 2 +=+′−′′ xyyy , ( ) 00 =y , ( ) 40 =′y .
10.239. 26 +=−′−′′ xyyy , ( ) 00 =y , ( ) 30 =′y . 10.240. 33 +=′+′′ xyy ,
( ) 00 =y , ( ) 30 −=′y . 10.241. 12 2 −=′−′′ xyy , ( ) 00 =y , .
10.242. xxeyy 4=+′′ , ( ) 20 −=y , ( ) 00 =′y . 10.243. xyy sin4=+′′ ,
( ) 10 =y , ( ) 20 =′y . 10.244. xyy 2sin−=′+′′ , ( ) 1=πy , ( ) 1=′ πy .
10.245. xyy 3cos69 =+′′ , ( ) 10 =y , ( ) 30 =′y . 10.246. =−′+′′ yyy 32
= xex248 , ( ) 10 =y , ( ) 2
3 0 −=′y . 10.247. xyy 2−=′+′′′ , ( ) 00 =y , ( ) 10 =′y ,
( ) 20 =′′y . 10.248. xeyy 8=−′′′′ , ( ) 10 −=y , ( ) 00 =′y , ( ) 10 =′′y , ( ) 00 =′′′y .
10.249–10.260 - .
10.249. x
yy 2sin
10.252. x
yy 3cos
12 + =
x
ex
ln 96 =+′+′′ .
10.260. xx eeyy 22 1−=′−′′ . 10.261–10.270 , -
.
( ) 00 =y ( ) 10 =′y
10.265. xexyy −+=′′+′′′ 6 . 10.266. xxeyy x cos+=−′′′′ .
10.267. x
eyy x
+−=+′−′′ .
10.269. 22cos xxyy +=′+′′ . 10.270. xxyy 2sin4 =+′′ . 10.271 – 10.279 () -
.
10.271.
11
15
11.1−−−−11.17 () , -
.
11.1. D — .
11.2. D — : .
11.4. D — : .
11.6. D :
11.11. − .
11.18–11.25 ,
D − , , .
11.18. 11.19. y y
. . . . . . . . . . . x . . . . . x . .
11.20. 11.21. . y . y . . . . . . . . . x . . . . . x . . . . . 11.22. 11.23. . y . y . . . . . . . . . . x . . . . . . x . . 11.24. 11.25. y y
. . . . . . . . . . . x . . . . . x . . . .
11.26 – 11.35.
, D - , , , - .
11.26. y 11.27.
( )∫∫ D
dxdyyxf ,
( )∫∫ D
dxdyyxf ,
17
. . . . . . . . . . x . . . . . x . . . . 11.28. y 11.29 , . . . . . . . . . x . . . . . x . . 11.30. y 11.31. y . . . . . . . . . . x . . . . . x . . . . 11.32. y 11.33. . . . . . . . . . . x . . . . . x . . . . 11.34. 11.35. y . . . . . . . . . . x . . . . . x . . . .
11.36 – 11.43
( ), - D - .
( )∫∫ D
dxdyyxf ,
18
. . . . . . . . . . . . x . . . . . . x
11.44. ( )∫ ∫ −3
.
11.76. ( )∫ ∫ 1
0 0
11.78. ∫ ∫ 4
0 0
22 y
dxdyyxf ,
D , .
11.96. ∫∫ D
= =
.
dxdyyxf ,
11.116. ∫∫ + D
11.124. , D:
∫∫ +D yx
dxdy 22
11.130. ( )∫∫ + D
11.133. ( )∫∫ −
+ D
11.138. 05,4 =−+= yxxy . 11.139. 6,4 2 =+−= yxyyx .
11.140. ( ) ( )0,14, 2
11.142. 2,,1 === xyxxy . 11.143. 4,4, 22 === yxyxy .
11.144. 2,4,1 === yxxy . 11.145. .
11.150. ( )0,0,cos,sin >=== xxxyxy . 11.151. 0,02,2 ==−+= yyxxy .
11.152 – 11.158 , ( - ).
11.152. )0(,2, 2222 ≥=+=+ yxyxxyx . 11.153. xyx 322 ≤+ ,
yyx 322 ≤+ . 11.154. 0,3,322 ===+ xxyyyx . 11.155. xyx 422 =+ ,
)( xy ≥ . 11.156. ( )2,4,0 2 ≥−== yyyxx . 11.157. ( )ρ cos12 −= .
11.158. ( )ρ cos12 += , ρ cos2= .
11.159. , 2=ρ -
( )ρ sin12 += .
11.160. – 11.172. , - .
11.160. 42 =++ zyx , 0,0,0 === zyx .
11.161. ,1,3 22 =++= yxyxz 0,0,0 === zyx .
11.162. ,4 2xz −= 0,5,0 === zyy .
11.163. ,2,2 =+= yxyz 0,0,0 === zyx .
11.164. xyxyzxz 2,,6,0 ===+= .
1,1 2 +==+ xyyx
11.167. 3,222 ≥+++= yxyxz , 3,3,0,0,0 ===== yxzyx .
11.168. ,6,122 xyyxz −=++= xyyz 2,1,0 === .
11.169. 0,0,9, 3
11.170. 1622 =+ yx , 0,,0 === zyzy .
11.171. 4=++ zyx , 0,422 ==+ zyx .
11.172. 10=++ zyx , 0,82,42 ==+=+ zyxyx . §4. -
11.173. , : , ,
, , ( )yx,ρ
, .
11.174. )1(ρ = , -
: .
11.175. , xyxy == ,2 ,
ρ ( )yx, ( ) yxyx 2, +=ρ .
11.176. R, ( )yx,ρ -
.
: .
: )0(,0,4 ≥=−= xyxy .
: 0,2,2 ==+= yxyxy .
1−=x 2=x
: , , , .
: 4,4,2 22 === xxyxy .
11.182. -
)1(ρ = , : 4,2,1,4 ==== xxxyxy .
11.183. -
, ( )1=ρ .
11.184. - )1(ρ = , : , , .
11.185. )1(ρ =
OX, : , , , .
)1(ρ = , : , , , .
)1(ρ = , : 0,42 =−= yyx .
12
12.1. . 12.2. . 12.3. +++ !3
2yx = 24 yx = 4=x 0≥y
1 94
xy =2 xy 42 = 1=y 3=y
xy =2 xy 42 = 1=y 3=y
+++ 7
6
5
4
3
2 +++
27
6
9
4
3
2
⋅ ++
⋅ +++
12.21. . 12.22. . 12.23. .
12.24. . 12.25. . 12.26. .
12.27- 12.34 , . - .
12.27. . 12.28. + ⋅
.
12.35. . 12.36. . 12.37. .
12.38. . 12.39. . 12.40. .
12.46- 12.61 , (
:
.
12.72- 12.83 , -
.
12.72. . 12.73. . 12.74. .
12.75. . 12.76. . 12.77. .
12.78. . 12.79. . 12.80. .
12.81. . 12.82. . 12.83. .
12.84- 12.91 , -
.
12.84. . 12.85. . 12.86. ∑ ∞
12.90. . 12.91. .
12.92 — 12.106 .
+++ 32 2
12.107. . 12.108. ( )
12.113. ( ) ( )∑
12.121. . 12.122. .
12.123. . 12.124. .
12.125. . 12.126. .
12.133. . 12.134. . 1 2.135. .
( ) .
12.151. . 12.152. . 12.153. .
12.154. . 12.155. . 12.156. .
12.157. . 12.158. . 12.159. .
12.160. . 12.161. ( ) n
n
nxn∑ ∞
=1 .
§5. . -
12.163- 12.174 -
, xe , xsin , sx, ( )x+1ln , ( )αx+1 .
12.163. . 12.164. . 12.165. . 12.166. .
12.167. . 12.168. . 12.169. . 12.170. .
12.171. . 12.172. . 12.173. . 12.174. .
( ) ∑ ∞
=
−⋅ +
( )α−xcos
1
12.177. , . 12.178. , .
12.179. , . 12.180. , .
12.181. , . 12.182. , .
12.183. , . 12.184. , .
12.185. , . 12.186. , .
12.187. , . 12.188. , .
12.189- 12.198 δ .
12.189. , . 12.190. , . 12.191. ,
12.194. 3 70, . 12.195. 5 40, . 12.196. , .
12.197. , . 12.198. , .
.
12.203. , . 12.204. , .
1
80 π=x
( ) xexf 2= 40 =x ( ) 2
x
001,0=δ 1sin 0001,0=δ
001,0=δ 01,0=δ 3 500 001,0=δ
e
01,0=δ ∫ −4,0
x
§ 1.
13.1. - . , ?
13.2. 8 ?
13.3. ( 1, 2 3 ) - 10 ?
13.4. 8 ( ). ?
13.5. 8 3 ( ). ? ( - ).
13.6. , , . - ?
13.7. . , - ?
13.8. 12 - , 3 ?
13.9. « » , ?
13.10. . ?
13.11. 2, 3, 5, 7, ?
13.12. 0, 1, 2, 3, 4, ?
35
13.13. 7 , – 9. - ?
§ 2. . 13.14. 2 . , : 1) 7; 2) 8, 4; 3) 8, , 4; 4) 5, 4.
13.15. 2 . , .
13.16. 6 , . - . , .
13.17. 1, 2, 3, 4, 5, – 8, 7, 8, 9, 10. . , : 1) 7; 2) 11; 3) 11?
13.18. , , 1000 , - , . , : 1) ; 2) ; 3) ; ) .
13.19. . , : 1) ; 2) , .
13.20. 4 5 . . , .
13.21. 4 5 . . . - . , .
13.22. 4 5 . , , . . . , , , .
13.23. , , , – . - ,
13.24. 20- 3 - . , : 1) ;
36
2) ; 3) .
13.25. , 4- 1 1 ?
13.26. 3 . - , , - , .
13.27. 90 - ( ). , - .
13.28. 86 ( ). , , 90 ( : 90, 89, 88, …, 6, 5).
13.29. «6 49» ( 6 49 ) : 1) ; 2) .
13.30. 15 , 10 - . . , : 1) 5, .. ; 2) 4, .. - ; 3) 3, .. - ; 4) 2, .. - .
13.31. 20 25. . , , . . , : ) ; ) ?
13.32. 100 . . 10 . , .
13.33. 1000 : 500 500 . - . , : 1) - ; 2) , – .
13.34. 50 28 . 15 . , 7 ( ).
13.35. 5 , . - . , - : 1) ; 2) - ; 3) .
37
13.36. 8 . , 2 , : 1) 8; 2) 12.
13.37. 5 ( 52 ). , : 1) ; 2) – 5 ; 3) – 4 .
§ 3. 13.38. OA L Ox
)(xB . , OB BA - , , 3/L . , .
13.39. R r . , , , . , .
13.40. 2 ( 51 =R , 102 =R ). . , - , . , - .
13.41. R . R . - . . , - .
13.42. - , . . , . , - .
13.43. , a2 . - ar < . , .
38
13.44.
2
a r < . , -
. , - .
13.45 12 13 . , . , , ( 12 13 ).
13.46. , - . , xy
, .
§4. .
13.47. , - . - : 8,0,9,0 21 == pp . . .
13.48. R . 4 . , : ) 4 ; ) - ; ) - .
13.49. . , , 0,1. - , .
13.50. 20 25 . - , .
13.51. 10 , 4 . . , - .
13.52. . 0,7, – 0,8. , .
a
4/1
i
39
13.53. 0,9, – 0,6, – 0,3. . , , ?
13.54. , , 0,03. , 4 ?
13.55. . . 0,15, - 0,3. - , .
13.56. . , , , : 0,3, 0,4, 0,6, 0,7.
13.57. , , . , - , 0,1. 0,15 0,2. , .
13.58. 10 6 3 . , : 1) ; 2) .
13.59. . - 0,2, – 0,6. ?
13.60. - 0,8. , .
13.61. 15 5 . - . , .
13.62. . 0,5 0,4. , , .
13.63. 100 . . . - , ?
40
13.64. . - . , . ?
13.65. 40 50. . . - , : 1) ; 2) ; 3) ; 4) .
§ 5. .
13.66. , , 5 . - 0,4; 0,8; 0,3; 0,2; 0,1. ?
13.67. 6 4 - . , , 0,95, – 0,8. . , - .
13.68. 2 . , - . , ( ).
13.69. 5 , 3 . , – 0,95, – 0,7. , , .
13.70. 10 , 8 ; 20 , 4 . - , . , .
13.71. 10 , 1,01 =p , – 4,02 =p . , .
13.72. 10 , . - , - , 0,95; 0,8. . : - ?
41
13.73. 45% , , 1- , 15% - 2-, – 3- . , , , - , 0,96, 0,84, 0,90 . - , .
13.74. - − . , 0,7, ( ) 0,92, - 0,65 . .
13.75. . 0,6 , − 0,7 , − 0,8 0,2, – 0,6 , – . - .
13.76. - . - . - : 1- , 2- − 2- 4 – , 3- − 3- 4- , 4- − 4 – . - 33,7% , 37,5% − , 20,9% − 7,9% − . , .
13.77. 1-, 2-, 3- 4:3:2. - 4% (.. 0,04), − 7%, − 9%. ?
13.78. 70%- . 20% 40% . - . , 1) , 2) -
13.79. , . - . 60% - , – 84%. . , - .
13.80. , , , , , 3:2. , -
42
, 0,1, – 0,2. - . , .
13.81. - . , - , 0,05, – 0,1. . , (- – ).
13.82. 10 , . - , - , 0,95; 0,8. . : - ?
13.83. 5 . , - 2- («») 0,9, («») 0,5. . : «» «»?
13.84. , 5% 0,25% . . , , , : 1) , 2) .
13.85. 97% - . - (.. ) 6% . . . - - , - ?
§ 6. , -,
13.86. 5 . , «» : ) ; ) ; ) .
13.87. . : 2 4- 3 6- ( - ).
13.88. . : ) 2- 2 4-; ) 2- 4- 3- 5-?
.
43
13.89. , . 0.1. .
13.90. ) , A 3- 4- , A 0,4. ) B , A 4- . B , 5 - , A 0,8.
13.91. AB C 2:1. 4 . , 2 C , – ( - ).
13.92. 0,4. , 5 ?
13.93. , 600 250 , 0,4.
13.94. , 1400 2400 - , 0,6.
13.95. 0,8. - , 100 75 .
13.96. 100 - 0,8. , : ) 75 90 ; ) 75 ; ) 74 .
13.97. - 0,7. , : ) - 1470 ; ) 1470 1500 ; ) 1470 .
13.98. 10% . , 400 349?
13.99. 10000 . - 100 .
A
2100
44
13.100. , , 0,004. 50 . , .
13.101. 60 . , 2 ?
13.102. , , 0,001. , 5000 .
13.103. - 10. , 60 3 ?
13.104. - 0,01. , 500 .
13.105. 1000 . - , , 0,003. - , .
14
§1.
14.1. 0,7. - L – . .
14.2. 0,4, 0,7, 0,8 . L – .
14.3. 5 3 . 3 . - .
14.4. , 4 , - , . - 0,7, 0,1. , - .
45
14.5. 0,8 0,7 . 5 . - , .
14.6. ( )cbaX ;;= ( )ba ; -
, ( )
,
− ⋅ ≤ −= − ⋅ ≥ −
, - . .
14.7. ( )0;1;1 +−=X ( )1;1 +−
( ) ( ) ( )
≥−⋅ ≤+⋅
, - . .
14.8. ( )π;0
( ) xAxf sin= .
( )xF . ( )xf , ( )xF -
14.9. ( )2;0
( ) Axxf = .
( )xF . ( )xf , ( )xF
( )5,1;5,0 .
14.10. ( )∞;0
( ) xexf λλ −⋅= , 0λ > . -
( )xF ( )xf , ( )xF . 1λ = -
( )2;0 . -
0,1.
46
14.11. ( )10;0 . ,
15=Z ? Z ( )20;0 .
14.12. 5 . - . - , 3- ?
14.13. , ( )λEX = , -
15 . , 20 .
14.14. -
2Y X= , ( )1;0RnX = − , (0,1).
§2.
14.15. 5 3 . 4 . - . , - .
14.16. - p=0,2. L – - . , .
14.17. - p . 1- -
. L – - . , .
47
( )ba ; . -
.
14.19. ( )1;1 +−
( ) ( )21 xAxf −⋅= . -
, - .
14.20. ( )1;1 +−
( )
, - .
14.21. ( ; ; )X a b c=
( )ba ;
, ( )
,
− ⋅ ≤ −= − ⋅ ≥ −
, - .
14.22. ( )X E λ=
(0; )∞ ( ) xexf λλ −⋅= , 0>λ . -
- , .
14.23. (0;3;1)X = ( )3;0 -
( )
≥− ≤
14.24. ( )0;3;3 +−=X
( )3;3 +−
48
x x
A xf
, - 0.9?
14.25.
( ) 22 xRAxf −⋅= . ,
, .
§3.
14.26. -
( ) ( )
a σ .
14.27. - , 60 / 10 /. , 80 /. - 50/ 70 /.
14.28. , , 15 1,442. - , - 2.
14.29. - 20 . , 15 .
14.30. 6 9 . 8 ? - .
14.31. - 5 0,81 2. - 0,99.
49
14.32. 15. - 0,04 2. 0,99.
14.33. ( )..50.,.350NX = . - 250.. , «». , , - , .
14.34. - 2XY = , ( )1;0NX = .
14.35 :
xi -1 0 1 pi 0.3 0.2 0.5 . )()()( YMXMYXM ⋅=⋅
14.36. : xi -4 0 4 pi 0.25 0.5 0.25 .
)()()( YDXDYXD +=+
§1. . - .
15.1. n=120, - , . - x2=4 ?
yi 0 1 3 pi 0.1 0.3 0.6
yi 2 4 6 pi 0.2 0.3 0.5
50
15.2. fj n=20. 4 6.
15.3. 110=n :
n6?
jf
51
§2.
15.5. :
?
15.6. :
?
15.7. :
mx - 0,95?
15.8. :
σx 0,99?
15.9. , - 15. ,
52
5 0,9.
15.10. ( ) ( ): 31; 33; 35; 36; 37. …
15.11. (32,6;41,1) . ?
15.12. (22,15;23,65) - . - …
15.13. 16 - 64 2. 0,99 - .
15.14. 14
2. 4 . .
§3. .
15.15. - ( )..50.,.350NX = .
. , , , 0,05.
15.16. 16 - 5,7 64 2. 0,01 , 8.
15.17. 16 - 5,7 72 2. 0,01 , - 642.
15.18. 15 25 . 15 16,3 , - 20 18.
53
- 0,05.
15.19. 20=n
0,05 - - . 4.
15.20. , 16 - - 0,5. 0,05.
15.21. - , D=0,81; D=0,64. : y+0,28x=1. ρ ?
15.22. - 25 36 15 , 0,75. 0,01.
15.23. , 60 105 , 69 140, 63 125 105 160. 0,05 - .
54
xy ++= 1 ln .
10.41. C x
− −
10.44. Cxy ln12 =+ . 10.45. Cyx ++= 3arctg 2 . 10.46. ( ) Cyyx ln2 =− .
10.47. Cx y +=
2
( )21
2
−= x
10.61. ( ) xy cos12 =− . 10.62. xy coscos = . 10.63. xey arccos= .
10.64. xy= . 10.65. x y
e−−=− 1
( )32 +− − xe x . 10.68. ( ) 2
1 1
10.70. 12
π arctg
x y
−=
y
x
=+ Cx
y
x
y
x
1= . 10.101. 322 yxy =− .
10.105. xxxy ln2 −= . 10.106.
10.125. ( )1−= − xey x . 10.126.
−⋅=
10.133. 213cos 9
10.135. 43
3 1
21 2
ln 1
211 14 CxCyC +=− . 10.151. Cy = ;
212 CxCy += . 10.152. ( )1 2arcsin Cey Cx −= + . 10.153. ( )12 cos CxCy +−= .
10.154. ( )211 sec CxCCy += . 10.156. 211
1 CxC
π cosln +−−= xxy .
10.160. xey 2
1
48
24
57
2
y −
=− 1
1 1 .
10.172. xy 3tg = . 10.173. xy −=ctg . 10.187. xx eeCy 2 2
3 1 += .
3 1
=
21
xeC −+ 3 . 10.197. ( )xeCy x 321 ++= − . 10.198. ( ++= − xeCy x 3cos2
2 1
10.201.
( )xxe x 3sin3cos 32 ++ − . .10.203. xeCxCxCy −+++= 4 2
321 .
10.209. xey x 2sin 2
1 −= . 10.210. xey 3
10.212. xy 5sin 5
10.215.
3
1 ++= − xx eeCy .
10.219. ( ) ( )12 32
58
2 −− −+= . 10.222. ++= xeCCy 2 21
xx 3sin 13
8 3cos
8
3
4
x2sin 3
1− . 10.228. ( )
321 xeCxCy x +++= .
10.231. xxx xeeCeCxxCy 2sincos 4321 ++++= − . 10.232. ++= xCy 21
xx xeeC −− ++ 3 . 10.233. ( )32 321 −+++= − xxeCxCy x . 10.234. += xeCy 1
( ) xx ex x
2
1
3
1
9
20
9
45
34
36
11
6
9
8
6
1
27
35
27
8
17
8
17
10.244. xxey 2sin 5
10.246. ( )5,134 23 +−+= − xxxeey xx . 10.247. 2sincos44 xxxy −+−= .
10.248. xxx xexxeey 2sin2cos3 ++++−= − . 10.249. +
xexC
22 −+ x . 10.263. ( ) xxxxCxCey x sin 25
3 cos
1 sin
xxxexxC x sin 4
1 21
xxxx 2
1 2sin2cos 21
xxxx 2sin 16
32 . 11.81. 18. 11.82.
1 2 −− ee .
11.94. 2π
188− . 11.103. 20
7
11.127. 2 π . 11.128. 12π18 − . 11.129. 4. 11.130.
2
4 + 11.149. 3
2
33 . 11.155. 2π− . 11.156. π . 11.157. π6 . 11.158. π5 . 11.159. 8π+ .
11.160. 3
32 . 11.161.
3
128
81 . 11.174. 4. 11.175.
11.184. , 4
15 51 . 11.187.
( ) 23
12.20. ( )
14
1 −
+= n
n
42 =u , 03 =u . 12.23. 31
2
2
( ) !2
∞= ∞→
n n
Slim , . 12.35. . 12.36. . 12.37. . 12.38. .
12.39. . 12.40. . 12.41. . 12.42. . 12.43. . 12.44. .
12.45. . 12.46. . 12.47. . 12.48. .
12.49. . 12.50. . 12.51. . 12.52. .
12.53. . 12.54. . 12.55. . 12.56. .
12.57. . 12.58. . 12.59. . 12.60. .
12.61. . 12.62. . 12.63. . 12.64. .
12.65. . 12.66. . 12.67. . 12.68. .
12.69. . 12.70. . 12.71. . 12.72. .
12.73. . 12.74. . 12.75. . 12.76. .
12.77. . 12.78. . 12.79. . 12.80. .
12.81. . 12.82. . 12.83. . 12.84. .
65
12.105. . 12.106. . 12.121. .
12.122. . 12.123. . 12.124. .
12.125. . 12.126. . 12.127.
. 12.128. . 12.129. .
12.130. . 12.131. . 12.132. -
. 12.133. 11 <<− x . 12.134. 11 <<− x . 12.135. 11 <<− x .
12.136 11 <<− x . 12.137. −3 < < 3. 12.138. 3
1
3
12.142. −∞ < < ∞. 12.143. −√3 − 4 < < √3 − 4. 12.144. 1 3
5 <<− x .
3 <<− x . 12.146. 1 < < 2. 2.147.−1 < < 0. 12.148. − − 1 <
< − 1. 12.150. −1 < < 1. 12.151. 11 <<− x . 12.152. 11 <<− x .
12.153. 2 ≤ < 4. 12.154. −3 ≤ ≤ 1. 12.155. −1 < ≤ 1.
12.156. −3 ≤ ≤ −1. 12.157. −5 ≤ < 3. 12.158. 13 ≤<− x .
66
12.163.
12.176. ( ) ( ) ( ) ( ) +−−+−−+−− −+ 112 32
12.179. ( )
12.192. 0,0175. 12.193. 0,158. 12.194. 4,125. 12.195. 2,09. 12.196. 7,937.
12.198. 0,7788. 12.199. 0,125. 12.200. 0,452. 12.201. 0,192. 12.202. 0,097.
12.203. 0,118. 12.204. 0,498. 12.205. 0,098. 12.206. 0,499.
13
13.1. 120. 13.2. 40320. 13.3 720. 13.4. 336. 13.5. 56. 13.6. 120.
13.7. 83. 13.8. 3 6
3 9
3 12 CCC ⋅⋅ . 13.9. 120. 13.10. 24. 13.11. 6. 13.12 96.
13.13. 756. 13.14 1) 1/6; 2) 1/18; 3) 0,5; 4) 1/18. 13.15. 0,75.
13.16. 1/720. 13.17. 1) 1; 2) 0,2; 3) 0,6. 13.18. 1) 0,008; 2) 0,096;
3) 0,384; 4) 0,512. 13.19. 1) 1/90; 2) 1/81. 13.20. 4/9. 13.21. 0,375.
13.22. 1/6. 3.23. 0,1. 13.24. n
m p = , ( ) ,
28 1 4
1 4 CCCm ⋅⋅= .
!4 . 13.29. 1) 6
CC 01765,0≈ .
13.30. 1) 24/91; 2) 45/91; 3) 20/91; 4) 2/91. 13.31. ) 62,0 ; ) 0,38.
13.32. 0,1. 13.33. 1) 499/1998; 2) 500/999. 13.34. 15 50
8 22
7 28
C CC .
13.35. 1) 0,6; 2) 0,3; 3) 0,9. 13.36. 1) 0,25; 2) 1/6. 13.37. n
m p = ,
44 2 42 CCm ⋅= ; 2) 5
134 Cm ⋅= ; 3) 62413 1 48 =⋅= Cm .
13.38. 1/3. 13.39. 22 / Rr . 13.40. 0,75. 13.41. π2
1 . 13.42. 0.5.
−− . 13.49. 0,18.
13.50. 5,0≈ . 13.51. 5/6. 13.52. P=0,38. 13.53. 0,306. 13.54. 0,88.
13.55. 0,955. 13.56. 0,95. 13.57. 0.388. 13.58. 1) 3/14; 2) 1/4. 13.59. 0,2.
13.60. 0,384. 13.61. 0,009. 13.62. 0,7. 13.63. 0,002. 13.64. 6,01 = ;
4,02 =P . 13.65. 1) 0,2; 2) 0,327; 3) 0,393; 4) 0,43. 13.66. 0,36.
13.67. 0,89. 13.68. 2/3. 13.69. 0,85. 13.70. 0,5. 13.71. 1/3. 13.72.
. 13.73. 0,918. 13.74. 0,95877. 13.75. 0,6448.
13.76. 0,574. 13.77. % 9
13.80. 3/7. 13.81. 1/3. 13.82. . 13.83. ,
«» . 13.84.: 1) 20/21; 2) 1/21.
13.85. 0,942. 13.86. ) 5/16; ) 3/16; ) 13/16. 13.87. 2 4.
13.88. ) ; ) 2 4. 13.89. 93,0≈ . 13.90. ) 0,18; ) 0,74. 13.91 8/27. 13.92. 0,922. 13.93. 0,024. 13.94. 0.004.
≈
) 4236,0≈ ; ) 5,0≈ . 13.98. 952,0≈ . 13.99. 0,0905. 13.100. 0,819.
13.101. 0,814. 13.102. 0,96. 13.103. 0,0308. 13.104. : 0,96. 13.105. 0.5.
14
14.6. ab
= 2 . 14.7. 1=A . 14.8. 5,0=A ; ( ) ( )xxF cos15,0 −= .
14.10. 11,1ln1,0 =x . 14.11. 0,875. 14.12. 0,4. 14.14. ( ) yyF 2= ; ( ) 1/f y y= .
14.15. 5,2=xm ; 784,6=xD ; 731,0=xσ . 14.16. pmx 5= ; ( )ppDx −= 15 .
14.17. p
mx 1= ;
0=xm ; 133,0=xD . 14.20. 0,5A = ; 0xm = ; 0,133xD = . 14.21. ab
A −
= 2 ;
3
λ 1=xm ;
λ =xD ;
λ 2
ln5,0 =x . 14.23. 667,0=A ; 333,1=xm ; 389,0=xD ; 268,15,0 =x .
14.24. 333,0=A ; 051,29,0 =x . 14.25. 2π
2
2R Dx = .
14.26. ,4=a 5=σ . 14.27. 0,977; 0,683. 14.28. 0,905. 14.29. 0,547.
14.30. 0,62. 14.31. 7,097. 14.32. 0,516. 14.33. 0,023.
14.34. ( ) ( )yOyF ˆ21+= ; ( ) ( )2/expπ2/1 yyyf −⋅= .
15
15.1. 375,02 =ν . 15.2. 63 =n . 15.3. 106 =n . 15.4. w2=0,2. 15.5. 5,1.
15.6. 2,808. 15.7. −0,777< xm <10,977. 15.8. 1,97< xσ <4,8. 15.9.
27. 15.10. 34.4. 15.13. 5,59< xσ <14,92. 15.14. 8. 15.15. 267,76.
15.16. , t−; −2,471<1,114< 2,471. 15.17. , 2χ -
; 4,6<18< 32,82,471. 15.18. F-
0,359<1,143<2,468; . t- -2,02<0,87< 2,02.
15.20. F- 0,151<2<4,236. 15.21. 0,315. 15.22.
70
t- 2,995<3,01. 15.23. 2χ - -
9,87>7,82.
1. ..
[ ] . 2 . . 1,2 / .. , .. , .
. – 6- . −: 21 , 2003.– 720 .− :http: //vipbook/info/nayka-i-ucheba/matematika/190523-danko-pe- popov-ag-vysshaya-matematika-v-uprazhneniyah-i-zadachah-chasti-12-6-e- izdanie.html
2. .. / .. .– : , 2004. – 336 .
3. : . / . ... – 2- ., .−:-,2008.–575 .
4. .., .., .., .. . 2 / .. , .. , .. , ... - 3- .,. .-−: – , 2003.- 576.
5. .. / .. .−: . ., 2003.-304.
6. .. / .. . −−−− -: , 2003.− 432.
7. .., .., .., .. 64 , 2. ,,2012, -284.
8. .. .—: , 2001, -400.
9. .. . , 2003,- 41.
71
2
6090 1/16 . . . . . 4,1. . . . 4,4. 500 .
____________________________________________________________________________
« - » 603950, , . , 65.
, 603950, ., , 65 http://www. nngasu.ru, [email protected]