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2. , . , 02076, . 1.
3. , .
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1.
1. .., .., . . -. 2001. . I-IV.
2. .., .. . . . 2002. . 1, 5.
3. .. . . . 2000. . 1. . III.
1. : (2; 4), (6; 2), (8; 7). : 1) ; 2) ; 3) ; 4) Q; 5) D; 6) , ; 7) , . .
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. 1) ( 1; 1) ( 2; 2)
= .
, :
= = 10.
= ; = . ,
= 10 + + .
C
A Q
2 D
E
2 0 2 4 6 8 10
B
2
2) , ( 1; 1) ( 2; 2)
.
,
3 + 4 10 = 0.
3 10 + 46 = 0, 9 2 58 = 0.
3)
.
. 1 , k 1 , k 2 - . ,
.
. , . 3 10 + 46 = 0 . tgA , :
.
.
4) , Q 2:1, . CE : . , Q
, .
, , l = 1.
, .
, (2; 1). Q, , l = 2:
.
Q (4; 3).
5) D , . ,
. , .
, 1( 1; 1) :
y 1 = k (x x 1).
(2; 4) , D
2 + 9 32 = 0.
6) L. L , , k L = k BC = . y 1 = k (x x 1) (2; 4) L L
9 2 + 26 = 0.
7) , , , .
,
3×8 + 4×7 10 > 0.
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, , ,
3 + 4 10 > 0.
,
9 2 58 < 0, 3 10 + 46 > 0.
,
1. ?
2. ?
3. .
4. ?
5. ? ?
6. ?
7. , ?
2.
1. .., .., . . -. 2001. . VII.
2. .., .. . . . 2002. . 7.
3. .. . . . 2000. . 1. . V.
2.
.
. = 3, . .
= = = .
3.
.
. = 3, . , . 3, 3.
= = =
= = = .
4.
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= = =
= = .
6.
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= = =
× = .
7.
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= = = .
1. .
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4. .
5. ? ?
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8. , , .
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11. .
12. ? ? ?
13. ?
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15. , .
16. ?
3.
1. .., .., . . -. 2001. . IX-XI.
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2. .., .. . . . 2002. . 8.
3. .. . . . 2000. . 1. . V.
.
8.
.
. y = u,
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= .
9.
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10.
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11.
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1. .
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2. :
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3. , . .
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. , , . , .
, . , .
(∞; 1) = 0.
. 5.
.
(1; 5) = 2.
.
(5; ∞) = 6.
.
, = 1 , = 5 . :
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max = y (1) = (13 9∙12 + 15∙1 3) = 1.
min = (5) = (53 9∙52 + 15∙5 3) = 7.
5. , . , .
, (∞; 1) (5; ∞), (1; 5). . 4 . 5 .
+ +
1 5
Max min
. 6.
6. ″()
″() = (3 2 18 + 15)′ = (6 18).
,
(6 - 18) = 0, = 3.
= 3 (∞; 3) (3; ∞). (∞; 3) , , = 0 :
″(0) = (6∙0 18) = < 0.
(3; ∞) , , = 4,
″(4) = (6∙4 18) = > 0.
f ″() , ( ). f ″() , ( ).
(-∞; 3) , (3; ∞) .
7. f ″() 0 , . , = 3 .
(3) = (33 9∙32 + 15∙3 3) = (12) = -3.
, (3; -3) - .
8. : = ( 3 9 2 + 15 3) (.7).
1
-2 -1 0 1 2 3 4 5 6 7
-2
-3
-4
-6
-7
. 7.
14. a , . , ?
. . (. 8).
. 8.
0 < < . 2 , .
V¢(x) = a 2 8 ax + 12 x 2, a 2 8 ax + 12 x 2 = 0,
, .
0 < < , , .
1. .
2. ?
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4. , , .
5. .
6. .
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12. .
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15. ?
4.
1. .., .., . . -. 2001. . III.
2. .., .. . . . 2002. . 6.
3. .. . . . 2000. . 1. . VII.
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15. .
. ,
.
:
= = = =
= = = .
16. .
. , , .
= = .
17. .
. , . . . ,
= = = = = = .
18. .
. , . . , :
= = = .
19. .
. , , . , t ,
= = = = .
20. .
. : .
, . , .
.
,
, . ,
= = = .
: .
21. .
. . u=x2 dv= sin x×dx. du=2xdx .
= -x 2cos x + = -x2 cos x + .
, u=x, dv=cosx dx. d u=dx v=sinx.
=
22. .
. , u= sin2 x, dv=ex dx. d u= 2cos2 x dx v=ex.
=
, u= cos2 x, dv=ex dx. d u=- 2sin2 x dx v=ex. :
=
:
=
:
+ 4 =
5 =
= .
1. ?
2. .
3. ?
4. .
5.
1. .., .., . . -. 2001. . IV, XV.
2. .., .. . . . 2002. . 6.
3. .. . . . 2000. . 1. . VIII.
.
23.
.
. t = ln x. . . = 1, t = ln1 = 0, = , t = lne = 1. .
= = = arcsin1 arcsin0 = .
24. .
.
= = = .
t = e x, d t = e x d x. . = 0, t = e0 = 1, = 1, t = e1 = e.
= = = =
= = .
25. .
.
.
u = ln(x 2 +4), d v = d x, , v = x.
= = =
= = =
= 2 ln(4 + 4) 0 2 (2 0 2arctg1 + 2 arctg0) =
= 2 ln8 2(2 2 + 2×0) = 2 ln23 4 + p = 6 ln2 4 + p.
26. , :
y = x 2, , y = 0, x = 2 (x > 0).
. , : S1 S2. (. 9).
S = S1 + S2. .
, 4 = 1, 1 = 1 2 = 1. x > 0, 1.
2,0
1,5
1,0
0,5 = 1/
0 0,5 1,0 1,5 2,0 2,5
. 9.
,
= ,
(..).
27. ,
y = x + 2.
. 1. , . . , , .
.
y ′ = 2 = 0. = 2. . . . , = 0. . , 1 = -2 2 = 6. (. 10).
8
= + 2
6
4
2
2 0 2 4 6
. 10.
2. = + 2 , . = 0, = 2, = 2, = 4. .
3. , , :