CÁLCULO Adams Essex

8/15/2019 CÁLCULO Adams Essex

1/112


2/112


3/112

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 :


4/112

C: AE; 7. E. 2

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 = .


5/112

C: AE; 7. E. 3

() 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.


6/112

C: AE; 7. E. 4

(, ∞)

(−∞, ]

(−∞, ∞) 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 − < + ≤ +


7/112

C: AE; 7. E. 5

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∩ = ∪ =


8/112

C: AE; 7. E. 6

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

≥= −


9/112

C: AE; 7. E. 7

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.


10/112

C: AE; 7. E. 8

() 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 .


11/112

C: AE; 7. E. 9

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

− ≥ −

.


12/112

C: AE; 7. E. 10

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


13/112

C: AE; 7. E. 11

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, .


14/112

C: AE; 7. E. 12

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 + ≤


15/112

C: AE; 7. E. 13

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

∆ − −= = = = −

∆ − − −


16/112

C: AE; 7. E. 14

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 ∆ ∆


17/112

C: AE; 7. E. 15

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 = − + , , = +


18/112

C: AE; 7. E. 16

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

= + = −


19/112

C: AE; 7. E. 17

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 , = .


20/112

C: AE; 7. E. 18

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


21/112

C: AE; 7. E. 19

( ) ( )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 + = + + ,


22/112

C: AE; 7. E. 20

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 () .


23/112

C: AE; 7. E. 21

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

= =


24/112

C: AE; 7. E. 22

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


25/112

C: AE; 7. E. 23

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,

).


26/112

C: AE; 7. E. 24

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

+ =


27/112

C: AE; 7. E. 25

− ≤ ≤ , − ≤ ≤

(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

− = + =


28/112


29/112

C: AE; 7. E. 27

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.


30/112

C: AE; 7. E. 28

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. ∞) .


31/112

C: AE; 7. E. 29

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 ,

.


32/112

C: AE; 7. E. 30

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

,


33/112

C: AE; 7. E. 31

( )( ) , 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 = + − .


34/112

C: AE; 7. E. 32

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

( ) =


35/112

C: AE; 7. E. 33

= 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

− .


36/112

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()).


37/112

C: AE; 7. E. 35

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().


38/112

C: AE; 7. E. 36

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


39/112

C: AE; 7. E. 37

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 ) − −


40/112

C: AE; 7. E. 38

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 ,

( )( ) ( ) =


41/112

C: AE; 7. E. 39

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 , / / .


42/112

C: AE; 7. E. 40

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 , .


43/112

C: AE; 7. E. 41

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

≥= −


44/112

C: AE; 7. E. 42

L ( ) , , = 0.

F .61 L H

EE 7 . L (F. P.62) :

1 0

( ) 1 0

N 0

>

= = −

2 1

( ) 1 2

2 2


45/112

C: AE; 7. E. 43

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


46/112

C: AE; 7. E. 44

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

= −

−


47/112

C: AE; 7. E. 45

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

≥=


48/112

C: AE; 7. E. 46

. 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 ,


49/112

C: AE; 7. E. 47

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 ,


50/112

C: AE; 7. E. 48

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

− − − − − − − − − − − +


51/112

C: AE; 7. E. 49

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:


52/112

C: AE; 7. E. 50

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 − − + .


53/112

C: AE; 7. E. 51

() 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− + + .


54/112

C: AE; 7. E. 52

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 .


55/112

C: AE; 7. E. 53

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).


56/112

C: AE; 7. E. 54

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


57/112

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


58/112

C: AE; 7. E. 56

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° = = = ° = = =


59/112

C: AE; 7. E. 57

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


60/112

C: AE; 7. E. 58

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.


61/112

C: AE; 7. E. 59

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 = − = +

+ = + =


62/112

C: AE; 7. E. 60

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


63/112

C: AE; 7. E. 61

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


64/112

C: AE; 7. E. 62

( ) ( )

( )

+ 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

= ° = = ° =


65/112

C: AE; 7. E. 63

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 ,


66/112

C: AE; 7. E. 64

( ) ( ) = . 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.


67/112

C: AE; 7. E. 65

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


68/112


69/112


70/112


71/112

CAPTULO 1

L

C

T ,

1642−−−−1727

, 1687

N L N, , , .

1821−−−−1894

E C . T :

• , ,

• , .

A . E C , , .

E . C .


72/112

C: AE; 7. E. 70

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 / .


73/112

C: AE; 7. E. 71

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

−= ≈ −

−


74/112

C: AE; 7. E. 72

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 .


75/112

C: AE; 7. E. 73

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 , , ,

.


76/112


77/112

C: AE; 7. E. 75

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


78/112

C: AE; 7. E. 76

( )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

→

−

−


79/112

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 =


80/112

C: AE; 7. E. 78

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, () :


81/112


82/112

C: AE; 7. E. 80

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,

( ) ( )

( ) ( )

→=


83/112

C: AE; 7. E. 81

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


84/112

C: AE; 7. E. 82

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 →

− − −

CÁLCULO Adams Essex

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