Presentador:

Prof. Doymo Morales- Universidad Interamericana

RECURSOS PARA FACILITADORES DEL PROGRAMA DE MATEMÁTICAS DEL

DEPARTAMENTO DE EDUCACIÓN DE PUERTO RICO (DEPR)

Materiales CRAIM

DIDÁCTICA DE LA MATEMÁTICA

Materiales CRAIM

DIDÁCTICA MATEMÁTICA

Las funciones

como modelos matemáticos

Presentador:

Prof. Doymo Morales

Universidad Interamericana, Barranquitas

RECURSOS PARA FACILITADORES DEL PROGRAMA DE MATEMÁTICAS

DEL DEPARTAMENTO DE EDUCACIÓN DE PUERTO RICO (DEPR)

Dr. Jorge M. López Director CRAIM, UPR Río Piedras

© Todos los derechos reservados

Publicaciones CRAIM

2012

Las funciones como modelos matemáticos

Didáctica matemática

Escuela secundaria

Recursos para facilitadores del PM del DEPR

Director Dr. Jorge M. López

Universidad de Puerto Rico, recinto de Río Piedras

Introducción

Los materiales de la serie Didáctica Matemática para los facilitadores del Programa de

Matemática (PM) y los maestros del nivel secundario del Departamento de Educación de Puerto

Rico (DEPR) se han confeccionado para atender temas medulares del nuevo currículo a la luz de

los documentos de Estándares curriculares y las Expectativas por grado. Esta serie recopila temas

identificados, por los maestros y facilitadores de matemática, como prioritarios en la matemática

del nivel secundario y se presentan con la profundidad que requiere el nuevo currículo. Estos

materiales de la serie Didáctica Matemática, cubren los siguientes temas específicos:

1) Funciones, modelación matemática, graficación. Transformación de funciones y sus

gráficas. Funciones polinómicas, racionales, logarítmicas y exponenciales.

2) Los modelos funcionales sinusoidales, leyes de senos y cosenos. Ondas sinusoidales y

el principio de superposición.

3) Empleo de las funciones como modelos matemáticos, graficación y geometría

analítica.

4) Probabilidad y estadística. Distribuciones binomial y normal. Regresión lineal y la

recta de cuadrados mínimos. Coeficiente de correlación.

El material didáctico presentado atiende especialmente temas prioritarios contenidos en los

siguientes cursos:

I. Matemática en acción (grado 10, primer curso)

II. Aventuras matemáticas (grado 10, segundo curso)

III. Funciones y modelos (grado 11, primer curso)

IV. Matemáticas contemporáneas (grado 11, segundo curso)

V. Precálculo (grado 12)

Esperamos que estos recursos sean de utilidad y hagan más efectiva la labor de los facilitadores

del Programa de matemáticas y los maestros del nivel secundario del DEPR.

Jorge M. López

Catedrático de Matemática

Universidad de Puerto Rico

Río Piedras

9 de septiembre de 2012

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x

y=3

1

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−1

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x

y

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,?* )( #-5)G&( #- )()*#H;01!2345, '(6()( ; I#) f : R −→ R 2)( 73# f : x 7→ x2; @*2&*#- () ,8).#* 1# f#- #( -36&*=3*2& 1# (&- J#)(#- >&/8)1& 5&/ 2&1&- (&- /#)(#- *& *#.)2,C&-' 73# -&*5/#,-)8#*2# (&- 3)1/)1&- 1# (&- *:8#/&- /#)(#-; K?2#-# 73#' #* #-2# )-&' () ,8).#*#- 1,-2,*2) )( &1&8,*,&;!; I#) g : R −→ R () >3*,?* 1#0*,1) 5&/ g(x) = 3;@* #-2# )-&' () ,8).#* 1# g &*-,-2# 1# 3* -?(& *:8#/&' im(g) = {3};L; I#) h : R −→ R 1#0*,1) 5&/ h(x) = x+ 1;A) ./+0) 1# #-2) >3*,?* )5)/## #* () 0.3/) !; @- >+,( C#/ 73# )1) y #- ,8).#* 1#)(.:* x' #M)2)8#*2# 1# x = y− 1; N/+0)8#*2#' #-2& #73,C)(# ) 1#,/ 73# )1) /#2)$&/,4&*2)( &/2) () ./+0) 1# h; I# 2,#*# 73# im(h) = codom(h);A&- #=#85(&- 83#-2/)* 73# () ,8).#* 1# 3*) >3*,?* 53#1# &,*,1,/ &* -3 &1&8,*,&' &53#1# -#/ 3* -36&*=3*2& 5/&5,& 1# -3 &1&8,*,&;

!"! #$%&'&(&)' #$ %*'(&)' !"...................................................................................................................................................................................................................................................................................

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#$%&'( )*+, y = x+ 1 !"!$%"&'!&() *"+,! -% .! /0!1%)-( $.(%/0 203$34/ 5/ 42523 623 7(62'/3 y 5/6 252.$0$2 8&9$(523 %':;(./04/ /0 /6 /2' 5205/ >(3( &0( '/4( ?2'$@204(6 A&/ 2'4( 6( %':;( 5/ f B >2' 62 ./023 &0( 7/@* C/'( 6( ;%&'( )* D*

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················································································································································································································································································································

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• !"#$ %& '( )*(+&" ··································· ··············

#$%&'( )* D, E&042 5/ 6( $.(%/0 !"!$%"&'!&() 2)!.3$&! -% .! /0!1%)45%06.78 9:;:9: * F/( f : R −→ R 4(6 A&/ f(x) = 7x− 5* !"#$%'()* G0 0H./'2 '/(6 b /34: /0 6( $.(%/0 5/ bB 33$B 6( /&($I0 f(x) = b 4$/0/326&$I0* E('( 5$?( /&($I0 4/0/.23 A&/f(x) = b =⇒ 7x− 5 = b =⇒ 7x = b+ 5 =⇒ x = b+ 5

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im(f) = R.

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im(f) = R \ {1}.F0+5.'3 2'8G()3 +:+)2&3 )0+5.'% 90,(3,+5 3, &% )(5)% '+*&%; 2+'3 3, -(9+'+,.+5 -3)(H,(3 E 3-3)(,(3# I+'+)35 /0+ &% '+&%(8, +,.'+ &% ()%*+, E +& 3-3)(,(3 .%)4(@, %)4(%-+2+,-(+,-3 -+ ,0+5.'% 5+&+

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x

y=x2

dom(f) = Rcodom(f) = R

im(f) = R+0

1

1

······································································································································································································································································

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·····································

··············

·····································

··············

#$%&'( )*6+ ,( -&.$0. &(7'48$( 345$(*

!"! #$ %$#&'($ !" !"!#! $% &'(*+( ,%-. /'%01%0% 234*% 5 f(x) = √x............................................................................................................................................................................................................................................................................................................................................................................................................................

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x

y=√x

1

1dom(f) = R+0

codom(f) = Rim(f) = R+0·······

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#$%&'( )*!+ ,( -&.$0. '(12 &(3'(3( 456$( !"!6! $% &'(*+( /78*% 234*%5 f(x) = x3.........................................................................................................................................................................................................................................................................................................................................................................

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x

y=x3

1

1dom(f) = R

codom(f) = Rim(f) = R

················································································································································································

······················································································································································

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··············

#$%&'( )*7+ ,( -&.$0. 84$( 456$(*

!" !"#$%&' () *+,-. !/ 01 -%2 .'21/ !"! ! #$ %&')*' +,-./0$ 123)$4 f(x) = 1/x.........................................................................................................................................................................................................................................................................................................................................................................

..............

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x

y=1/x

1

1dom(f) = R \ {0}

codom(f) = Rim(f) = R \ {0}················································································································································································································································

················································································································································································································································

·························�