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Eixo das Abscissas
Eixo das Ordenadas
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4º Quadrante3º Quadrante
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Conceitos basicos parte2

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Conceitos basicos parte2

  1. 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. 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,
  3. 3. 02, )** %
  4. 4. * 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
  5. 5. 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
  6. 6. 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 ,
  7. 7. /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
  8. 8. 9? 0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
  9. 9. 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 ))%% $$+# #$*% % %$ +$*% $4% 6 +# +$54% %) % #$*% % %$ +$*% )*3 ))%% #) +# #$*% % %$ +$*%
  10. 10. 6 +# +$54% %) *%% % #$*% % %$ +$*% )*3 ))%% )%#$* +# #$*% % %$ +$*% 0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8 0C84JI9 /0 ?8JI9 A07:69 A07:69 98=4/00 , 1?8JI9
  11. 11. /9 0A07:69 ,8049 007480 /97M849 /0
  12. 12. .985?89 /0 .302,/, /0
  13. 13. 0
  14. 14. .98,/97M849 /0
  15. 15. 96?JI9
  16. 16. 0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
  17. 17. 0C84JI9 /0 ?8JI9 %0,6 /0 ),4H@06 %0,6 0C84JI9 (7, 1?8JI9 /4B =0 ?7, ! 1 @ ;?,8/9 ;?0 9 /97M849 ;?0 9 .98,/97M849 =I9 =?-.985?89= /9 .985?89 /9= 8P709= 0,4= R 4=9 K R 0
  18. 18. R : R R - . - -=0@,JI9 %#7$% +# +$54% ( ,(3, ( ) $4% 6 )0% 6 % #%( %$ +$*% ,%() R ( %) '+) )+ -())4% $7* *# )$*% )*% 6 6 % %$ +$*% *%%) %) ,%() *(+7,) 2 ,(3, $$$* - (*,#$* %) '+) ) %(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
  19. 19. 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 - - - -
  20. 20. - - 96?JI9 R %) $4% -)* ,)4% %( /(% %#% - 6 % $%#$%( )* $4% %(3 )( $+% $4% -)* ,)4% %( /(% %(*$*% - - %% R
  21. 21. %#% - )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
  22. 22. /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=
  23. 23. - . 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
  24. 24. %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
  25. 25. '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
  26. 26. 0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
  27. 27. %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
  28. 28. %0:0=08,JI9 HC., A07:69 ?8JI9 0C84/, :9 %,79= A07:69 98=4/00 , 1?8JI9 : R R ,6 ;?0 - - - - - 007480
  29. 29. 0 98=?, 9 2HC.9 /, 1?8JI9 7,.,8/9 ,62?8= :989= 0 /007480 9 /97M849 /, 1?8JI9 96?JI9
  30. 30. -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
  31. 31. #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 ! - .
  32. 32. - .
  33. 33. x y (x,y)(-x,y) ·· ! - .
  34. 34. - .
  35. 35. x y (x,y) (x,-y) · · !
  36. 36. - .
  37. 37. - .
  38. 38. x y (x,y) (-x,-y) · · 0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
  39. 39. #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
  40. 40. #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
  41. 41. #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
  42. 42. #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
  43. 43. #9:40/,/0= /,= ?8JO0= 48,6 0C84JI9 05, : R R 0 /0=428079= :9 ?7, :,0 /9 =0? /97M849 K 0=:
  44. 44. 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 %)*, # ∞
  45. 45. $*, #
  46. 46. ∞ - -
  47. 47. - )4% %) /(%) 0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
  48. 48. #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
  49. 49. #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
  50. 50. #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
  51. 51. #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
  52. 52. #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
  53. 53. #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
  54. 54. #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 *# %$, ,%* ( -% #
  55. 55. ,%* ( # #
  56. 56. *# +# %$*% $1-4% # - 0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
  57. 57. #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
  58. 58. ?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
  59. 59. 0:0=08,JI9 2HC., K ?7, 0., 394B98,6
  60. 60. 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
  61. 61. ?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=
  62. 62. R 0:0=08,JI9 2HC., K ?7, 0., 9-6M;?, .97 ?7 P84.9 B09 07 #
  63. 63. # 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
  64. 64. =-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
  65. 65. 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
  66. 66. %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
  67. 67. ?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 ∆
  68. 68. @,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
  69. 69. ∆ ∞
  70. 70. ∞ ∆ 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
  71. 71. ?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
  72. 72. 98/0 $ K ?7 8P709 48049 :9=44@9 (7, 1?8JI9 :9648974,6 /0 2,?
  73. 73. /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
  74. 74. ?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
  75. 75. R +09 07 - !I9 K =9-050.4@, 807 4850.4@, !I9 K 6474,/,
  76. 76. 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., - -
  77. 77. 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

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