matematica

1. %$*%) 3)%)
#,0 ?8JO0= %0,4= /0 ),4H@06 %0,6
,4,8, 4,=
'0P-,6 #
,8049
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
89/?JI9
?=9 /0 1?8JO0= :9/0 =0 08.98,/9 07 48P70,= =4?,JO0= /, @4/,
.9080 :9 0A07:69 8, ,-06, /0 :0J9= /0 ?7, 695, 98/0 , .,/,
:9/?9 .90=:98/0 ?7 /00748,/9 :0J9 9? 89 :0J9 , =0 :,29
8?7, .98, /0 6?B ;?0 /0:08/0 /, ;?,84/,/0 /0 08024,
.98=?74/, !, ,8H64=0 .408MC., /0 108N7089= 07 M=4., 496924, 9?
.98974, :9 0A07:69 3H , 80.0==4/,/0 /9 ?=9 /0 1?8JO0=
.98.049 -H=4.9 /0 1?8JI9 =?20 ;?,8/9 89= /0:,,79= .97 ,
80.0==4/,/0 /0 0=,-060.0 ?7,
#
080 /94=
.985?89= /0 9-50.9= ;?0 1,J, , ,==9.4,JI9 /0 9/9 9 0607089 /9
:47049 .985?89 , ?7 P84.9 0607089 /9 =02?8/9
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

2. 89/?JI9
A07:69=
A07:69
!, =02?480 ,-06, /0 10;?L8.4,= 1,BE=0
.90=:98/0 , /00748,/,= 074==9,=
, 0=:0.4@, 10;?L8.4, 07 B
'
808,
808,

4. 02,
)**
%

5. *
970.4,6
A07:69
!9 =02?480 ;?,/9 ,==9.4, =0
, .,/, ?7 /0 .48.9 ,6?89= /0
?7, ?7, , .4/,/0 /0 ;?0 K
8,?,6
,8?06 0P-,6
%4, 67,/,
H49 4=-9,
,6, 05,
,8/, 4=-9,
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

6. 0C84JI9 /0 ?8JI9
0C84JI9
05,7 0 .985?89= 8I9 @,B49= (7, ! 9?
! /0C84/,
07 .97 @,690= 07 9? ?7, 1?8JI9 /0 07 K ?7,
.90=:98/L8.4, ;?0 , .,/, 0607089 -
1,B .90=:98/0 ?7 P84.9
0607089 .
47-964.,7080 0=.0@079= -
.
: . -
F 3,-4?,6 0:0=08, =0 , 1?8JI9 .979
:
- . -
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

7. 0C84JI9 /0 ?8JI9
0C84JI9
.,/, 0607089 -
/H =0 9 8970 /0
=0 ?7 0607089 -
0=4@0 ,==9.4,/9 , ?7 0607089 .
/4B =0 ;?0 . K ,

8. /0 -
/089,8/9 =0 . - 979 - 0 . L7 @,690= ;?0 @,4,7 89=
.985?89= 0 0.0-07 9 8970 /0
@,4H@06 - K .3,7,/,
0 , @,4H@06 . K .3,7,/,
:94= 9 @,69 /0 . /0:08/0 /9 @,69 /0 - 3,7, =0
! $
/0
?7, 1?8JI9 , ?7, 0A:0==I9 ;?0 ,/?B, , 02, ;?0 ,==9.4, 9= 9-50.9=
G= 0=:0.4@,= 47,208= 0:0=08,8/9 =0 :9 -
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
0C84JI9 /0 ?8JI9
0C84JI9
.985?89 98/0 0=H /0C84/, K 9 $
/0 9? .985?89 /0
:,4/, 0 0:0=08, =0 :9 .985?89 K 9 $
/0 9?
.985?89 /0 .302,/, 9 .985?89 /H =0 9 8970 /0
$
/0 9? .985?89 /,= 47,208= 0 0:0=08, =0 :9

9. 9?
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

10. 0C84JI9 /0 ?8JI9
A07:69
A07:69
-=0@,8/9 9= =02?480= /4,2,7,= 48/4;?0 ;?,4= /,= 06,JO0= =I9 1?8JO0=
1
2
3
4
1
2
3
4
5
6
X Y
1R
1
2
3
4
1
2
3
4
5
6
X Y
1R
1
3
5
7
-1
0
1
2
3
X Y2R
1
3
5
7
-1
0
1
2
3
X Y2R
2
4
6
8
0
1
4
3
5
X Y
6
9
3R
2
4
6
8
0
1
4
3
5
X Y
6
9
3R
96?JI9
$4% 6 +# +$54% %) % #$*%
% %$ +$*% $4% )*3 ))%% $$+#
#$*% % %$ +$*%
$4% 6 +# +$54% %) % #$*% % %$ +$*% )*3 ))%% #) +#
#$*% % %$ +$*%

11. 6 +# +$54% %) *%% % #$*% % %$ +$*% )*3 ))%% )%#$* +#
#$*% % %$ +$*%
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
0C84JI9 /0 ?8JI9
A07:69
A07:69
98=4/00 , 1?8JI9

12. /9 0A07:69 ,8049 007480
/97M849 /0

13. .985?89 /0 .302,/, /0

15. 0

16. .98,/97M849 /0

17. 96?JI9

18. 0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

19. 0C84JI9 /0 ?8JI9 %0,6 /0 ),4H@06 %0,6
0C84JI9
(7, 1?8JI9 /4B =0 ?7, !
1 @ ;?,8/9
;?0 9 /97M849 ;?0 9 .98,/97M849 =I9 =?-.985?89= /9 .985?89 /9=
8P709= 0,4= R 4=9 K R 0

20. R
: R R
- . -
-=0@,JI9
%#7$% +# +$54% ( ,(3, ( ) $4% 6 )0% 6 %
#%( %$ +$*% ,%() R ( %) '+) )+ -())4% $7*
*# )$*% )*% 6 6 % %$ +$*% *%%) %) ,%() *(+7,) 2 ,(3,
$$$* - (*,#$* %) '+) ) %(59) $) # -
)4% %))7,)
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
97M849
%06,4@,7080 ,9 /97M849 /0 ?7, 1?8JI9 /0@0 0 07 .98,
-
-
-
:,
- -
692 - -
tg - - π
!π !
Z
cotg - - !π !
Z
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

21. 97M849
A07:69
A07:69
98=4/00 ,= =02?480= 1?8JO0= 0,4= /0 @,4H@06 0,6 48/4.,8/9 :,, .,/,
?7, 9 =0? /97M849
-
-
-
-

22. -
-
96?JI9
R %) $4% -)* ,)4% %( /(%
%#% -
6 % $%#$%( )* $4% %(3 )( $+% $4% -)* ,)4% %(
/(% %(*$*% -
-
%% R

23. %#%
-
)8 6 %))7, # R ) -
- $*4%
∞
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
%0:0=08,JI9 HC.,
0C84JI9 /0 HC.9 0 #6,89 ,0=4,89 9298,6
2HC.9 /0 ?7, 1?8JI9 :0740 @0 7?49 1,.467080 9/, , 0@96?JI9 /,
1?8JI9 #9 4==9 .,/, @0B 7,4= 89 7?8/9 ,.?,6 K 47:9,80 :9==?4 ,62?7,=
.97:0L8.4,= 06,4@,= G 604?, 0 480:0,JI9 /0 2HC.9= = 2HC.9= 0=I9
:0=080= :9 0A07:69 89= 0A,70= 6,-9,94,4= 8,= =98/,208= 8,= ,.JO0= /,
-96=, /0 @,690= 0.
0C84JI9
05, : R R ?7, 1?8JI9 0,6 /0 @,4H@06 0,6 /0 /97M849

24. /0 ?7, 1?8JI9 89 :6,89 K 9 .985?89 /0 :,0= 9/08,/9=
- .
R ,4= ;?0 . - 0 -
47-964.,7080 0=.0@079=

25. - .
R : -
. -
0 x
y
Eixo das Abscissas
Eixo das Ordenadas
1º Quadrante2º Quadrante
4º Quadrante3º Quadrante
1 2-2 -1
1
2
-2
-1
P = (2,1)
0 x
y
Eixo das Abscissas
Eixo das Ordenadas
1º Quadrante2º Quadrante
4º Quadrante3º Quadrante
1 2-2 -1
1
2
-2
-1
P = (2,1)
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

26. %0:0=08,JI9 HC.,
97M849 0 98,/97M849
,/9 9 2HC.9 .,0=4,89 /0 ?7, 1?8JI9 : R R :9/0 =0 /4B0 ;?0
:950.JI9 /, .?@, =9-0 9 04A9 -- /H 89= 9 /97M849 /, 1?8JI9
:950.JI9 /, .?@, =9-0 9 04A9 .. /H 89= 9 .98,/97M849 /, 1?8JI9

27. '9/, , 0., @04.,6 ( ;?0 :,==, :9 ?7 :989 /9 /97M849 /, 1?8JI9
480.0:, 9 2HC.9 /, 1?8JI9 07 89 7HA479 ?7 :989
A07:69
98=4/00 , 1?8JI9 : R R .97 , =02?480 0:0=08,JI9 2HC.,
6
x
y
1 4
2
y = f(x)
r
P
6
x
y
1 4
2
y = f(x)
r
P
007480 9 /97M849 0 9 .98,/97M849 /0
96?JI9

28. 0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

29. %0:0=08,JI9 HC.,
A07:69
#,, .98=?4 9 2HC.9 /0 ?7, 1?8JI9 -,=, ,4-?4 @,690= /9 /97M849 G @,4H@06 -
8, =0808J, 7,07H4., ;?0 /0C80 , 1?8JI9 0 .,6.?6, 9= @,690= .90=:98/080= /,
@,4H@06 .
A07:69
98=?, 9 2HC.9 /, 1?8JI9 : R R ,6 ;?0 - -
- . - -
210-1-2
2
1
-1
-2
x
y
-1/2
1/2
1/2
1/4
( )
2
x
xfy ==
210-1-2
2
1
-1
-2
x
y
-1/2
1/2
1/2
1/4
( )
2
x
xfy ==
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

30. %0:0=08,JI9 HC.,
A07:69 ?8JI9 0C84/, :9 %,79=
A07:69
98=4/00 , 1?8JI9 : R R ,6 ;?0 -
-
-
- -
007480

31. 0
98=?, 9 2HC.9 /, 1?8JI9 7,.,8/9 ,62?8= :989= 0 /007480
9 /97M849 /, 1?8JI9
96?JI9

32. -2 -1 1 2
-2
-1
1
2
x
y
·
°
R
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
80=0.JI9 /0 ?7 HC.9 .97 9= 4A9=
!
:989= /9 4:9
!
:989= /9 4:9
x
y
·
!083?7, 480=0.JI9 .97 9 04A9 --
(7, 480=0.JI9 .97 9 04A9 ..
x
y
·
··
?,= 480=0.JO0= .97 9 04A9 --
(7, 480=0.JI9 .97 9 04A9 ..
x
y
· ··
'L= 480=0.JO0= .97 9 04A9 --
(7, 480=0.JI9 .97 9 04A9 ..
x
y
!083?7, 480=0.JI9
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

33. #989= /0 480=0.JI9 /0 /94= 2HC.9=
(7
! /9= 2HC.9= /0 /?,= 0;?,JO0= K ?7 :989 ;?0
=,4=1,B ,7-,= ,= 0;?,JO0=
%0=96@0 =0 9 =4=07,
. -
. -
x
y
f
g
·
·
94= :989= /0 480=0.JI9
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
4704,= /0 ?7 2HC.9
!
- .

34. - .

35. x
y
(x,y)(-x,y)
··
!
- .

36. - .

37. x
y
(x,y)
(x,-y)
·
·
!

38. - .

39. - .

40. x
y
(x,y)
(-x,-y)
·
·
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

41. #9:40/,/0= /,= ?8JO0=
?8JI9 9-050.4@,
0C84JI9
(7, 1?8JI9 : /4B =0
=0 9 .98,/97M849 .948.4/0
.97 9 .985?89 /0 .302,/, =9 0;?4@,60 , ,C7, ;?0 9/9 0607089 /0
K 47,207 /0 :069 7089= ?7 0607089 /0 .
-
: . -
4,2,7, ?8JI9 =9-050.4@, ?8JI9 8I9 =9-050.4@,
X Y
f
X Y
f
210-1-2
2
1
-1
-2
x
y f(x) = 2x
210-1-2
2
1
-1
-2
x
y f(x) = 2x
420-2-4
4
2
-2
-4
x
y g(x) = x 2
420-2-4
4
2
-2
-4
x
y g(x) = x 2
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
#9:40/,/0= /,= ?8JO0=
?8JI9 850.4@,
0C84JI9
(7, 1?8JI9 : /4B =0
=0 , 9-50.9= /410080=
.90=:98/07 47,208= /410080= -
-
: -
-
-
-
4,2,7, ?8JI9 4850.4@, ?8JI9 8I9 4850.4@,
X YfX Yf
210-1-2
2
1
-1
-2
x
y f(x) = 2x
210-1-2
2
1
-1
-2
x
y f(x) = 2x
420-2-4
4
2
-2
-4
x
y g(x) = x 2
420-2-4
4
2
-2
-4
x
y g(x) = x 2
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

42. #9:40/,/0= /,= ?8JO0=
?8JI9 450.4@,
0C84JI9
1?8JI9 K
=0 K =47?6,80,7080 4850.4@, 0 =9-050.4@,
.
-
: . -
4,2,7, ?8JI9 -450.4@, ?8JI9 8I9 -450.4@,
X YfX Yf
210-1-2
2
1
-1
-2
x
y f(x) = 2x
210-1-2
2
1
-1
-2
x
y f(x) = 2x
420-2-4
4
2
-2
-4
x
y g(x) = x2
420-2-4
4
2
-2
-4
x
y g(x) = x2
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
#9:40/,/0= /,= ?8JO0=
#,4/,/0
0C84JI9
(7, 1?8JI9 : R R /4B =0 8?7 =?-.985?89 /9 =0?
/97M849 =0 - - :,, 9/9 9 - 0 - 07 ,C.,7080 =0 ,
1?8JI9 K :, 9 =0? 2HC.9 K =47K4.9 06,4@,7080 ,9 04A9 ..
- -
: - -
420-2-4
4
2
-2
-4
x
y g(x) = x2
420-2-4
4
2
-2
-4
x
y g(x) = x2
-=0@,JI9
6 +# +$54% ( $*4% $4% 6 $ *,
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

43. #9:40/,/0= /,= ?8JO0=
#,4/,/0
0C84JI9
(7, 1?8JI9 : R R /4B =0 $ 8?7 =?-.985?89 /9 =0?
/97M849 =0 - - :,, 9/9 9 - 0 - 07
09704.,7080 =0 , 1?8JI9 K M7:, 9 =0? 2HC.9 K =47K4.9
06,4@,7080 G 94207 /9 01008.4,6 - -
: - -
420-2-4
4
2
-2
-4
x
y g(x) = x3
420-2-4
4
2
-2
-4
x
y g(x) = x3
-=0@,JI9
+# +$54% $4% 6 ( $# 7#( / ) '+ $4% %))+ (
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

44. #9:40/,/0= /,= ?8JO0=
?8JI9 #04N/4.,
0C84JI9
(7, 1?8JI9 : R R /4B =0
%
$ =0 :,,
9/9 9 - :008.080 , 0A4=0 :069 7089= ?7 8P709 0,6 8I9 8?69
,6 ;?0 - - 9 7089 8P709 :9=44@9 ;?0 =,4=1,B ,
0;?,JI9 .3,7, =0 :0M9/9 /0 -
: - -
?8JI9 8I9 :04N/4., ?8JI9 :04N/4.,
420-2-4
4
2
-2
-4
x
y f(x) = x
420-2-4
4
2
-2
-4
x
y f(x) = x
0
2
1
-1
-2
x
y
2 3--2-3
( ) seng x x= 2
0
2
1
-1
-2
x
y
2 3--2-3
( ) seng x x= 2
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

45. #9:40/,/0= /,= ?8JO0=
48,6
0C84JI9
05, : R R 0 /0=428079= :9 ?7, :,0 /9 =0? /97M849
K
0=:

46. 07 =0 - 0=: -
:,, 9/9 9 - :008.080 , : -
: - 0=:
-
: -
-
/4B =0 ?7
9? $ /0 =0 , =?, 47,207 19 8?6,
-
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
#9:40/,/0= /,= ?8JO0=
48,6 A07:69
A07:69
-=0@0 , 0:0=08,JI9 2HC., /, 1?8JI9 0 0=?/0 ;?,89 ,9 =48,6
-2 2 4
- 2
-1
1
2
3
x
y
-1 3 51
( )f x
-2 2 4
- 2
-1
1
2
3
x
y
-1 3 51
( )f x
96?JI9
6 %)*, # ∞

47. $*, #

48. ∞
-
-

49. - )4% %) /(%)
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

50. #9:40/,/0= /,= ?8JO0=
989984,
0C84JI9
05, : R R 0 /0=428079= :9 ?7, :,0 /9 =0? /97M849
/4B =0
0=: 0=4,7080 .0=.080 07 =0 :,,
;?,4=;?0 -
0 - :008.080= , ,4= ;?0 -
- =0 07
-
- 0=: -
-
-
-
: -
-
-
-
/4B =0
0=: 0=4,7080 /0.0=.080 07 =0 :,,
;?,4=;?0 -
0 - :008.080= , ,4= ;?0 -
- =0 07
-
- 0=: -
-
-
-
: -
-
-
-
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
#9:40/,/0= /,= ?8JO0=
989984,
0C84JI9
(7, 1?8JI9 /4B =0 % 07 =0 K .0=.080 9? /0.0=.080 07
?8JI9 0=4,7080 .0=.080
x2x10 x
y
f(x 1)
f(x 2)
f(x)
x2x10 x
y
f(x 1)
f(x 2)
f(x)
?8JI9 0=4,7080 /0.0=.080
0
x
y
x2x1
f(x 2)
f(x 1)
f(x)
0
x
y
x2x1
f(x 2)
f(x 1)
f(x)
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

51. #9:40/,/0= /,= ?8JO0=
989984,
0C84JI9
05, : R R 0 /0=428079= :9 ?7, :,0 /9 =0? /97M849
/4B =0
07 =0 -
- ;?,4=;?0 ;?0
=05,7 9= @,690= /0 -
0 - :008.080= ,
-
-
: -
-
,C.,7080
0
x
y
x2x1
f(x 1) = f(x 2) f(x)
0
x
y
x2x1
f(x 1) = f(x 2) f(x)
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
#9:40/,/0= /,= ?8JO0=
A079= %06,4@9=
0C84JI9
05, : R R
/4B =0 ?7
9?
/0 - 9? 07 ?7
7HA479 69.,6 07 -
=0 0A4=0 ?7, @4B483,8J, /0
,6 ;?0
-
: -
/4B =0 ?7 $
9?
/0 - 9? 07 ?7
7M8479 69.,6 07 -
=0 0A4=0 ?7, @4B483,8J, /0
,6 ;?0
-
: -
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

52. #9:40/,/0= /,= ?8JO0=
A079= -=96?9=
0C84JI9
05, : R R
/4B =0
/0 - 9? 07 ?7 7HA479
,-=96?9 07 -
=0
-
: -
/4B =0 $
/0 - 9? 07 ?7 7HA479
,-=96?9 07 -
=0
-
: -
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

53. #9:40/,/0= /,= ?8JO0=
A079= A07:69
A07:69
0 x
y
a b c d
f(a) m ínimoabsoluto
f(b) m áximorelativo
f(c) m ínimorelativo
f(d) m áximoabsoluto
0 x
y
a b c d
f(a) m ínimoabsoluto
f(b) m áximorelativo
f(c) m ínimorelativo
f(d) m áximoabsoluto
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

54. #9:40/,/0= /,= ?8JO0=
98.,@4/,/0
0C84JI9
05, : R R 0 /0=428079= ?7, :,0 /9 =0? /97M849
07 ,
07 =0 9 2HC.9 /0 0=H
,.47, /, 0., ,82080 07 ;?,6;?0 ?7 /9= :989= /0
07 ,
07 =0 9 2HC.9 /0
0=H ,-,4A9 /, 0., ,82080 07 ;?,6;?0 ?7 /9= :989= /0
?8JI9 .97 , .98.,@4/,/0
@96,/, :,, .47,
?8JI9 .97 , .98.,@4/,/0
@96,/, :,, -,4A9
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

56. #9:40/,/0= /,= ?8JO0=
#989= /0 8D0AI9
0C84JI9
(7, 1?8JI9 : R R /4B =0 ;?0 07 ?7
! 07
- =0 9 =084/9 /, .98.,@4/,/0 G 0=;?0/, /0 0 G /404, /0 K
/410080
PontodeInflexão
Concavidade
voltadaparacima
Concavidade
voltadaparabaixo
PontodeInflexão
Concavidade
voltadaparacima
Concavidade
voltadaparabaixo
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

57. #9:40/,/0= /,= ?8JO0=
98.,@4/,/0 0 #989= /0 8D0AI9 A07:69
A07:69
-=0@0 , 0:0=08,JI9 2HC., /, 1?8JI9 48/4.,8/9 ,= .98.,@4/,/0= 0
:989= /0 48D0AI9 /0
-3 -2 -1 1 2 3 4
-6
5
11
x
y
-15
( )f x
-3 -2 -1 1 2 3 4
-6
5
11
x
y
-15
( )f x
96?JI9
*# %$, ,%* ( -% #

58. ,%* ( #
#

59. *# +# %$*% $1-4% # -
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

60. #9:40/,/0= /,= ?8JO0=
?8JI9 474,/,
0C84JI9
05, : R R 0 /0=428079= :9 ?7, :,0 /9 =0? /97M849
1?8JI9 /4B =0
07 =0 0A4=07 # 0 0,4= ,6
;?0 # - :,, 9/9= 9= @,690= - :008.080= ,
#
R-
: # -
0C84JI9
(7, 1?8JI9 /4B =0
07 =0 8I9 19 6474,/, 07
?8JI9 474,/, 07
-1 0 1
1
x
y
( ) 2
f x x=
-1 0 1
1
x
y
( ) 2
f x x=
?8JI9 6474,/, 07
0 1 2 3
1
2
3
x
y
( )
1
g x
x
=
0 1 2 3
1
2
3
x
y
( )
1
g x
x
=
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

61. ?8JO0= 60708,0=
?8JI9 98=,80
0C84JI9
(7, 1?8JI9 -
R /0=428, =0 :9 !
(7, 1?8JI9 .98=,80 07 ,= =02?480= :9:40/,/0=
R 0

62. 0:0=08,JI9 2HC., K ?7, 0., 394B98,6

63. 0 , 1?8JI9 K :9=44@, 0 =0 , 1?8JI9 K 802,4@,
!I9 K =9-050.4@, 807 4850.4@,
F 6474,/,
0
b
x
y
y = b
b0
0
b
x
y
y = b
b0
0
b
x
y
y = b
b0
0
b
x
y
y = b
b0
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

64. ?8JO0= 60708,0=
?8JI9 C7
0C84JI9
= 1?8JO0= /, 197, - #- 98/0 # =I9 /0=428,/,= :9
(7, 1?8JI9 ,C7 07 ,= =02?480= :9:40/,/0=

65. R
0:0=08,JI9 2HC., K ?7, 0., 9-6M;?, .97 ?7 P84.9 B09 07
#

66. # K 9 /0.64@0 9? 48.648,JI9 /, 0., 07 06,JI9 ,9 04A9 /,= ,-.4==,=
K , 9/08,/, 8, 94207
F -450.4@, =9-050.4@, 0 4850.4@, !I9 K 6474,/,
0 # , 1?8JI9 K .0=.080 0 =0 # , 1?8JI9 K /0.0=.080
0
b
x
y
m0
0
b
x
y
m0
y mx b= + y mx b= +
b
m
−
b
m
−0
b
x
y
m0
0
b
x
y
m0
y mx b= + y mx b= +
b
m
−
b
m
−
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

67. =-9J, 9 2HC.9 /0 ?7, 1?8JI9
%0.,
;?,JI9 20,6 /, 0., - .
;?,JI9 /0 ?7, 0., ;?0 :,==, :069 :989 - . 0 07 /0.64@0 # :
. . # - -
;?,JI9 0/?B4/, 0.,= 8I9 @04.,4= /0.64@0 # 0 9/08,/, 8, 94207 :
. #-
%0., @04.,6 -
%0., 394B98,6 .
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

68. 0.64@0 .90C.4080 ,82?6, /0 ?7, 0.,
/0 ?7, 0., 8I9 @04.,6 ;?0 :,==, :9 -
.
0
- . K # . .
- -
- -
#,, 0.,= @04.,4= 9 /0.64@0 K 48/0C84/9
x
y
m=0
m0
m0
m indefinido
/0.64@0 /0 ?7, 0., :9/0 =0 480:0,/9 .979 ,BI9 04A9= .97 , 70=7,
?84/,/0 /0 70/4/, 9? .979 ,A, /0 @,4,JI9 04A9= .97 /410080= ?84/,/0= /0
70/4/,
#
@,4,JI9 @04.,6
@,4,JI9 394B98,6
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

69. %0.,= #,,606,= 0 #0:08/4.?6,0=
?,= 0.,= /4=48,= 8I9 @04.,4=
=I9
==0 #
# x
y
m1
m2m1=m2
?,= 0.,= /4=48,= 8I9 @04.,4=
=I9
==0 #
#
x
y
m1
m2
m1=-1/m2
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
?8JO0= 60708,0=
?8JI9 $?,/H4.,
0C84JI9
= 1?8JO0= /, 197, - - - 98/0
R 0
/0=428,7 =0 :9
98=4/0,8/9 ?7, 1?8JI9 ;?,/H4., - - - ,= =?,= ,MB0=
0,4= :9/07 =0 9-4/,= ,,@K= /, N7?6, %0=96@080
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

70. ?8JO0= 60708,0=
?8JI9 $?,/H4.,
(7, 1?8JI9 ;?,/H4., 07 ,= =02?480= :9:40/,/0=
R
0:0=08,JI9 2HC., K ?7, :,H-96, .?5,= .99/08,/,= /9 @K4.0
=I9
∆
98/0 ∆

71. @,69 ,-=96?9 /0 /00748, , ,-0?, /, 1?8JI9 9? =05,
;?,89 7,49 19 7,4= 10.3,/, K , :,H-96,
!I9 K =9-050.4@, 807 4850.4@, !I9 K 6474,/,
98.,@4/,/0 @96,/, :,, .47, 98.,@4/,/0 @96,/, :,, -,4A9

72. ∆
∞

73. ∞ ∆
0 x
y
a
b
2
−
a4
∆
−
V
0 x
y
a
b
2
−
a4
∆
−
V
0 x
y
a
b
2
−
a4
∆
− V
0 x
y
a
b
2
−
a4
∆
− V
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

74. ?8JO0= 60708,0=
?8JI9 #9648974,6
= 1?8JO0= ,8,64=,/,= ,80497080 .98=,80 ,C7 0 ;?,/H4., =I9
.,=9= :,4.?6,0= /0 1?8JO0= :9648974,4=
0C84JI9
= 1?8JO0= 0:0=08,/,= /, =02?480 197,
- $-$
$
-$
-
/0=428,7 =0 :9
= 8P709= 0,4= $ $
. . .
=I9 /0=428,/9= :9
=0 $ , 1?8JI9 :9648974,6 /4B =0 /0

75. 98/0 $ K ?7 8P709 48049 :9=44@9
(7, 1?8JI9 :9648974,6 /0 2,?

76. /0=428, =0 :9 1?8JI9 .P-4., $?,6;?0
1?8JI9 :9648N74,6 /0 2,? $ 07 89 7HA479 $ ,MB0= 0,4= /4=48,=
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8

77. ?8JO0= 60708,0=
?8JI9 N/?69
0C84JI9
9? % /0 ?7 8P709 0,6 - K /0=428,/9 :9 - 0
/0C80 =0 .979
-
- -
- -
1?8JI9 7N/?69 07 ,= =02?480= :9:40/,/0=
R 0

78. R +09 07 -
!I9 K =9-050.4@, 807 4850.4@, !I9 K 6474,/,

79. 0.0=.080 07 R 0 .0=.080 07 R M8479 42?,6 , 07 -
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
x
y
( )f x x=
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
x
y
( )f x x=
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
=-9J, 9 2HC.9 /0 ?7, 1?8JI9
62?7,= ?8JO0= H=4.,=
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
?8JI9 4/084/,/0
- -
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
x
y
?8JI9 ;?,/H4.,
- -
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
?8JI9 .P-4.,
- -

80. 0 1 2 3 4
0
1
2
3
4
x
y
?8JI9 ,4B ;?,/,/,
-
-
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
x
y
?8JI9 7N/?69
- -
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
?8JI9 ,.498,6
-
-
-10 -8 -6 -4 -2 2 4 6 8 10
-1.0
-0.5
0.5
1.0
x
y
?8JI9 =089
- sen -
-10 -8 -6 -4 -2 2 4 6 8 10
-1.0
-0.5
0.5
1.0
x
y
?8JI9 .9=089
- .9= -
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8