Agenda Ver 5
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abrilL M M J V S D
1 2 3 4 56 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24 25 2627 28 29 30
febreroL M M J V S D
12 3 4 5 6 7 89 10 11 12 13 14 1516 17 18 19 20 21 2223 24 25 26 27 28
marzo 2015
9
10
11
12
13
14
Lunes Martes Miércoles Jueves Viernes Sábado Domingo23 24 25 26 27 28 1
2 3 4 5 6 7 8
9 10 11 12 13 14 15
16 17 18 19 20 21 22
23 24 25 26 27 28 29
30 31 1 2 3 4 5
imprimido: 15/03/15 17:04
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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mayoL M M J V S D
1 2 34 5 6 7 8 9 1011 12 13 14 15 16 1718 19 20 21 22 23 2425 26 27 28 29 30 31
marzoL M M J V S D
12 3 4 5 6 7 89 10 11 12 13 14 1516 17 18 19 20 21 2223 24 25 26 27 28 2930 31
abril 2015
14
15
16
17
18
Lunes Martes Miércoles Jueves Viernes Sábado Domingo30 31 1 2 3 4 5
6 7 8 9 10 11 12
13 14 15 16 17 18 19
20 21 22 23 24 25 26
27 28 29 30 1 2 3
imprimido: 15/03/15 17:04
LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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junioL M M J V S D1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30
abrilL M M J V S D
1 2 3 4 56 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24 25 2627 28 29 30
mayo 2015
18
19
20
21
22
Lunes Martes Miércoles Jueves Viernes Sábado Domingo27 28 29 30 1 2 3
4 5 6 7 8 9 10
11 12 13 14 15 16 17
18 19 20 21 22 23 24
25 26 27 28 29 30 31
imprimido: 15/03/15 17:04
LUNES ___ / ____ / ____
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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julioL M M J V S D
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mayoL M M J V S D
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junio 2015
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Lunes Martes Miércoles Jueves Viernes Sábado Domingo1 2 3 4 5 6 7
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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VIERNES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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MARTES ___ / ____ / ____
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VIERNES ___ / ____ / ________________________________________
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agostoL M M J V S D
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junioL M M J V S D1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30
julio 2015
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Lunes Martes Miércoles Jueves Viernes Sábado Domingo29 30 1 2 3 4 5
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MIERCOLES ___ / ____ / ____
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VIERNES ___ / ____ / ________________________________________
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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VIERNES ___ / ____ / ________________________________________
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MIERCOLES ___ / ____ / ____
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julioL M M J V S D
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Lunes Martes Miércoles Jueves Viernes Sábado Domingo27 28 29 30 31 1 2
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agostoL M M J V S D
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septiembre 2015
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Lunes Martes Miércoles Jueves Viernes Sábado Domingo31 1 2 3 4 5 6
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noviembreL M M J V S D
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septiembreL M M J V S D
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octubre 2015
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Lunes Martes Miércoles Jueves Viernes Sábado Domingo28 29 30 1 2 3 4
5 6 7 8 9 10 11
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26 27 28 29 30 31 1
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diciembreL M M J V S D
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octubreL M M J V S D
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Lunes Martes Miércoles Jueves Viernes Sábado Domingo26 27 28 29 30 31 1
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eneroL M M J V S D
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Lunes Martes Miércoles Jueves Viernes Sábado Domingo30 1 2 3 4 5 6
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MARTES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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LUNES ___ / ____ / ____
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MIERCOLES ___ / ____ / ____
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JUEVES ___ / ____ / ________________________________________
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VIERNES ___ / ____ / ________________________________________
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SABADO ___ / ____ / ________________________________________
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DOMINGO ___ / ____ / ____
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
_________________
_________________
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________________
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________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
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________________
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________________
________________
________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
_________________
_________________
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________________
________________
________________
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________________
________________
________________
________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
_________________
_________________
_________________
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________________
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________________
________________
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________________
________________
________________
________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
_________________
_________________
_________________
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________________
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________________
________________
________________
________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
_________________
_________________
_________________
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________________
________________
________________
________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
_________________
_________________
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________________
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________________
________________
________________
________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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________________
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_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
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________________
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________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
_________________
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________________
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________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
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________________
________________
________________
_______________
HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
_________________
_________________
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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HRSCRS
OBJETIVO DE APRENDIZAJE
ACTIVIDADES INDIC. DE EVALUACIÓN
RECURSOS EVALUACIÓN
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CURSO: ________ ASIGNATURA:_____________ TRIMESTRE: _______
APELLIDO NOMBRES X123456789
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OBS: _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
CURSO: ________ ASIGNATURA:_____________ TRIMESTRE: _______
APELLIDO NOMBRES X123456789
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CURSO: ________ ASIGNATURA:_____________ TRIMESTRE: _______
APELLIDO NOMBRES X123456789
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APELLIDO NOMBRES X123456789
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APELLIDO NOMBRES X123456789
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APELLIDO NOMBRES X123456789
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APELLIDO NOMBRES X123456789
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APELLIDO NOMBRES X123456789
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APELLIDO NOMBRES X123456789
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Área, perímetro y volumen de figuras del plano y del espacio www.vaxasoftware.com
A = Área, P = Perímetro, V = Volumen Figuras del plano
Cuadrado
2aA = °= 90 interno Ángulo α
aP 4= °= 90 externo Ángulo β 2 diagonales Núm. =ND
Rectángulo
hbA ·=
hbP 22 +=
Paralelogramo
hbA ·=
abP 22 +=
Rombo
2·DdA =
aP 4=
2224 Dda +=
Trapecio
hBbA2+
=
cBbaP +++=
www.vaxasoftware.com
Trapecio recto
hBbA2+
=
hBbaP +++=
222 )( hbBa +−=
Triángulo equilátero
2
43
2· ahaA == °= 60 interno Ángulo α
aP 3= °=120 externo Ángulo β
ah23
= 0 diagonales Núm. =ND
Triángulo isósceles
2sen ··
2· AbahbA ==
baP += 2 , Aah sen · =
222 44 bha +=
Triángulo escaleno
2·hbA =
))()(( csbsassA −−−= 2
cbas ++=
cbaP ++= CaAch sen · sen · ==
Triángulo rectángulo
2·abA = BcAca cos · sen · ==
cbaP ++= AcBcb cos · sen · ==
222 bac +=
Pentágono regular
°=+== 72sen ·255210
85
2·5 22 rrbaA
bP 5= 222 44 bar += °=108 interno Ángulo α
°=−= 36sen ·252102
rrb °= 72 externo Ángulo β
°=+= 36 cos·5264
rra 5 diagonales Núm. =ND
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Hexágono regular
°== 06sen ·32
33 22 bbA °=120 interno Ángulo α
bP 6= °= 60 externo Ángulo β
°== 30·cos23 bba 9 diagonales Núm. =ND
Octágono regular
( )12
25,22tan
2 8282,52 ·tan·8··422
22
−=
°=−=°==
bbaabaA
°== 5,22·tan·16·8 abP °=135 interno Ángulo α
°= 5,22·cosra °= 45 externo Ángulo β
°= 2,52sen·2rb 20 diagonales Núm. =ND
Polígono regular de n lados
nanbanA °
==180 ·tan·
2·· 2
nn °−
=180)·2(
:interno Ángulo
α
nanbnP °
==180·tan·2·
αβ −°=180:externo Ángulo
nra °
=180·cos
nrb °
=180sen·2
2)3·(
:diagonales Núm.−
=nnND
Círculo
2 π rA =
rP π2=
Sector circular
°=
360 π 2 αrA
°=
180 π αrL
LrP += 2 α en grados sexagesimales
www.vaxasoftware.com
Segmento circular
⎟⎠⎞
⎜⎝⎛ −
°=
2sen
360·π2 ααrA
⎟⎠⎞
⎜⎝⎛ −=
2cos1 αrh
2sen·2 αrc =
°=
180 π αrL
cLP += h
chr82
2
+= α en grados sexagesimales
Triángulo circular
2sen 2 αrA =
2sen·2 αrc =
crP += 2 α en grados sexagesimales
Trapecio circular
( )°
−=360
π 22 αrRA
)(2360
)( π2 rRrRP −+°
+=α
α en grados sexagesimales
Corona circular
( )22 π rRA −=
)( π2 rRP +=
Elipse
baA · π=
)( π baP +≅
∫ +=2/π
0
2222 cossen4 dttbtaP
www.vaxasoftware.com
Figuras del espacio
Cubo (hexaedro)
2 6 aA =
3aV =
Prisma recto
cbcabaA ·2·2·2 ++=
cbaV ··=
Esfera
2π·4 rA =
3·π4 3rV =
Cilindro
)( π2 rhrATOTAL +=
2 π2 rABASES = hrALATERAL · π2=
hrV ·π· 2=
Cono
2 π· π rgrATOTAL +=
2 π rABASE = grALATERAL · π=
3· π 2 hrV = 222 rhg +=
www.vaxasoftware.com
se clasifican
propiedades propiedades relaciones
aplicaciones de
am/n = n√am orden
tipos utilidad aplicaciones
ej. tipos
Nº REALES
IRRACIONALES √aRACIONALES (a/b)
• (a/b)m . (a/b)n = (a/b)m+n
• (a/b)m : (a/b)n = (a/b)m-n
• [(a/b)m ]n = (a/b)m.n
• (a/b . c/d )m = (a/b)m. (c/d)m
• n√ an = a
• n√ a . n√b = n√ab
• n
n
b
a = n
b
a
• (n√a)m = n√am
• n√m√an= nm√a
• n√an= nm√an
• Productos ycocientes
• Suma• Racionalización
INTERVALOS
• Abiertos ( )• Cerrados [ ]• Semiabiertos ( ]
VALOR | |ABSOLUTO
DISTANCIA
ERRORES• Error Absoluto ∆x = | x- x |• Redondeo• Notación científica• Error Relativo δx= ∆x/x CIFRAS
SIGNIFICATIVAS
operaciones combinación
se relaciona con operaciones
ejemplo
factorización
de
polinomios
P(x) = (x- α1) . (x- α2) ….. (x- αk) C´(x)
POLINOMIOSP(x)= an x
n + an-1 xn-1 + …..+ a1 x0 + a0
• Suma : an xn + bn x
n = ( an + bn) xn • Resta: (- P(x) )
• Producto : (amxm) . (bn xn )= am bn x
m+n
• Cociente : P(x) = Q(X) . C(x) + R(x) ⇒ gr R(x) < gr Q(x)
• Regla de Ruffini ⇒ P(x) : (x-a)• Teorema del Resto ⇒ R = P(a)• Raíces de un polinomio
RAZONES ALGEBRAICASP(X) / Q (X)
• Reducir a comúndenominadorm.c.m
SUMA
• Producto
• Cociente
igualdades desigualdades
resolución gráfica
Tipos clasificación solución tipos
tipos
tipos
IGUALDADES Y DESIGUALDADES
ECUACIONES (=)
• Primer grado, ax + b = 0 ⇒ a ≠0 ⇒ x= - b/a
• Segundo grado, ax2 + bx + c = 0 X= ( -b ± √b2 - 4ac ) / 2a
• Bicuadradas, ax4 + bx2 + c =0 ⇒ x2=z
• Ecuaciones con radicales
SISTEMAS DEECUACIONES
LINEALES
COMPATIBLES
INCOMPATIBLES
DETERMINADOS
INDETERMINADOS
MÉTODO DEGAUSS
INECUACIONES
• Primer grado, ax + b ≤ 0
• Segundo grado, ax2 + bx + c = a(x-x1 ) . (x-x2 )
• Otros grados, (x-a) . (x+b) …..≥ 0
clasificación combinación
nomenclatura operaciones
producto escalar
operaciones ejemplo
módulo
producto escalar
operaciones
representantes
VECTORESQ P
VECTOR FIJO• Q , origen• P , extremo• QP , desplazamiento• , módulo
Suma : a + b
Resta : a - b
Multiplicación : ra
VECTOR LIBRE
DIRECCIÓN SENTIDO MÓDULO
Suma; regladel polígono
Multiplicación
rPA
BASESMATRIZ DECOMPONENTES
BASESCARTESIANAS v= √[ ( v1)2 + (v2)2]
a.b = a1.b1 + a2. b2
a.b=1/2(a 2+b 2-b-a2)
tipos
identificación relación con posición
punto
aplicación
asocia
forman
relación
aplicación
son
aplicación
GEOMETRÍA VECTORIAL DEL PLANO
VECTOR LIBRE
ORIGEN
VECTORPOSICIÓN
BASE
COORDENADAS(X,Y)SISTEMA DE
COORDENADAS
AB = OB - OA
COORDENADASDEL PUNTOMEDIO
RECTA
VECTORIALr = OA + t. v
PARAMÉTRICAX= a1 + t . v1Y= a2 + t . v2
CONTINUAx -a1/v1 = y-a2/v2
GENERALAx +By + C = 0
PUNTO - RECTA EXPLÍCITAy= mx + hPENDIENTE, m
PARALELAS
PERPENDICULARES
SECANTE COINCIDENTE NOCOINCIDENTE
DOSPUNTOS
DISTANCIA
soporte estructura se identifican caracteriza
se
define consta
relación entre ellas operaciones formas de
Expresión
operaciones
operaciones
Nos COMPLEJOS
TRIGONOMETRIA
RAZONESTRIGONOMÉTRICAS
IDENTIDADES
REDUCCIÓN AL PRIMER CUADRANTE
AFIJO VECTOR DEPOSICIÓN (x ,y)
X, PARTEREAL
Y, PARTEIMAGINARIA
CUERPO
SUMA
PRODUCTOPOR Nº REAL
PRODUCTO
COCIENTE
POTENCIA
RAÍZ
BINÓMICAZ= x+yi
(a+bi) . (c+di) = (ac-bd)+(ad+bc)i
Multiplico conjugado
Multiplicaciones sucesivas
z . z´= ( r. r´ ) α+β
z : z´= (r / r´ ) α - β
zm = ( rm )mα
n√r α = { R= n√r , θ = α/n + k360º/n
POLAR r α
r = |z |x = r cos αy = r sen α
son operaciones
relación
son nomenclatura
aplicación
Equivalencia
TRIGONOMETRÍA
Sen α = y
cos α = x
Tang α = y/x
Cosec α= 1/y
Sec α = 1/x
Cotangα = x/y
RAZONESTRIGONOMÉTRICAS
IDENTIDADES BÁSICAS
sen2 α + cos2 α = 1 1 + tang2α = sec2α 1 + cotang2α = cosec2α
Suma de ángulos• Sen(α+β)=senα .cos β+ cos α . senβ• Sen(α-β)=senα .cosβ - cos α . senβ• cos(α+β)=cos α.cosβ - sen α . senβ• cos(α+β)=cosα .cosβ +sen α . senβ
Ángulo doble• Sen2α = 2 senα . cosα• cos2α = cos2α - sen2α
Ángulo mitad
• Sen α/2= ± √ (1 - cos α) / 2
• Cos α/2= ± √ (1 + cos α) / 2
Suma de razones• SenA + senB = 2sen (A+B)/2 . cos (A - B)/2• SenA - senB = 2cos (A+B)/2 . sen (A - B)/2• cosA + cosB = 2cos (A+B)/2 . cos (A - B)/2• cosA + cosB = 2sen (A+B)/2 . sen (A - B)/2
ECUACIONESTRIGONOMÉTRICAS
RADIAN180º = Π rad
relación entre lados y ángulos resolución
utilizando aplicación
aplicación utilizando
TRIÁNGULOS C B a
 c B
TEOREMA DEL COSENOa2= b2 + c2 - 2abc cosA
TEOREMA DEL SENOa/senA = b/senB = c/senC
CIRCUNFERENCIACIRCUNSCRITAa/senA= 2R
TRIÁNGULOSRECTANGULOS
RAZONES TRIGONOMÉTRICASRELACIONES TRIGONOMÉTRICAS
CASOGENERAL
ÁNGULOS DEELEVACIÓN YDEPRESIÓN
CÁLCULO DEDISTANCIASINACCESIBLES
asocia se define tipos
asigna clasificación
propiedades
operaciones
FUNCIONES
DOMINIO; Dx⇒y= f (x)
COJUNTO IMAGEN
Imf= {f(x) x∈D }
GRÁFICA( X, f(x) )
INYECTIVAf (x1 ) = f (x2 ) ⇒ x1=x2
SIMETRÍAS
PARf (-x) = f(x)
IMPARf (-x) = - f(x)
F . PLINÓMICAf(x)= an xn + an-1 xn-1 +….+a1 x + a0
F . CONSTANTEf (x)= c
F . IDENTIDADf(x) =x
F . PRIMER GRADOf(x) = ax + b
pendiente o.o
F. SEGUNDO GRADOf(x)=ax2 + bx + c
F. PROPORCIONALIDADINVERSAy = c ⁄ x
SUMAf (x) + g(x) = (f+g) (x)
RESTAf (x) - g(x) = (f-g) (x)
PRODUCTO POR cte(C. f)(x) = C . f(x)
PROCUCTO(f . g) (x) = f(x) . g(x)
COCIENTE(f/g) (x) = f(x)/g(x)
COMPOSICIÓN(g 0 f) = g(f(x))
tipos indeterminaciones
clasificación propiedades
propiedades indeterminaciones
Tipos
LÍMITES
LÍMITES ENEL INFINITO
limx à ± ∞ =
VALORES MUYGRANDES
limx à + ∞ =
VALORES MUYPEQUEÑOS
limx à - ∞ =
• ∞/ ∞• ∞−∞• 0 . ∞• 0/0
• limxà+∞ =[f(x)+g(x)]= limxà+∞ f(x) +
limxà+∞ g(x) = l + m
• limxà+∞ =[f(x).g(x)]=[ limxà+∞ f(x) ].
[ limxà+∞ g(x) ]= l .m
• limxà+∞ =[f(x):g(x)]=[ limxà+∞ f(x) ]:
[ limxà+∞ g(x) ]= l/m
Cte + ∞ = ∞ (+∞)+(+∞)= +∞ (-∞)+(-∞)= -∞ Cte . ±∞ =± ∞ ±∞ .± ∞=± ∞
l/± ∞=0l ≠0
l/0 = ∞l ≠0
±∞ / l = ± ∞l ≠0
± ∞ / 0 = + ∞
LÍMITE EN UN PUNTOlimx à ± a =
0 /0
CONTINUA
limx àa f(x) = f(a)DISCONTINUA
1ª ESPECIE 2ª ESPECIEEVITABLE
• limx àa c =c
• limx àa x = a
• limx àaf(x)=l l>0 , entonces
limx àa√f(x) = √l
aplicación
definición técnicas de derivación
conclusión
1º Criterio
estudia 2º Criterio
son
1º Criterio
2º Criterio
implica
DERIVADAf(x) derivada f´(x)
f ´(x0) = lim h→0 f (x0 +h ) - f (xo) h
f(x) = mx + b → f´(x)=m RECTA TANGENTEy- f(x0 )= f´(xo) . (x - xo)
GRÁFICAS
ASÍNTOTAS ASINTOTA VERTICALLim x→ a f(x)= ∞
ASINTOTA HORIZONTALLim x→ ∞ f(x)
CRECIMIENTO
F . CRECIENTE X1 < X2
f(x1) < f(x2)
F . DECRECIENTE X1 < X2
f(x1) > f(x2 )
OPTIMIZACIÓNDE FUNCIONES
EXTREMOSRELATIVOSf ´(x)=0
MÁXIMO
Creciente→Decreciente
f ´(x0)=0 → f ´´(x0) < 0
MÍNIMO Decreciente→Creciente
f ´(x0)=0 → f ´´(x0) > 0
CONCAVIDAD YCONVEXIDAD
PUNTOS DE INFLEXIÓNf ´´ (x0 )=0
CONVEXA ; f ´´(x)<0 ⇒ f ´(x)decreciente
CONCAVA ; f ´´(x)>0 ⇒ f ´(x)creciente
f(x) ± g(x) = [f ´(x) ± g ´(x)]
f ´(x) . g(x) = [f ´(x) . g(x) +g ´(x) . f(x)]
f ´(x) : g(x) = [f ´(x) . g(x) +g ´(x) . f(x)] : g 2 (x)
[ x n ] ´ = n x n - 1 → n√pm = p m/n
h (x)= ( u(x) )n ; n∈Qh ´(x) = n ( u(x) )n-1 . u ´(x)
definición teorema
fundamental
del
sabiendo cálculo propiedades
cálculo
integral
aplicaciones
ejemplo
INTEGRAL ; ∫b
a
∫b
af(x) dx = lim n→∞ =[f (ξ1 ) ∆x1 + f (ξ2) ∆x2 +
… + f (ξn ) ∆xn ]
Sn = f (ξ1) ∆x1 + f (ξ2) ∆x2 + … + f (ξn ) ∆xn
a , límite inferiorb , límite superior
∫b
af(t) dx es una primitiva
de f(x)
FUNCIÓN PRIMITIVA→H(x) primitiva f(x)
∫b
af(x) dx = G(b) - G(a)
G(x) primitiva
∫b
af(x) dx = - ∫
b
af(x) dx ; a<b
∫b
a
f(x) dx = 0
LINEALIDAD; ∫b
aC f(x) dx = C ∫
b
af(x) dx
⇓
∫b
a [f(x)+g(x)] dx = ∫
b
af(x) dx + ∫
b
ag(x) dx
ADITIVIDAD DE INTERVALOS
∫b
af(x) dx = ∫
b
af(x) dx + ∫
b
af(x) dx
a<c<b y f(x) ≥0
CÁLCULO DE ÁREAS
A=
∫b
a
f(x) dx , f(x) ≥0 , x ∈ (a,b)ENERGÍAPOTENCIAL
datos método de cálculo ej.
subconjunto
Representación
características número de datos grande
clasificación
ordenación
de datos
tipos representación gráfica tipos
medidas medidas
de de
dispersión centralización
ejemplo
son
ESTADÍSTICAPOBLACIÓN
MUESTRA
VARIABLEESTADÍSTICA, X
INTERVALOS Xi= (Li-1 + Li ) / 2
HISTOGRAMAS
POLÍGONOS DEFRECUENCIA
CUANTITATIVA
CUALITATIVA
DISCRETA
CONTINUA
DISTRIBUCIÓN DEFRECUENCIAS
FRECUENCIAABSOLUTA, fi
FRECUENCIARELATIVAhi =fi /N
DIAGRAMAS
BARRAS
POLIGONAL
SECTORES
MEDIA
DIVISIÓN DEDATOS
CUARTILES MEDIANA
DECILES
PERCENTILES
RANGO
VARIANZA
DESVIACIÓN TÍPICA
DESVIACIÓN MEDIA
COEFICIENTE DE VARIACIÓN
SUMATORIO,
∑FRECUENCIASACUMULADASABSOLUTAS YRELATIVAS
definición definición definición
relación
propiedades
propiedades
tipos
propiedades
propiedades
derivada
propiedades
derivada tipos relación
FUNCIONESTRASCENDENTES
FUNCIÓN EXPONENCIALf(x)= ax
FUNCIÓN LOGARÍTMICAy= loga x
FUNCIONESTRIGONOMÉTRICASsen x, cos x
ap/q = q√a p donde p,q ∈Z , q≠o
a >1 CRECIENTEINYECTIVA
limx→+∞ ax= +∞ limx→-∞ ax= 0ax>o ⇒x
o<a<1 DECRECIENTE
limx→∞ ax= 0 , limx→-∞ ax= +∞ax>o ⇒x
BASE e COMÚN a >1
(ex)´= ex ⇒ [e u/x] ´ = u ´ (x) eu (x)
(ax ) ´= ax ln a ⇒ [au(x)] ´= au(x) u´(x) Ln a
loga x =y ⇔ ay =x
loga 1 =0 →a0=1
loga a =1 →a1=a
loga (ax) =x si x ∈ R
aLogax =x si x ∈ R
loga (x.y) = loga x + loga y
loga (1/x) = - loga x
loga (y/x) = loga y - loga x
loga (xy) = y loga x
DECIMAL ; BASE 1O
NEPERIANO ; BASE e
CAMBIO BASELog b x =(1/ loga b). loga x
(Ln x)´= 1/x ⇒ [au(x)]´= au(x). u´(x) Ln a
Cos(t) =0 , Sen (t)=y
Tang (t)
-1 ≤cos t≤ 1- 1 ≤sen t≤ 1-
Cos(t+2Π)=costSen(t+2Π)=sentTang(t+2Π)=tan
• (senx)´=cosx[senu(x) ]´=u´(x).sen u(x)• (cosx)´=-senxcosu(x)]´=- u´(x). sen u(x)• (tangx)´=1/cos 2x[tangu(x)]´= u´(x)/ cos2 u(x)
derivada
ejemplo caracteriza
representación relación
relación relación grado de medidaentre dependencia devariables medidas aritméticas aproximación
tipos medida
interpretación
relación clasificación
VARIABLES ESTADÍSTICAS BIDIMENSIONALES
VARIABLESBIDIMENSIONALES(X , Y)
DEPENDIENTES
FUNCIONALCORRELACIÓN
INDEPENDIENTES
Media X = ∑ xi /N
Media Y = ∑ yi /N
Media XY= ∑ xi. yi
/N
Covarianza σ xy = XY - mx .my
DIAGRAMA DE DISPERSIÓN
LINEAL
NO LINEAL
REGRESIÓN LINEAL
CORRELACIÓNLINEAL
COEFICIENTE DECORRELACIÓN, r
r = σxy /σxσy- 1 < r < 1 → Intervalo
r >0
r < 0
RECTACRECIENTE
RECTADECRECIENTE
CORRELACIÓNDIRECTA
CORRELACIÓNINVERSA
ERRORCUADRÁTICOMEDIO E2 = (y - a - bx2)
r =±1→ E2=0
Y sobre X : Y -m y = σxy / σ2
x ( X - mx )X sobre Y : X -m x = σxy / σ
2y ( Y - my )
en general planteamiento de posibilidades posibilidades de combinación
característica
concepto asociado
fórmulas concepto asociado características
TÉCNICAS DE CONTAR
DIAGRAMADE ÁRBOL
DIAGRAMADE CAJAS
PRINCIPIO DEMULTIPLICACIÓN
VARIACIONESVm
k= m.(m-1)…(m-k1)
VARIACIONES CONREPETICIÓNVRm
k= mk
IMPORTA ORDEN
PERMUTACIONESPm= Vm
mPERMUTACIONESCON REPETICIÓNPRn
k1 k2 = n!
FACTORIALn! = n . (n-1) . (n-2)…
COMBINACIONESCm
k= Vmk / Pk NO REPETICIÓN
NO IMPORTA ORDEN
NÚMERO COMBINATORIO(m
k) = m! / k! (m-k)!
BINOMIO DE NEWTON
TRIÁNGULO DE PASCAL
TABLAS DE CONVERSION DE UNIDADES
LONGITUD
metro milímetro pulgada pie yarda milla (statute)
m mm in (¨) ft yd mi
1 1000 39,3700787 3,2808399 1,0936133 0,00062137
0,001 1 0,0393701 0,0032808 0,0010936 0,00000062137
0,0254 25,4 1 0,08333 0,02777 0,000015782
0,3048 304,8 12 1 0,333 0,00018939
0,9144 914,4 36 3 1 0,00056818
SUPERFICIE
metro cuadrado hectárea pulgada cuadrada pie cuadrado yarda cuadrada acre
ha
1 0,0001 1550,0031 10,76391 1,19599 0,00024711
10000 1 15500031 107639,1 0,0001196 2,4710538
0,0006,4516 0,00000006451 1 0,006944 0,0007716 0,00000015942
0,09290304 0,000009290351 144 1 0,111 0,000022957
0,8361274 0,000083613 1296 9 1 0,00020661
4046,856 0,4046856 6272640 43560 4840 1
m2 in2 ft2 yd2
VOLUMEN
metro cúbico litro pie cúbico galón (USA) galón imperial (GB) barril de petróleo
gal gal bbl (oil)
1 1000 35,3146667 264,17205 219,96923 6,2898108
0,001 1 0,0353147 0,2641721 0,2199692 0,0062898
0,0283168 28,3168466 1 7,4805195 6,2288349 0,1781076
0,0037854 3,7854118 0,1336806 1 0,8326741 0,0238095
0,0045461 4,5460904 0,1635437 1,20095 1 0,028594
1589873 158987295 56145833 42' 34,9723128 1
dm3m3 ft3
1 gal (USA) =3,78541dm3
1 ft3=0,0283 m3
UNIDADES DE PRESIONUNIDADES DE PRESION
kilopascal atmósfera técnica milímetro de c. Hg metros de c. agua bar
(0ºC) (4ºC) 100000 Pa
kPa atm mm Hg psi bar (hpz)
1 0,0101972 7,5006278 0,1019745 0,1450377 0,01
98,0665 1 735,560217 1000028 14,2233433 0,980665
0,1333222 0,0013595 1 0,0135955 193367 0,0013332
9,8063754 0,0999972 73,5539622 1 1,4222945 0,0980638
6,8947573 0,070307 51,7150013 0,7030893 1 0,0689476
100 1,0197162 750,062679 10,1974477 14,5037738 1
in H2O (60ºF=20ºC)=0,248641 kPain H2O (60ºF=20ºC)=0,248641 kPa
1 atmósfera física (Atm)= 101,325 kPa=760 mm Hg1 atmósfera física (Atm)= 101,325 kPa=760 mm Hg1 atmósfera física (Atm)= 101,325 kPa=760 mm Hg
in Hg (60ºF=20ºC)=3,37685 kPain Hg (60ºF=20ºC)=3,37685 kPa
1 Torr= (101,325/760) kPa1 Torr= (101,325/760) kPa
kN /m2 Kgf/cm2 lib/in2libras por pulgada2
m H2O
1 in H2O (60ºF = 15,55ºC) = 0,248843 kPa
Tablas.-431
ENERGIA (Calor y Trabajo)ENERGIA (Calor y Trabajo)
Kilojulio kW/hora Hourse power/hora Caballo/hora Kilocaloría (IT) British
USA 550 ft.lbf/seg 75 m.Kgf/seg Kcal(IT) Thermal Unit
kJ kW h hp. h CV.h Kcal (IT) Btu (IT)
1 0,0002777 0,000372506 0,000377673 0,2388459 0,9478171
3600 1 1,3410221 1,3596216 859,84523 3412,1416
2684,5195 0,7456999 1 1,0138697 641,18648 2544,4336
2647,7955 0,7354988 0,9863201 1 632,41509 2509,6259
4,1868 0,001163 0,00155961 0,00158124 1 3,9683207
1,0550559 0,000293071 0,00039301 0,000398466 0,2519958 1
1 termia = 1000 Kcal
1 therm = 100.000 Btu1 therm = 100.000 Btu
1 But (IT) = 1055,0558 J1 But (IT) = 1055,0558 J
1 kilogramo fuerza.metro (m.Kgf) = 0,00980665 kJ1 kilogramo fuerza.metro (m.Kgf) = 0,00980665 kJ1 kilogramo fuerza.metro (m.Kgf) = 0,00980665 kJ
IT se refiere a las unidades definidas en International Steam TableIT se refiere a las unidades definidas en International Steam TableIT se refiere a las unidades definidas en International Steam Table
MACROUNIDADES ENERGETICASMACROUNIDADES ENERGETICAS
Terajulio Gigavatio hora Teracaloría (IT) Ton. equivalente Ton. equivalente Barril de petróleo
de carbón de petróleo día-año
TJ GW h Tcal (IT) Tec Tep bd
1 0,2727 0,2388459 34,1208424 23,8845897 0,4955309
3,6 1 0,8598452 122,8350326 85,9845228 1,7839113
4,1868 1,163 1 142,8571429 100 2,0746888
0,0293076 0,008141 0,007 1 0,7 0,0145228
0,041868 0,01163 0,01 1,4285714 1 0,0207469
2,0180376 0,560568 0,482 68,8571429 48,2 1
POTENCIA
Kilowatio Kilocaloría/hora Btu (IT)/hora Horse power (USA) Caballo vapor Tonelada demétrico refrigeración
kW Kcal (IT)/h Btu (IT)/h hp CV
1 859,84523 3412,1416 1,3410221 1,3596216 0,2843494
0,001163 1 3,9683207 0,0015596 0,0015812 0,0003307
0,00029307 0,2519958 1 0,00039301 0,00039847 0,000083335
0,7456999 641,18648 2544,4336 1 1,0138697 0,2120393
0,7354988 632,41509 2509,6259 0,9863201 1 0,2091386
3,5168 3023,9037 11999,82 4,7161065 4,7815173 1
1 caballo vapor (métrico> = 75 m kgf/seg = 735,499 W1 caballo vapor (métrico> = 75 m kgf/seg = 735,499 W1 caballo vapor (métrico> = 75 m kgf/seg = 735,499 W
1 Horse power (USA) mecánico = 550 ft Ibf/seg1 Horse power (USA) mecánico = 550 ft Ibf/seg1 Horse power (USA) mecánico = 550 ft Ibf/seg
TEMPERATURA
Temperatura en ºC = (ºF -32)/1,8
Temperatura en ºF = 1,8 ºC + 32
Temperatura en ºK = ºC + 273,14
PREFIJOS DEL SISTEMA INTERNACIONAL DE UNIDADESPREFIJOS DEL SISTEMA INTERNACIONAL DE UNIDADESPREFIJOS DEL SISTEMA INTERNACIONAL DE UNIDADESPREFIJOS DEL SISTEMA INTERNACIONAL DE UNIDADESPREFIJOS DEL SISTEMA INTERNACIONAL DE UNIDADESPREFIJOS DEL SISTEMA INTERNACIONAL DE UNIDADESPREFIJOS DEL SISTEMA INTERNACIONAL DE UNIDADESPREFIJOS DEL SISTEMA INTERNACIONAL DE UNIDADES
Prefijo exa peta tera giga mega kilo hecto deca deci centi mili micro nano pico femto atto
Símbolo E P T G M k h da d c m m n p f a
Factor 1e +18 1e +15 1e +12 1e +9 1e +6 1000 100 10 0,1 0,01 0,001 1e-6 1e-9 1e -12 1e -15 1e -18
Tablas.-432