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CAPTULO P
P
E . L , , , , , , .
D , . S , ; , .
1
E . L ; ,
3413
5 5.000000
0.750000
0.3333
2 1.4142
3.14159
=
− = −
=
=
π =
…
…
…
…
…
E , . P
, , . P 2 π, .
L ,
, F. P.1. E ℝ , ,
.
F 9.1 L
L : , . Y ; ,, ( ) .
L . S , . E
< < , . L ≤ < = . L :
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S , , , :
1. < ⇒ + < +
2. < ⇒ − < −
3.
0 < > ⇒ <
4. < < ⇒ > − > − 0 ; ,
5. 1
0 0
> ⇒ >
6. 1 1
0
< < ⇒ <
L R 1−4 6 ( > 0) < > ≤ ≥.
O ( ) . S , ; , .
L . U : ,
≤ , ( ), , , ( ). E , , . E . T , , T 8 9 C 1. (E A III ; .) S , C 9, .
E :
() , , 1, 2, 3, 4,
() , , 0, ±1, ±2, ±3,
() , , /,
≠ 0.
L :
() , , , , 3/4 = 0.750000
() , , ; , 23/11 = 2.090909
= 2.09 . (L .)
L .
EE 1 D () 1.323232 1.32=… () 0.3405405405 0.3405=…
.
() S = 1.323232 E 1 0.323232 − = …
100 132.323232 132 0.323232 132 1 = = + = + −… …
P , 99 131 = 131 99 = .
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() S 0.3405405405 = … . E 10 3.405405405 = … 10 3 0.405405405 − = … . T
10000 3405.405405405 3405 10 3 = = + −…
P , 9990 3402 = 3402 9990 63 185 = = .
E . P , 2. P ,
2 1. C , .
U . P ,
6 > , ≠ 0 (P ?). C .
S < ,
() E , (, ), < < .
()
, [, ], ≤ ≤ .
() [, ), ≤ < .
() (, ], < ≤ .
F. P.2. O , . L .
(, )
[, )
[, ]
(, ] F .2 I
L F. P.2 ; − . L , . L F. P.3
. O ℝ (−∞, ∞). E ∞ ()
, ∞ .
1 ¿C 2 ? S, , 2 . E 2 = ,
≠ 0. P / ;
. A, 2 2 2 = , 2 22 = , . P , . (E
.) P , = 2 , . P , 2 24 2 = 2 22 = ,
. P , . E 2 /
; . E , 2.
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(, ∞)
(−∞, ]
(−∞, ∞) F .3 I
EE 2 R . E .
() 2 1 3 − > + () 2 13
− ≥ − ()
25
1 ≥
−
() 2 1 M 33
6 3 S 6 7 3 D 7
7 E ( . 3 7)
− ≥ − −
≤ − +≤
≤ −∞
() 2 1 3 S 1
2 4 R
4 E (4, )
− > +
> +
> ∞
() S 5 :
( )2 2 5 1 7 55 0 0 0
1 1 1
− − −− ≥ ⇔ ≥ ⇔ ≥
− − −
L
7 5
1
−
− = 1 0 = 7/5. E
, . E :
P , (1, 7/5]. V F. P.4 .
A .L .
EE 3 R
() 3 2 1 5 ≤ + ≤ () 3 1 5 3 2 15 − < + ≤ +
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F .4 L E 2
() U E 2, 3 2 1 ≤ + 2 2 ≤ , 1 ≥ . E , 2 1 5 + ≤ 2 4 ≤ , 2 ≤ . P , () [1, 2].
() R :
3 1 5 3 5 3 2 15
1 1 5 3 5 2 15 3
4 2 3 12
2 4
− < + + ≤ + − − < − − ≤ −
− < ≤ − < ≤
E (−2, 4].
R .
EE 4 D C R
() 2 5 6 0 − + < () 22 1 4 + > .
()
E 2 5 6 − + ( )( )2 3 − − , . P 3 2 − < − , 3 0 − < 2 0 − > . P ,
3 < 2 > ; (2, 3).
() L 22 1 4 + > 22 4 1 0 − + > . L 22 4 1 0 − + = , 2 0 + + = :
2 4 4 16 8 21
2 4 2
− ± − ± −= = = ±
( )( )1 12 21 2 1 2 0 − + − − >
. P ,
12
1 2 < − 12
1 2 > + . E
( ) ( )1 12 2, 1 2 1 2,−∞ − ∪ + ∞
O ∪ . U . T . U
. U ∩ . P ,
( ] [ ] [ ) [ ) [ ] [ ]1, 3 2, 4 2, 3 1, 3 2, 4 1, 4∩ = ∪ =
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EE 5 R 3 2
1 < −
− .
N ( )1 − , . (C ?) M , :
( )
3 2 3 2 5 2 0 01 1 1
−< − ⇔ + < ⇔ <
− − −
E :
E , , ( ) ( ), 0 2 5, 1−∞ ∪ . V F.
P.5.
( ) ( ) , 0 2 5, 1−∞ ∪ F .5 E E 5
E A
E , , , ( ),
0 0
≥= −
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L :
1. − = . U .
2. =
= . E ( ) ( )
.
3. ± ≤ + ( ). E
.
L
. L 2 2 2 ± ≤ = . P ,
( )
( )
2 2 2 2
22 2
2
2
± = ± = ± +
≤ + + = +
() , ± ≤ + . E
. P , 0,
, ,
− . E .
E D () A
L = ( > 0) , = = −:
. L
, . P ,
− < , − + . (,
, − + .) S ,
= ⇔ = − =
< ⇔ − < <
≤ ⇔ − ≤ ≤
> ⇔ < − >
M ,
− = ⇔ = − = +
− < ⇔ − < < +
− ≤ ⇔ − ≤ ≤ +
− > ⇔ < − > +
EE 7 R: () 2 5 3 + = () 3 2 1 − ≤ .
() 2 5 3 2 5 3 + = ⇔ + = . P , 2 3 5 8 = − − = − 2 3 5 2 = − = . L = 4 = 1.
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() 3 2 1 1 3 2 1 − ≤ ⇔ − ≤ − ≤ . R :
1 3 2 3 2 1
1 2 3 3 1 2
1 3 1
− ≤ − − ≤
− + ≤ ≤ + ≤ ≤
P , [1/3, 1].
A () E 7, :
2 23 2 3 3
3 3
− = − = −
P ,
2 2 13 1
3 3 3 − ≤ − ≤
2/3 1/3. P , 1/3 1, ( F. P.7).
F .7 E E 7()
EE 8 R 1 3 + = − .
L −1 3. P , −1 3:( )1 3 2 1 = − + = . A, 1 + = ( )1 3 + = − − . L
; = 1.
EE 9 Q 2
5 3
− < ?
T
2 25 3 3 5 3 R 5
2 8 2 D 2
1 4 1 T
− < ⇔ − < − <
− < − < − −
> >
1 14
< <
E , . O . S , . L . L
( )1 4, 1 .
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EECC .1
E E 1−7, . U .
1.2
9 2.
1
11
E E 3−4, .
3. 0.12 4. 3.27
5. E 1/7, 2/7, 3/7 4/7 . (U .) O ? E 5/7 6/7 .
6. P ? Q 0.999 = 0.9 ?
E E 7−12, .
7. 0 5 ≥ ≤ 8. 2 3 < ≥ − 9. 5 6 > − < − 12. 4 2 < ≥
E E 13−26, , .
13. 2 4 − > 14. 3 5 8 + ≤ 15. 5 3 7 3 − ≤ − 16.6 3 4
4 2
− −≥
17. ( ) ( )3 2 2 3 − < + 18. 2 9 < 19.1
32
<−
20.1
2
+≥
21. 2 7 0 − ≤ 22. 26 5 1 − ≤ − 23. 3 4 > 24. 2 2 − ≤
25.4
12
≥ + 26.
3 2
1 1 <
− +
R E 27−32.
27. 3 = 28. 3 7 − = 29. 2 5 4 + = 30. 1 1 − =
31. 8 3 9− = 32. 1 12
− =
E E 33−40, .
33. 2 < 34. 2 ≤ 35. 1 2 − ≤ 36. 2 1 + <
37. 3 7 2 − < 38. 2 5 1 + < 39. 1 12
− ≤ 40.
12
2 2
− <
E E 41−42, .
41. 1 3 + > − 42. 3 2 − <
43. N − = . P ? P ?
44. R 1 1 − = − .
45. D
− ≥ −
.
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2 C C
L 0 . E .U, , ; . E (
) .S , , S
, . E , . E (, ) , , . N (, ) (, ) ( F. P.8). O . L ; . E , R D, ( ). C , . O (,) . S
, .
F .8 L (, )
L F. P.9 . O , 0. U . E , = 0, .
L . E I IV, F. P.10. E ; . A III ; II; IV.
F .10 L
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E E
C , . S, , , 1 1 .
C
, . E, . I , , . A , , .
L . L , , . C . .C .S ( ). A
. C , / .
D
C , . . U
. S 1 2, 2 1 ∆ = − .
EE 1 H (3, −3) (−1, 2).
L ( F. P.11) :
( )1 3 4 2 3 5 ∆ = − − = − ∆ = − − =
F .11 I
S ( )1 1, ( )2 2, ,
, F. P.12. L
2 1 2 1 ∆ = − ∆ = −
. P T P, .
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F .12 L ( ) ( )2 2
2 1 2 1 = − + −
F D
L ( )1 1, ( )2 2,
( ) ( ) ( ) ( )2 2 2 2
2 1 2 1 = ∆ + ∆ = − + −
EE 2 L (3, −3) (1, 2) F. P.11
( ) ( )( ) ( )2 22 21 3 2 3 4 5 41 − − + − − = − + =
EE 3 L (0, 0) ( , )
( ) ( )22 2 20 0 − + − = +
G
L ( ) ( , ) ( ).
EE 4 L 2 2 4 + = ( , ) 2 2 4 2 + = = .
E 2 . E 2 2 4 + =
( F. P.13()).
F .13 () E 2 2 4 + = . () E 2 2 4 + ≤
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EE 5 L ( , ) 2 2 4 + ≤
2 . L 2 ( F.P.13()).
EE 6 C 2 = . A (0, 0), (1, 1),
(−1, 1), (2, 4) (−2, 4). E ( ) ( F. P.14).
F .14 L
2
=
D ( )1 1 1, ( )2 2 2, , 2 1 ∆ = − 2 1 ∆ = − ,
, 1 2. D ( ) . L 12.
C
2 1
2 1
−∆= = =
∆ −
( )1 1 1, ( )2 2 2, ( F.P.15). L = ∆ ∆ .
F .15 ′ ′∆ ∆ = ∆ ∆ 1 2 1 2 ′ ′ ′ .
EE 7 L (3, −3) (−1, 2)
( )2 3 5 5
1 3 4 4
∆ − −= = = = −
∆ − − −
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L . U ; . M ,
. P ∆ , ; .
L . L
. E F. P.16, φ . L
φ 0 ≤ φ < 180. L 0
90.
F .16 L φ
S ,
φ F. P.16:
∆= = φ
∆
L S P.7.
L . S , , , .I, .
S , 1 2, , 1 2 1 2 1 = − ,
:
1 2
2 1
1 1
= − = −
(E .) P , F. P.17
1 2
= = −
P ∆ ∆,
=
1 2 1
= − = −
2 1
F .17 ∆ ∆
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E
L , . T . P , = . E , = .
EE 8 L (3, 1) (F. P.18) = 1 = 3,
.
= 1
= 3
F .18 L = 1 = 3
P ,
( )1 1 1, . S ( ), ,
1
1
− =−
( ) ( )1 1 1 1 − = − = − +
L
( )1 1 = − +
( )1 1, .
EE 9 H −2 (1, 4).
S 1 = 1, 1 = 4 = −2
( )2 1 4 2 6 = − − + = − +
EE 10 H (1, −1) (3, 5).
L ( )5 1
33 1
− −
= =−
. P
. S (1, −1),
( )3 1 1 = − − 3 4 = −
S (3, 5),
( )3 3 5 = − + 3 4 = −
E , 3 4 = − .
L ( F. P.19). E , . U (0, ),
( )0 = − + , , = +
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F .19 R
U (, 0) , ,
( ) = −
L = +
.
L ( ) = −
.
EE 11 H 8 5 20 + = .
A ,
20 8 84
5 5
−= = − +
A = + ,
8 5 = − = 4.
P , = 0 ; 8 20 = = 5/2. L
5 2 = .
L 0 + + = ( ) ,
.
M . U , , .
EE 12 L F ( ) C ( ) = + . E = 32 = 0, = 212
100 = ° . P ,
32 0 212 100 = + = +
= 32 ( )212 32 100 9 5 = − = . L
( )9 5
32 325 9
= + = −
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EECC .2
E E 1−4, . H ∆ ∆ . H .
1. ( )0, 3 , (4, 0) 2. ( )1, 2 , (4, 10) − − 3. ( )3, 2 , ( 1, 2) − − 4. ( )0.5, 3 , (2, 3)
5. U (−2, 3) ∆ = 4 ∆ = −7. H .
6. U (−2, 2) ∆ = −5 ∆ = 1. D ?
D E 7−12.
7. 2 2 1 + = 8. 2 2 2 + = 9. 2 2 1 + ≤ 10. 2 2 0 + =
11. 2 ≥ 12. 2 <
E E 13−14, () () .
13. ( )2, 5 3− 14. ( )2, 1.3−
E E 15−18, .
15. ( )1, 1 , 1 − = 16. ( )2, 2 , 1 2 − = 17. ( )0, , 2 = 18. ( ), 0 , 2 = −
E E 19−20, , , ?
19. ( )2, 1 , 2 3 6 + = 20. ( )3 2, 1 , 7 − − =
E E 21−24, .
21. (0, 0), (2, 3) 22. ( 2, 1), (2, 3)− − 23. (4, 1), ( 2, 3)− − 24. ( 2, 0), (0, 2)−
E E 27−30, .
27. 3 4 12 + = 28. 2 4 + = − 29. 2 3 2 − = 30. 1.5 2 3 − = −
E E 31−32, () () .
31. (2, 1), 2 = + 32. ( 2, 2), 2 4 − + =
33. H 3 4 6 + = − 2 3 13 − = .
34. H 2 8 + = 5 7 1 − = .
35. (E ) S ,
1
+ = , .
36.
D 12 3
− = .
37. H (2, 1) (3, −1).
38. U (−2, 5) ( , 1) 3. H .
39. E $ , = + . S 10000 $5000 15000 $6000, 100000 ?
40. (F C) E , ( )5
329
= −
F C E 12. E , = .
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E C F? S , .
G
41.
U , (2, 1),
(6, 4) (5, −3) .
42.
D (0, 0), ( )1, 3 (2, 0) .
43. D (2, −1), (1, 3) (−3, 2) .
44. H 12 ( )1 1 1, ( )2 2 2, .
45.
H ( )1 1 1, ( )2 2 2,
1 2.
46. E 2 = − . E (2, 1) . H
.
E E 47−48, , , .
47. ( )2 22 4 − + =
48. ( ) ( )2 22 22 2 − + = + −
49. P 2 3 + = 4 1 + = ? P
?
50. H (1, 2) 2 3 + = 2 3 1 − = −
3 G E C
E , , , , .
C D
E .
L ( , ) (, ) ( ) ( )2 2
− + − , > 0
(, )
( ) ( )2 2
− + − =
S .
E E C
E (, ) > 0
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( ) ( )2 2 2
− + − =
E , (0, 0)
2 2 2 + =
EE 1 E 2 (1, 3) (F. P.20) ( ) ( )2 2
1 3 4 − + − = .
F .20 C ( ) ( )2 2
1 3 4 − + − = .
EE 2 E ( ) ( )2 2
2 1 7 + + − = (−2, 1) 7 ( F. P.21).
F .21 C ( ) ( )2 2
2 1 7 + + − =
S ( ) ( )2 2 2
− + − = ,
2 2 2 2 22 2 − + − = − −
U
2 2 2 2 + + + =
, . P ,
. C 2 2 + ( )2 2 22 + = + + ,
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2 . (O 2
.) E , 2 . E
( ) ( )22 2 2
+ + + = + +
S 2 2 0 + + > , (−, − ) 2 2 + + . S 2 2 0 + + = ,
(−, −). S 2 2 0 + + < , .
EE 3 H 2 2 4 6 3 + − + = .
O 2 4 − ( )2 22 4 4 − = − +
2 6 + ( )2 23 6 9 + = + + . P , 4 + 9
( ) ( )2 22 24 4 6 9 3 4 9 2 3 16 − + + + + = + + − + + =
(2, −3) 4.
E ; . E ( F. P.22.) E , . L
( ) ( )2 2 2
− + − ≤
(, ).
F .22 E () .
EE 4 I :
() 2 22 8 + + ≤ () 2 22 8 + + < () 2 22 8 + + >
P 2 22 8 + + = :
( )
2 2
2 2
2 1 8 1
1 9
+ + + = +
+ + =
P , 2 ( −1, 0). L () () ( F. P.23). L () ( ). L () .
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F .23 E 2 22 8 + + ≤
E
U . E ; . L . E .
F .24 L 24 = (0, ) = − .
O . S , , .
EE 5 H (0, ) = − .
S ( , ) , ( F. P.24) ( )
( ) ( )
( ) ( )( )
22 2 2 2
22 2 2
0 2
2
= − + − = + − +
= − + − − = + +
P , = :
2 2 2 2 22 2 + − + = + +
, ,
22 4 ( )
4
= =
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L F. P.24 > 0; , . S
0 < , (0, ) = − . E ,
.
L F. P.25 2 = .
F .25 A 2 =
EE 6 U (0, 1) = −1 2 4 = 2 4 = ( = 1
).
EE 7 H 2 = − .
L ( )2 4 = 4 1 = − . P , 1 4 = − . E
(0, −1/4) = 1/4.
S ,
22 4 ( )
4
= =
( , 0) = − . E .
U . L , F. P.26. E , . E . E S 8.1.
F .20 R
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E G
L . S , , 1/ ( F. P.27). S , .
F .27 E : () 21 − . () G () .
() G () .
P , , :
, . P
. S 24 = 4 4 , ,
( ) ( )2
4 4 4 = 2 = . P , 24 =
2 = , F. P.28.
F .28 L . C .
D
P , − ( < 0, ).
P , − ( < 0,
).
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EE 8 L ( )2
3 = − 2 = 3 . L
( )21 = + 2 = 1 ( F. P.29()).
F .29 () D 2 = . () D 2 =
EE 9 L 2 1 = + ( 21 − = ) 2 = 1 . L 2 3 = − ( ( ) 23 − − = ) 2 = 3 ( F. P.29()).
EE 10 E ( ) ( )2 2 2
− + − = (, )
2 2 2 + = . E
− − .
L 2 = + + . L > 0.
P ( )2
= − + (, ).
EE 11 D 2 4 3 = − + .
L 2 4 3 = − + . P ,
:
( ) ( ) ( )2 22 4 4 1 2 1, , 1 2 = − + − = − − − − = −
E 2 = 2 1 . P ,
(2, −1) = 2. C 2 = (0, 1/4), (0 + 2, (1/4) − 1)
( )2, 3 4− ( F. P.30).
E
S , 2 2
2 2 1
+ =
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− ≤ ≤ , − ≤ ≤
(P ?). S = , . S ≠ , .
= 2
(2, −3/4)
F .30 L 2 4 3 = − +
L (, 0), (0, ), (−, 0) (0, −) ( F. P.31). L (−, 0) (, 0) (0, −) (0, ) ; .
F .31 L 2 2
2 2 1
+ =
EE 12 L 2 2
19 4
+ = (−3, 0) (3, 0)
(0, 2)− (0, 2).
L
2 2
2 2 1
− =
(−, 0) (, 0) ( F.P.32). L ( ). C ( ) . L
0 0
− = + =
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16. 2 2 2 4, 1 + − + > + =
17. E (−1, 2) 6 .
18. E (2, −3) 4.
19. E (0, 0) 2 (1, 0).
20.
E (0, 0) 2, (1, 3) .
E E 25−28, , .
21. F: (0, 4) D: = −4
22. F: (0, −1/2) D: = 1/2
23. F: (2, 0) D: = −2
24. F: (−1, 0) D: = 1
E E 25−28, ,
.
25. 2 2 = 26. 2 = − 27. 2 4 = − 28. 2 16 =
29. L F. P.34 2 = . E
.
V ()
V ()
V ()
V ()
F .34
30. Q = () (, );
(, ).
E E 31−34, 1 = + . D
.
31. D 3.
32. D 4.
33. D 4 3.
34. D 4 2.
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E E 35−38, .
35. 21 = − 1, 1
36. 2 2 25 + = 2, 4
37. ( )2
1 1 = − − 1, 1
38. = 2, 4
H E 39−42.
39. 2 3, 3 1 = + = +
40. 2 26, 4 = − = −
41. 2 2 25, 3 4 0 + = + =
42.
2 22 2 5, 1 + = =
E E 43−50, .
43.2
2 14
+ = 44. 2 29 16 144 + = 45.
( ) ( )22 231
9 4
+−+ = 46. ( )
( )22 11 4
4
+− + =
47.2
2 14
− = 48. 2 2 1 − = − 49. 4 = − 50. ( )( )1 2 1 − + =
51. C
() − ?
() − ?
52. C − − ?
53.
D 1 + = .
.4 F G
E . L . E .
S , . P,
2 = π
. L ( ) .
E . E . C
, [0. ∞) .
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E , . S , , .
E C .P ,
( ) =
. E , L E, . T, , , , , , ( ) .
DEFC 1
U ( )
.
E , = D( ) . E R ( )
( ) . P (F. P.35) ( ) .
DD( )
RR ( )
F .35 U
H . L
2 :
() 2 = , ;
() , , 2( ) = ,
() , , 2 → , ( 2 ).
E () () . E , ( ) . S , , ( )
. A
; 2( ) = = π 2( ) = = π . E ,
.
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EE 1 E
34( )3
= π
≥ 0. E, 3 ()
( )3 34
(3) 3 36
3
= π = π
O 3 = 3.
EE 2 U
8 ) 2 3 = +
H 0, 2, + 2 (2).
E , :
( ) ( )
( )( )
(0) 2(0) 3 0 3 3
(2) 2(2) 3 4 3 7
2 2 2 3 2 7
2 (7) 2(7) 3 17
= + = + =
= + = + =
+ = + + = +
= = + =
C D
U . P , 2( ) =
≥ 0, 2( ) =
, . E C 1−9, ( ). C ,
. A, 2 , ( ) .
C , ( ) .
E , ( ) .
S 0 .
EE 3 . E ( ) = [0, ∞),
. T (0) = 0, (4) = 2, (10) ≈ 3.16228. O ,
4, , −2 2, , 2, 4 ( ; ). L
. L 2 4 = 4 2 = =
4 2 = − = − .
EE 4 E 2
( )4
=
− = −2 = 2. E
,
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( )( ) , 2 ( 2, 2) (2, ) = −∞ − ∪ − ∪ ∞D
L .
EE 5 E 2( ) 1 = − 21 0 − ≥ . A
2 1 ≤ 1 1 − ≤ ≤ . E [−1, 1].
G F
U . E : .
L ( ) = . E
( , ) − . A, ( , ) ( ) = .
E ( )( ),
, .
EE 6 G 2( ) = .
H ( , ) 2 = ( T 1). A
( F. P.36()).
1
F 9.36 () G 2
( ) = ; () G 2( ) =
C , , F. P.36()? P , , ? E C 4, C .
A
. A F. P.37−P.46. E ; .O, , , ( ) = , F. P.46. E
= − < 0 = ≥ 0.
S ,
F. P.37−P.46.
EE 7 D 1 4 = + − .
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F .37 L
( ) =
F .38 L ( ) = F .38 L
( ) = 2
=
F .40 L
( ) =
F .41 L
( ) = 3F .42 L
1 3( ) =
F .43 L
( ) = 1 /
F .44 L
( ) = 1 / 2F .44 L
2( ) 1 = −
F.46 L
( ) =
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= F. P.40 4 (
− 4) 1 ( F. P.47).
F .47 L = 4
1
EE 8 D ( ) ( )( ) 2 1 = − − .
N . P
, 2 − 1 − −1 1:
( )2 1 1 1 1 11
1 1 1 1
− − + + − − += = = − +
− − − −
P , 1/ F. P.43 1 1 ( F.P.48).
F .48 L ( ) ( )2 1 − −
N . U ( )
, . S
, = ( )( ), . E
2 2 1 + = F. P.49
. S , , ,
2 21 1 = − = − −
, , () .
F ;
C . L
− .
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C: AE; 7. E. 34
F .49 E 2 2
1 + =
DEFC 2
F
S − . D
( ) ( ) − =
D
( ) ( ) − = −
L 0 1 = , 2 4 2 4, , , , − −… …
, 1 = , 3 1 3, , , − −… … . O,
, ( )4 4
− = ( ) 3 3
− −− = − .
C ( )2 2
− = , 2 . P ,
2 = = .
L . U , , ( F. P.50).
L . U , , . S
0 = , (0) = 0 ( F. P.50().
S ( ) ( ), ( ) , 2 ( )
5 ( ) − . L ( ) ; ( )
. P , 4 2( ) 3 5 1 = − − , : 4 2 03 , 5 1 − − = − .
E , ( )34 2 − . L 2( ) 2 = −
.
T . P , 2( ) 2 = −
( )2( ) 1 1 = − − . ( )1 ± , (F. P.51())
= 1; 2 = 1 1 . E
, = +3( ) 1 (0, 1) (F. P.51()).
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F .50 () L ; () L
.
F .51 () L 2
( ) 2 = − = 1. () L
3( ) 1 = = + (0, 1).
L .A, .
D , , , . L .
C :
1. L − .
2. L − .
3.
L − = /2.
4.
L − = /2.
5.
E = .
EE 9 D 2 3 = − − .
L 2 = − = (F. P.40) = 1. L
2 3 = − − 2 . V F. P.52().
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F .52 () C 2 3 = − − ; () T =
.
EE 10 E 2 3 = − + F. P.52()
.
P (F. P.46)
3 2 . L =
− = = − . D 3 , 3 = − + . F, 2
2 3 = − + , .
EECC .4
E E 1−6, .
1. 2( ) 1 = + 2. ( ) 1 = − 3. ( ) 8 2 = − 4. ( )( ) 1 1 = −
5.1
( )2
=−
6.1
( )1 2
=− −
7. C F. P.56 ( ) = ? P ?
() ()
() ()
F .56
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8. L F. P.57 : () 4 − , () 3 4 − , () ( )2
1 − , () 2 3 − . C ?
() ()
() ()
F 9.57
E E 9−10, ( ) = 0,
1 2 = ± , 1 = ± , 3 2 = ± 2 = ± .
9. 4( ) = 10. 2 3( ) =
E E 11−22, Q ( ) ? E , ?
11. 2( ) 1 = + 12. 3( ) = + 13.2
( )1
=
− 14.
2
1( )
1
=
−
15.
1
( ) 2 = − 16.
1
( ) 4 = + 17.
2
( ) 6 = − 18.
3
( ) 2 = −
19. 3( ) = 20. ( ) 1 = + 21. ( ) 2 = 22. ( )2
( ) 1 = −
D E 23−38.
23. 2( ) = − 24. 2( ) 1 = − 25. ( )2
( ) 1 = − 26. ( )2
( ) 1 1 = − +
27. 3( ) 1 = − 28. ( )3
( ) 2 = + 29. ( ) 1 = + 30. ( ) 1 = +
31. ( ) = − 32. ( ) 1 = − − 33. ( ) 2 = − 34. ( ) 1 2 = + −
35.2
( )2
=+
36.1
( )2
=−
37. ( )1
=
+ 38. ( )
1
=
−
E E 39−46, [0, 2] [0, 1], F. P.58.D .
39. ( ) 2 + 40. ( ) 1 − 41. ( )2 + 42. ( )1 −
43. ( ) − 44. ( ) − 45. (4 ) − 46. 1 (1 ) − −
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F .58
C . E E 47 −48, ( ) , 2 .
47.2
2( )
2 3
+=
+ + 48.
2
1( )
−=
+
E E 49−52, . E (, ) . C / ? T .
49. 4 3 2( ) 6 9 1 = − + − 50.2
2
3 2( )
2 2
− +=
− + 51.
1( )
2
−=
− 52.
2
2
2 3( )
4 5
+=
+ +
53. Q ( ) , ℝ, ?
.5 C F F F
L .
C , , , , .
, D, , C
I , , , ( ) .
DEFC 3
S , + , − ,
( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( )
( ) ( )
( )
, ( ) 0
+ = +
− = −
=
= ≠
U . S ,
( )( ) ( ) =
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EE 1 L F. P.59() 2( ) = , ( ) 1 = − ( ) 2( ) 1 + = + − . O
+ .
F .59 () ( )( ) ( ) ( ) + = + ; () ( ) 0.5 ( ) =
EE 2 L F. P.59() 2
( ) 2 = − ( ) 0.5 ( ) = . O .
EE 3 L
( ) ( ) 1 = = −
H 3 , + , − , , ,
.
L T 2.
2 C
F F D
( ) = [0, ∞)
( ) 1 = − (−∞, 1]
3 ( )3 ( ) 3 = [0. ∞)
+ ( )( ) ( ) ( ) 1 + = + = + − [0, 1]
− ( )( ) ( ) ( ) 1 − = − = − − [0, 1]
( ) ( )( ) ( ) ( ) 1 = = − [0, 1]
( )( ) ( ) 1
= = − [0, 1)
( ) 1( )
( )
− = =
(0, 1]
O
[0, ) ( , 1] [0, 1]∞ ∩ −∞ =
. S , / / .
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F C
E , , .
DEFC 4
S ,
( ) ( )( )( ) =
E ( ) . E
, ,
.
C F. P.60,
.
D( )
( )
( )
( )
( )
R
F .60 ( ) ( )( )( ) =
A ( ) ( )( ) ( ) = , ( ) . L
. P ,
( ) ( )( ) ( ) = , , , ,
. L , .
EE 4 D ( ) = ( ) 1 = + , ( )( ) , ( )( ) ,
( )( ) ( )( ) .
U , .
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3 C
F F D
( ) = 0. ∞)
( ) 1 = + ℝ
( )( ) ( )( ) ( )1 1 = = + = + [−1, ∞)
( )( ) ( )( ) ( ) 1 = = = + [0, ∞)
( )( ) ( )( ) ( ) 1 4 = = = = [0, ∞)
( )( ) ( ) ( )1 1 1 2 = + = + + = + ℝ
P , , [−1, ∞), ( ) 1 = +
1 0 + ≥ , , ≥ −1
EE 5 S 1( )1
−=+
, ( )( ) .
C
( ) ( )( )
11
1 11( ) G1 1 111
−−
+ − ++= = = =− + + −
++
P , , ,
ℝ. E ! P , :
() ,
() ( ) .
E = −1. S = −1 ,
(). O ( ) 1 = − , ( )1 1 − = − +
1 1= − . P , ( ) ()
. E (−∞, −1) ∪ (−1, ∞), , −1.
F D
A . U :
0 0
≥= −
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L ( ) , , = 0.
F .61 L H
EE 7 . L (F. P.62) :
1 0
( ) 1 0
N 0
>
= = −
2 1
( ) 1 2
2 2
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F 9.63 F
A , (−1), . L (2).
L ,
.
EE 10 . L
. , , [ ]
[ ] . L = F. P.64(). O
2.4 2, 1.9 1, 0 0, 1.2 2
2 2, 0.2 0, 0.3 1, 2 2
= = = − = −
= = − = − − = −
EE 11 . L
. S . S
F. P.64(). P , , , $1 .
F .64 () L ; () L
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EECC .5
E E 1−2, + , − , / , .
1. ( ) , ( ) 1 = = − 2. ( ) 1 , ( ) 1 = − = +
D E 3−6 .
3. 2 − 4. 3 − 5. + 6. 2 + −
7. S ( ) 5 = + 2( ) 3 = − , :
() ( )(0) () ( )( )0
() ( )( ) () ( )( )
() ( )( 5) − () ( )( )2
() ( )( ) () ( )( )
E E 8−10, .
() ( )( ) () ( )( )
() ( )( ) () ( )( )
8. ( )( ) 2 , ( ) 1 = = − 9. ( )( ) 1 1 , ( ) 1 = − = −
10. ( ) ( ) ( )( ) 1 1 , ( ) = + − =
H T 4 (E 11−16).
4
( ) ( ) ( )( )
11 2 + 1
12 + 4
13
14 1 3 2 3 +
15 ( )1 +
16 − 1 21
17. U
= 2 = +
2 3 = + + ( )1 2 3 = + +
D .
18.
R
2 = 2 1 = − 1 2 = −
1 2 = − 1
1 2
=
−
11
1 2
= −
−
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E E 19−24, [0, 2] [0, 1], F. P.65.D .
19. 2 ( ) 20.1
( )2
− 21. (2 ) 22. ( 3)
23. ( )1 2 + − 24. ( )( )2 1 2 −
F .65
E E 25−26, .
25. 0 1
( )2 1 2
≤ ≤=
− < ≤
26. 0 1
( )2 1 2
≤ ≤=
− < ≤
27. H ( ) = + :
() ( )( ) ( ) = .
() ( )( ) = .
F
28. P () 0 = ? () 0 = ?
29.
Q = ?
30. F : = − ?
31. D = − .
32. D
0( )
0
≥=
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. S ( )( ) ( ) ( ) 2 = + − .
D ( ) ( ) − = , . D, ( ) ( ) ( ) = − .
() D .
U (). S 1 1( ) ( ) ( ) = + , 1 1 ,
1 1 − = − E 34 1 = 1 = .
.6 F .
E C . .
DEFC 5
U
1 21 2 1 0( )
−−= + + + + +⋯
1 2 1 0, , , , , − … , , , > 0, ≠ 0. E ,
, ( ).
P
3 0
2 − 1
32 17 1 − + 3
G, , , ; . L . P , , , , . L . L + . P ,
( )( )2 3 5 21 2 2 2 + − − = − − −
2 3 , , 5.
E , :
3 2
2
2 3 3 4
1
− + +
+
C ,
0 ≤ < / ( ) ( ) . P ,
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7 12
3 3= +
E , , , > ,
( ) − −
, ( )
< ;
( ) ( )( )
( ) ( )
−= + (E A D)
L .
EE 1 E 3 2
2
2 3 3 4
1
− + +
+.
D . U :
3 2
3 2
2
2
2 3 3 4 2 3
2 2 1
3 4
3 3
7
− + + −
+
− + +
− −
+
P ,
3 2
2 2
2 3 3 4 72 3
1 1
− + + += − +
+ +
E 2 3 − 7 + .
D . U ; ,
:
( ) ( )
3 2 3 2
2 2
2 3 3 4 2 2 3 3 3 4 2 3
2 1 3 1 7
− + + = + − − + + − +
= + − + + +
3 2
2 2
2 3 3 4 72 3
1 1
− + + += − +
+ +
, C F. U ( ) 0 = . P ,3( ) 4 = − : 0, 2 −2; ( ) 0 = . E
, . T ( ) 0 = ( ) 0 = ,
. P , , ( ) 0 = .
E T F ( A II) 1
( ). P , ( 1) +
− ( ) 0 − + = . U ( )
, .
L ; 2 4 + ,
2 −2 ,
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2 1 = − ( A I ). L 2 −2 . C ( A II ).
E C, , 1 2 ( ). E .
EEA 1 E F
E 1 − ( ).
P , 0 ( , )
( )( )
= +
− −
P , ( )( ) ( ) = − + ( ) 0 = = 0, ( )( ) ( ) = − − ( ).
D T 1 T F ≥ 1 . (S
≥ 2, = ( )( ) ( ) = − , ( ) 1 1 − ≥ , , .) P , . E
4. ( )34 3 2( ) 3 3 1 = − + − = − : 0 1. D
1 3 ( ) ( )3
1 − .
S 1 = + , ≠ 0, ,
, 1, , 2 = − . (A, 1 2
.) P , − − − + ( ) , ,
( )( ) ( )2 2 2 2 22 − − − + = − + = − + +
. S
( ) ( ) .
EE 2 C ( )23 2( ) 2 5 = + + ? C ,
?
S , ( )2
3 2 7 = ; 7. E
( )33 0 = − 0 3. L
2 2 5 + + , 2. A,
( ) ( )
( )( )[ ]
22 22
2
2 5 5 4
1 2 1 2
+ + = + +
= + + + −
P , :
0, 0, 0 0 3,
1 2 , 1 2 1 2 2
1 2 , 1 2 1 2 2
− − − − − − − − − − − +
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F C
E .
F C. L
2 + +
, ≠ 0, 2 4
2
− ± −=
P , :
2
2 22
2 2
2 2 2
2
0
2
2 4 4
4 4
2 2 24
+ + =
+ + = −
− − + = ⇒ + = ±
L 2 4 = − . L :
() S > 0, 2 = , ( ) ( )2 − +
( ) ( )2 − − .
() S = 0, ( )2 − 2 ( )
() S < 0, 2 = − ,( ) ( )2 − + ( ) ( )2 − − .
EE 3 H :
() 2 1 + − () 29 6 1 − + () 22 1 + +
U .
() = 1, = 1, = −1:
2
1 1 4 1 5
2 2 2
1 5 1 51
2 2 2 2
− ± += = − ±
+ − = + − + +
() = 9, = −6, = 1:
( )2
22
6 36 36 1 ( )
18 3
19 6 1 9 3 1
3
± −= =
− + = − = −
() = 2, = 1, = 1:
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2
1 1 8 1 7
4 4 4
1 7 1 72 1 2
4 4 4 4
− ± −= = ±
+ + = + − + +
E ( 3) , , , . M , ( ) .
F . A ( ) . A :
() F C: ( )2 + = + .
() D C: ( )( )2 2 − = − + .
() D C: ( ) ( )( )3 3 2 2 − = − + + .
() M , :
( )( )1 2 2 3 2 1 − − − − −− = − + + + + +⋯
O − − .
() T , + + . P ,
( )( )
( )( )
3 3 2 2
5 5 4 3 2 2 3 4
+ = + − +
+ = + − + − +
F, . P
( )( ) ( )
( )( ) ( )
( )( ) ( )
2
2
2
+ + = + + +
− − = − + +
+ − = + − −
2 + +
. M ,
( )( )2 + + = + +
+ = . P , , , , .
EE 4 ( )( )
( )( )( )( )
( )( )
2
2
2
2
5 6 3 2 3, 2, 6 5
7 6 6 1 6, 1, 6 76 3 2 3, 2, 6 1
2 10 2 5 2 2, 5, 1, 2
− + = − − = = = + =
+ + = + + = = = + =+ − = + − = = − = − + =
+ − = + − = = = = −
2, 10, 1 = = − + =
EE 5 H :
() 3 2 4 4 − − + () 4 23 4 + − () 5 4 2 − − + .
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() H :
( )( ) ( )( )( )3 2 24 4 1 4 1 2 2 − − + = − − = − − +
L 1, 2 −2.
() 2 :
( )( ) ( )( )( )( )4 2 2 23 4 4 1 2 2 1 1 + − = + − = + − + −
L 1, −1, 2 −2 .
() C :
( ) ( )( )
( ) ( )
5 4 2 4 3 3
2 2
1 1 1
1 1
− − + = − − + = − −
= − + +
P , 0 1 . L 2 1 + + ,
1 1 4 1 3
2 2 2
− ± −
= = − ±
EECC .6
H E 1 −12. S , . T, .
1. 2 7 10 + + 2. 2 3 10 − − 3. 2 2 2 + + 4. 2 6 13 − +
5. 4 216 8 1 − + 6. 4 3 26 9 + + 7. 3 1 + 8. 4 1 −
9.6 4 2
3 3 1 − + − 10.5 4
16 16 − − + 11.5 3 2
8 8 + + + 12.9 7 6 4
4 4 − − + E E 13−16, .
13.3
2
1
2
x
x
−
− 14.
2
2 5 3
x
x x+ + 15.
2
2 2 3
x
x x+ + 16.
4 2
3 2 1
x x
x x
+
+ +
17. D − 1 .
18. Q + 1
?
19. E z u iv= + ( )
z u iv= − . E A I ( ) ( ) . U
z u iv= + , , z .
20. C , z u iv= + ( )
, 2 2 22x ux u v− + + .
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21. U E 20 z u iv= + ( )
, z .
.7 F
L cost sen t . S , ( F. P.66),
ady opucos y sen ( )
hip hipt t= = ∗
E , , .
ady
hipopu
F . 66
=
=
E C , . T .
S 1; 2 2 1 + = . S (1, 0) . P
, ,
> 0 < 0. P , 2π, 2
π
; (0, 1).
U . V F. P.67.
DEFC 6
L :
∠ =
E . P π (−1, 0), (π ) ,
π = 180
P , π/180; , 180/π.
C
E C
. C π/3, π/3 ( 60), π/3 .
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F .67 S , =
EE 1 . U . H .
L 2 π
2π ( 360). P ,
( )2 2
= π =
π
E , (F. P.68) 2 π :
( )2 2 21 2 2
= π =
π
(E S 1.1 2 π .)
s
r
r
t
F .68 L =
= 2 2
S , , . D ( F. P.69).
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DEFC 7
C
P , ( ) ( ) :
=
=
C , . O ( (*) ). E F. P.69.
Longitud de arco
F .69 L ( ) ,
EE 2 E 0, 2π , π 2 3 2 −π π= F. P.70, :
30 1 0 1 0
2 2 2
30 0 1 0 1
2 2 2
π π π = = π = − − = =
π π π = = π = − = = −
F .70 A
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C: AE; 7. E. 55
A
M
2 2 1 + = .
E . P ,
1 1 1 1 − ≤ ≤ − ≤ ≤
. L = = . P ,
,
2 2 1 + =
. P 2π, 2π
: 2 + π = . P , ,
( ) ( ) 2 2 + π = + π =
2π.
. . C 2 2 1 + =
, − (F. P.71).
( ) ( ) − = − = −
F .71 ( )
( )
− =
− =
. D π/2 (90). L ( )2 π −
= (F. P.72), . P
2 2
π π
− = − =
. D π ( 180). C
, π− ( F. P.73). P
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F .72 ( )( )
( )( )
2
2
π − =
π − =
F .73 ( )
( )
π − = −
π − =
( ) ( ) π − = − π − =
A E
EE 3 H π/4 ( , 45).
E 4π = . P , =
2 2 1 + = 22 1 . P , 1 2 = = ( F. P.74)
( ) ( )1 1
45 , 45 4 42 2
π π° = = ° = =
F .741
4 4 2
π π= =
EE 4 H π/3 ( 60) π/6 ( 30).
E 3π (0, 0) (1, 0) 1
( F. P.75). P , 3π 1/2, ( )2
1 1 2 3 2− =
( ) ( )1 3
60 , 60 3 2 3 2
π π° = = ° = =
C6 2 3π π π= − ,
( ) ( )3 1
30 , 30 6 3 2 6 3 2
π π π π° = = = ° = = =
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F .75 3 1 2
3 3 2
π =
π =
L T 5 30 45 0 80. L 120, 135 150 ;
( ) ( )2 1
120 603 3 3 2
π π π ° = = π − = − = − ° = −
5. C
S
G
R
C
EE 5 H: () ( ) 3 4π () ( ) 4 3π .
P . V F. P.76.
() ( ) ( )( ) 3 4 4 1 2π = π − π = .
() ( ) ( )( ) 4 3 3 1 2π = π + π =
F .76 U
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A
, 0, π/6, π/4, π/3 π/2. .
C , ( , , ), . L
F. P.77 P.78. E , = 0 = 2π
. O
π/2.
F .78 L
F .78 L
!
C ,
: .
F A
L .
EEA 2 F A C
( )
( )
( )
( )
+ = −
+ = +
− = +
− = −
L : S
( ) ( ) ( )( )
( ) ( )
, ,
, 1, 0
−= = − −
= =
F. P.79.
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F .79 = −
E −= − = , − . P ,
( ) ( )2 2 −= . E :
( ) ( ) ( )( ) ( )
( ) ( )( )
22 2 2
2 2 2 2
2 2
1
2 2
2 1
− + − = − − + −
− + + − +
= − − − + + −
C 2 2 1 + = ,
( ) − = +
S − ( ) − = ( ) − = − ,
( ) + = − S :
( ) ( ) 2
2
2 2
π + = − +
π = − −
π π = − + −
= +
− .
EE 6 H ( ) 12 15π = ° .
12 3 4 3 4 3 4
1 1 3 1 1 3
2 22 2 2 2
π π π π π π π = − = +
+ = + =
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A ,
. H = ( ) + ( ) +
( )
2 2
2 2 2
2
2 2
2
2 1 1
1 2
=
= −
= − + =
= −
S 2 2 ,
2 21 2 1 2 2 2
+ −= =
2 . M .
F
H : (), (), () ()
. S F. P.80 P.83.
DEFC 8 , ,
1
1 1
= =
= = =
O ( ) 0. O ,
. C 1 < 1 < , 1 > 1 ≥ .
F .80 L F .81 L
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F .82 L F .83 L
L , , , . L ; . L F. P.84 . T , A. D , S, T C.
F .84 L
EE 7 H θ 3
,π
π
1
3θ = − .
D 2 2 1θ + θ = ,
2 1 8 8 2 2 1 9 9 9 3
θ = − = θ = ± = ±
E θ [π 3π/2] θ . P ,
2 2 32 2 2 2
2 11 3
−θθ = − θ = = = −
θ
I , 2π. S ,
π
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( ) ( )
( )
+ π π + π −+ π = = = =
+ π π − π −
S 2 2 1θ + θ = 2 2 , , :
2 2 2 21 1 + = + =
L . P ,
( ) ( )
( )
+ ++ = =
+ −
A,
( )
1
++ =
−
R −
( ) 1
−− =
+
L
. C , θ
, ( θ), ( θ)
(). V F. P.85. L θ ,
, ,
θ = θ = θ =
ady
hipopu
θ
F .85
EE 8 H F. P.86.
F .86
A 30. L 5 . P,
1 330 30
5 2 5 2
= ° = = ° =
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5
2 =
5 3
2 = .
EE 9 P F. P.87, θ.
F .87
E θ . E θ . P ,
= θ = θ
= θ
= = θ
θ.
C ( ),
, . V F. P.88. L , , , C.
F .88 E ,
EEA 3 :
= =
C: 2 2 2
2 2 2
2 2 2
2
2
2
= + −
= + −
= + −
F .89
V F. P.89. S . D
( ( ) π − = ), = = . P ,
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( ) ( ) = . P ( ),
( ) L S. P L C,
( )
( )( )
( ) ( )
22
2
22
22
2 2 2 2 2
2 2
2
2
2
2
π+ − ≤
= π + + π − >
= + − π − = −
= + − +
= + −
L L C .
EE 10 U = 2 = 3 = 40. H .
D L C
2 2 2 2 4 9 12 40 13 12 0.766 3.808 = + − = + − ° ≈ − × =
E 3.808 1.951=
. U L S,
3 0.6428 40 3 0.988
1.951 1.951
×°= ≈ × ≈ ≈
U ( ):
1. ( , E 10);
2. , :
3. ;
4.
.E T P L
S C 180 ( π ).
U ; , .
EE 11 E , = 30, = 2 = 3. H .
. P L C,
( )
2 2 2
2
2
4 9 6 3 2
= + −
= + −
P , 2 3 3 5 0 − + = . R ,
3 3 27 20
2 1.275 3.921
± −
=
≈
H , F. P.90.
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F .90 D = 2, = 3 = 30
EECC .7
H E 1−6 . N .
1.3
π
2.3
4
π −
3.
2
3
π 4.
7
12
π
5. 512
π 6. 1112
π
E E 7−12, .
7. ( ) π + 8. ( ) 2 π − 9.3
2
π
−
10.3
2
π
+
11. + 12.
−
+
E E 13−16, .
13. ( )4 4 2 − = 14.1
1 2
−= =
+
15. 21
1 2
−=
+ 16.
2 2
−= −
+
17. E 3 .
18. E 3 .
E E 19−22, . C ?
19. ( ) 2 = 20. ( ) 2
= 21. ( )( ) = π 22. ( )
2
π=
23. D 2 3
π
= − .
24. D 1 4
π
= + +
.
E E 25−30, , θ θ θ . H θ
.
25.3
, ,5 2
π θ = θ π
26. 2, 0,2
π θ = θ
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CAPTULO 1
L
C
T ,
1642−−−−1727
, 1687
N L N, , , .
1821−−−−1894
E C . T :
• , ,
• , .
A . E C , , .
E . C .
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1.1 , C
E .
L . L
.
1 ( ) L ,
24.9 =
() C 2 ?
() C = 1 = 2 ?
L 1, 2 ∆
∆ :
[ ]2 2
2 11 22 1
4.9 4.9 ,
−∆= =∆ −
() E 2 ( 0, 2),
( ) ( )2 24.92 2 4.9 09.8 /
2 0
−∆= =
∆ −
() E 1, 2,
( ) ( )2 24.92 2 4.9 114.7 /
2 1
−∆= =
∆ −
2 C E 1 () = 1? () = 2?
P , . S , = 1? P , 1, 1 + = 1 :
[ ] ( ) ( )2 24.9 1 4.8 1
V 1, 1
∆ + −+ = =
∆
N = 1 = 0 , . P
. L T 1 ∆ /∆
. E , 9.8 / . E 9.8 .
E , T 2 2, 2 + , = 2. L 19.6 / .
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1 V 1, 1 + 2 V 1, 1 +
E E 2, , +
( )2 24.9 4.9
∆ + −=
∆
P , ( ) = 1 = 2, . D, . E
( )2 2
0
4.9 4.9
→
+ −=
L 0" " → … . N
= 0 . S , .
3 S , + ( )2
+ . D , .
L , +
( ) ( )
( )
2 2 2 2 2
2
4.9 4.9 4.9 2
4.9 2
9.8 4.9
+ − + + −=
+=
= +
L . S 9.8 4.9(0) 9.8 + = 0. P
, , 9.8 /. E , =1 = 2, 9.8/ 19.6 /, .
C C A
E , 4 . E ( ) = , F. 1.1.
O 0.1 2 10 1.7 2
40, 1.7 − 0.1 = 1.6 2 40 − 10 = 30 . L 10 40
21.7 0.1 1.6 0 053 /40 10 30
−= ≈ −
−
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1.1 L
E ( ) =
= 10 = 40. E , . T .
4 C 60?
P , 60. L , , , ( ) = = 60. E
F. 1.1; (2, 0) (69, 5),
25 0 0.0746 /69 2
−≈
−
60.
C
T ; . L
2 ( ) . E π :
22
= π = π
E π :
2 = π
C π?
L , , .
S ( F. 1.2). E , , , , . (D , F. 1.2 180 , 2 . E 180 .) E .
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1.2 U
U ,
, . U , ∆, F. 1.2. C 2π ( 1 2π),
2π/ . S , . S , :
2
2 2
1 1 2
2 2
π= =
π π = =
π π
=
E :
2
2
π=
π π=
D ( ) 2 π = ,
2
π =
A, = π/ 0 , ( ) π = , 1. P
2 = π , ( ) 22 2 (1) π = π , , ,
.
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1.2
P , , . E, . A , .
1 D 2 1
( )1
−=
− = 1.
O ( ) = 1 ( ).
P ≠ 1, ( )
:
( )( )1 1( ) 1 1
1
− += = + ≠
−
L 1 = + , , (1, 2). E
F. 1.3. A (1) , ( )
2 1. P , ( )
2 1, , , ( ) 2 1. E
2
1 1
1 ( ) 2 2
1
→ →
−= =
+
1.3 L
2 1( ) 1
−=
−
2 Q ( )212( ) 1
= + ?
L ( ) = 0. D ,
2 . (R = /, > 0,
.) I ( ) 2 . N ( ) E 1. S , () 0. (L T 4 .)
4
E , ( ) , 2.71828, 0. E S 3.4
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( )212
0 0 ( ) 1 2.718281828459045
→ →= + = = …
E .
O . E
( )21 0.00001 1.0000000001+ = 1 , , 100000000001 1= . C , . E ,
.
L S 1.1 .
C 1
S ( ) , , ( )
, , ,
( )
→
=
E ; . P , . P , . S , . S ,
S 1.5, , T 2−4, .
3 H ()
→
()
→
( ).
E , () : A ? L :
→
=
E , () :A ? L ; :
→
=
E E 3 ( ) → ( ) . ( )
= ( )( ), . E
( ) → ( ) . E ( ) 0
= .
4 E:
()2
22
2
5 6
→−
+ −
+ + ()
1 1
→
−
− ()
24
2
16
→
−
−
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C: AE; 7. E. 77
C 0 .
()( )[ ]
( )( )
2
22 2
2
2 2 1 C
2 35 61
E 23
→− →−
→−
+ − + −=
+ ++ +
−= = −
+
2 1 32 3
− −= = −− +
()
( )
( )
1 1
F . S
C
→ →
→
−−
= =− −
− −=
−
2
1 1
→
−= = −
()
( )( )
( )( )
24
24
2 4
16
2 2
16 2
→
→
−=
−
− +=
− +
( )( )( )
( )( ) ( )( )
4
4
4
4 4 2
1 1 1
4 4 2 2 324 2
→
→
−=
− + +
= = =+ ++ +
U = = . P , ( ) 1 = 0. C F. 1.4(), 1
0; .E , ( ) = , ( )
( ) .
5 S1 2
( )0 2
≠=
= (V F. 1.4(.) E 2 2 ( ) 2 → →= = (2) 1 = .
1.4 ()0
1
→ ; ()
2 ( ) 2
→= , (2) 1 =
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L ; ( ) → = ( ) → = , = ( E 31 S 1.5.) A
, , . (V F. 1.5.)
C 2
S ( ) (, ) = , ( )
, ( )
=
( )
−→
=
S ( ) (, ) = , ( )
, ( )
=
( )
+→
=
O + ( ) − ( ).
6 L ( ) = ( F. 1.6) −1
1 = 0:
0 0 ( ) 1 ( ) 1
− +→ →
= − =
( ) −1 ( −1) 0, 1
0.
1.6 0
( )
→
, 0 0
( ) 1, ( ) 1
− +→ →
= − =
C E 6, () :
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3. : ( ) ( )
→
=
4. : ( )
→
=
5. :( )
0( )
→= ≠
S ,
6. : [ ] ( )
→= , > 0 , ≠ 0 < 0.
S ( ) ( ) ≤ ,
7. : <
L R 1−6 . T R 7, ( ) ( ) ≤ .
E , 1 T 2 . E , 5 ,
. T .P () → = ( ) () → =
.
9 H ()2
3 2
4
2 7
→
+ +
− + ()
2 2 1
→+ .
() L 2
3 2
4
2 7
+ +
− + () , ,
. E T 2 , . P ,
2 23 2
3 2 3 2
4 4 2 7 0
2 7 2 7
→
+ + + += − + ≠
− + − +
() E () ( )2 2 1 2 2 1 5 → + = + = . E R P ( 6
T 2)
2 2 1 5
→+ =
E T 2. (V S P.6 .)
A 3
1. S ( ) ,
( ) ( )
→
=
2. S ( ) ( ) () ≠ 0,
( ) ( )
( ) ( )
→=
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E . S . E , ( F. 1.8).
1.8 L
A 4
S ( ) ( ) ( ) ≤ ≤ ,
= . S
( ) ( )
→ →
= =
E ( )
→
= . E .
10 D ( ) 0 → = , ( ) 0 → = .
P ( ) ( ) ( ) − ≤ ≤ ( ) − ( ) 0 , ,
T E, ( ) .
1.2
1. H: ()1
( )
→ −
, ()0
( )
→
()1
( )
→
, F. 1.9.
1.9
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2. P ( ) = F. 1.10, .
()1
( )
→
()2
( )
→
()3
( )
→
1.10
E E 3−6, F. 1.10.
3.1
( )
−→
4.1
( )
+→
5.3
( )
+→
6.3
( )
−→
E E 7−36, .
7. ( )24
4 1
→
− + 8. ( )( )2
3 1 2
→
− − 9.3
3
6
→
+
+ 10.
2
4
4
→ − −
11.
2
1
1
1
→
−
+ 12.
2
1
1
1
→ −
−
+ 13.
2
23
6 9
9
→
− +
− 14.
2
22
2
4
→ −
+
−
15.22
1
4 → − 16.
2
2 30
3 4
→
+
− 17.
9
3
9
→
−
− 18.
0
4 2
→
+ −
19.( )2
→ π
− π
π 20.
2 2
→ −− 21.
0
2
2
→
−
− 22.
2
2
2
→
−
−
23.2
21
1
2 1
→
−
− + 24.
2
2
4 4
2
→
− +
− 25.
1
4 4
→ + − − 26.
2
1
1
3 2
→
−
+ −
27.( ) ( )
2
2 20
3
2 2
→
+
+ − −
28.( ) ( )2 2
0
1 1
→
+ − − 29.
21
4 3
1
→
− +
−
30.3
1
1
1
→ −
+
+
31.4
32
16
8
→
−
− 32.
2 3
1 38
4
2
→
−
− 33.
22
1 4
2 4 →
− − −
34.22
1 1
2 4 →
− − −