UNIVERSIDAD LATINA CAMPUS HEREDIA FACULTAD DE INGENIERÍA CURSO: MATEMÁTICA MATEMÁTICA INTRODUCTORIA, CALCULO I Y ECUCACIONES DIFERENCIALES TEMA: LA PARÁBOLA PROFESOR: ELIÉCER VINDAS ARIAS PROPONENTES: RODOLFO CALDERÓN MADRIGAL JOSÉ ALBERTO LEPIZ LEP IZ ALVARADO ALVARADO JUAN JOSÉ SANCHO GONZALES AÑO 2!"
T#$%# T# $%# &' C()*')+&(
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Contenido Tabla de Contenidos ..................................................................................................2 Tabla de Gráficos .....................................................................................................3 Introducción...................................... ......................................................... ....................................... ........................................ ............................... ........... 4 Objetivo General ......................................................................................................5 Objetivos Específicos .............................................................................................5 Desarrollo..................................... ......................................................... ........................................ .................................... .............................. ................. ... 6 Reseña Histórica ...................................................................................................6 Definición de arábola ...........................................................................................8 Características Geo!"tricas de la arábola .................................................................10 Ecuaciones de la parábola ......................................................................................12 Ecuación de la parábola con v"rtice en el ori#en $% eje &' %( ) *p+, ..............................12 Ecuación de la parábola con v"rtice en el ori#en $eje -' +.) +.) *p%, .................................14 arabola con /ertice fuera de Ori#en .....................................................................15 Tan#ente de parábolas ..........................................................................................17 E!pal!es entre arábolas .....................................................................................17 Tan#ente Tan#ente de la arábola desde el punto interior ........................................................19 ........................................................19 Tan#ente Tan#ente a la arábola paralela a una deter!inada dirección ........................................19 0plicaciones de la arábola ...................................................................................20 Conclusión...........................................................................................................22 1iblio#rafía..........................................................................................................23
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Grafico 53Ecuación de la parábola con v"rtice en el ori#en & .............................................13 Grafico *3arabola con v"rtice en el ori#en % eje - ..........................................................14 Grafico 63Tan#ente 63Tan#ente de la parábola ...............................................................................17 Grafico 73 E!pal!e de parábolas en el punto $v, ...........................................................18 Grafico 8Tan#ente 8Tan#ente de la arábola desde el punto Interior ..................................................19 ..................................................19 Grafico 93Tan#ente 93Tan#ente a la arábola paralela a una deter!inada dirección .................................20
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En esta investi#ación se :ará un análisis % una breve reseña :istórica de lo ;ue es la parábola' instru!ento utili
n! co!>n!ente ente se piensa piensa en las ecuacio ecuaciones nes % los conceptos de las !is!as ' pero sin e!bar#o desde tie!pos !u% anti#uos :asta la actualidad la parábola tiene diversos usos 'se podría decir ;ue :an estado presente no solo en la !ate!ática 'sino ta!bi"n en la #eo!etría 'la ar;uitectura 'la in#eniería 'entre otras ciencias ;ue la :an utilitiles en el te!a de la teleco!unicación e industria' pero ta!bi"n se las puede encontrar en el universo= osible!ent osible!ente' e' a !edida ;ue la tecnolo#ía tecnolo#ía va%a pro#resando' co!o co!o es al#o tan co!>n en el si#lo &&I' se tendrá a>n !ás presente el te!a de la parábola con la intención de !ejorar lo pree+istente % darle una !a%or utilidad %a sea en la vida cotidiana co!o en las
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En esta investi#ación se :ablara de la parábola' se definirá % se e+plicara cuáles son sus aplicaciones e i!portancia en lo ;ue respecta no solo a la ?ate!ática sino ta!bi"n en su relación relación con la in#eniería in#eniería 'co!o así ta!bi"n ta!bi"n los usos de ella en la vida cotidiana' cotidiana' %a ;ue son fuente de inspiración para la creatividad 'i!a#inación e inventiva del ser :u!ano en su :abitad % en su diario vivir=
O$3'*+4( G')'-#%
Investi#ar las características de la !ate!ática funda!entadas en la parábola' conocer la :istoria % la i!portancia de la !is!a a trav"s de los si#los en sus diferentes conte+tos % en relación con otras ciencias' co!o así ta!bi"n en el cotidiano vivir del ser :u!ano=
O$3'*+4( E5'06/+0(
0nali
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D'#--(%%(
R''7# H+*-+0# El !ate!ático #rie#o ?enec!o @56A a=C=B descubrió estas curvas en su estudio frente al inconveniente de la duplicación del cubo donde e+pone la e+istencia de una solución !ediante el corte de una parábola con una :ip"rbola' pero fue el !ate!ático #rie#o 0polonio @.7.2A 0=C=B de er#a' el pri!ero en estudiar detallada!ente las curvas cónicas % encontrar la propiedad plana ;ue las definía= 0polonio revela ;ue las cónicas= son curvas ;ue se pueden obtener cortando un cono' se podían clasificar en tres tipos a los ;ue dio el no!bre de3 elipses' :ip"rbolas % parábola' no!bre con ;ue se las conoce actual!ente % las describió de esta for!a3
4as elipses son las curvas ;ue se obtiene cortando una superficie cónica con un
plano ;ue no es paralelo a nin#una de sus #eneratrices= # eneratrices= 4as :ip"rbolas son las curvas ;ue se obtiene al cortar una superficie cónica con un
plano ;ue es paralelo a dos de sus #eneratrices #ener atrices @1ase % aristaB= 4as parábolas son las curvas ;ue se obtienen al cortar una superficie cónica con un plano paralelo a una sola #eneratri< @0ristaB= i bien estas nociones no pudieron ser aprovec:adas % aplicadas en la ciencia de su
"poca' su si#nificación :a per!anecido co!pleta!ente reconocida con el paso del tie!po= Estos t"r!inos ta!bi"n fueron utili
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Elipse si#nificaba deficiencia= Hip"rbola si#nificaba e+ceso= arábola si#nificaba e;uiparación=
Entre el si#lo &/I % &/II otros pensadores ta!bi"n dieron ra<ón de la parábola a trav"s de diferentes estudios' por eje!plo3 Galile Galileoo Galilei Galilei @267*27 @267*27*.B' *.B' estudi estudiand andoo el !ovi!ie !ovi!iento nto de un pro%ect pro%ectil' il' con una co!ponente :ori
En donde 89 es la #ravedad, 84 la velocidad inicial de la bala % 8#%/# la inclinación del tiro por !edio de esto Galileo for!ó a partir de la dicción anterior la inclinación para lo#rar la !á+i!a distancia ;<== 9A años despu"s' Isaac Feton @27*.28.8B probaba ;ue las órbitas elípticas de los
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pueden representar por ecuaciones en las variables por + ' %' por lo ;ue la parábola ;uedo así relacionada con su ecuación 8>?# <@ ero ta!bi"n cabe traer a la !e!oria ;ue el científico 0r;uí!edes por su parte incendio las naves ro!anas utili
D'/+)+0+) &' P#-.$(%#
4a ecuación de la parábola vista en el íte! anterior servirá para deducir a partir de su definición co!o un lu#ar #eo!"trico de un deter!inado punto ;ue se despla
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del v"rtice al foco o de del v"rtice a la directri< se le denota !ediante la letra p= @Clap:a!' 29=p *A9B @0la!ar enad"s Roi# alas' .AA9B
G-#/+0( !:L# 5#-.$(%#
4a distancia ;ue se encuentra de cual;uier punto @+' %B ;ue corresponda de la parábola al foco es3
&! ? ;< 5= @ ;> =@
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;< 5= @ >@ ? ;< 5=@ Desarrollando esto >lti!o se obtiene3
<@ 2 <5 5@ >@ ? <@ 2 <5 5@ % si se eli!inan t"r!inos ;ueda de esta !anera3
2 <5 > ? 2 <5 o lo ;ue es i#ual a3
> ? 2 <5 2 <5 ;ue es i#ual a3
>@ ? 5< Da co!o resultado una ecuación conocida co!o $ecuación ordinaria o canónica de la parábola con v"rtice en el ori#en,=
C#-#0*'-6*+0# G'(*-+0# &' %# P#-.$(%#
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FOCO ;F=: unto fijo no correspondiente a la parábola % ;ue se ubica en el eje focal al interior de las ra!as de la !is!a % a una distancia p del v"rtice=
DIRECTRIZ ;&=: 4ínea recta perpendicular al eje focal ;ue se sit>a a una distancia p del v"rtice % fuera de las ra!as de la parábola=
DISTANCIA FOCAL ;5=: proporción de la distancia entre v"rtice % foco' así co!o entre v"rtice % directri<=
CUERDA: e#!ento de recta ;ue une dos puntos cuales;uiera' correspondientes a la parábola=
CUERDA FOCAL3 Cuerda ;ue pasa por el foco= LADO RECTO ;AB=: Cuerda focal ;ue es perpendicular al eje focal=
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E01#0+()' &' %# 5#-.$(%#
E01#0+) &' %# 5#-.$(%# 0() 4-*+0' ') '% (-+9') 8> '3' , >@ ? 5< Jna parábola cu%o v"rtice se encuentra en el ori#en % concuerda con el eje de las ordenadas' tiene una ecuación de la for!a %)a+( donde el pará!etro $a, especifica la escala de la parábola' errónea!ente descrita co!o la representación de la parábola' %a ;ue todas las parábolas tienen la !is!a for!a= Cuando el pará!etro es positivo % la parábola tiene su apertura es K:acia arribaL % cuando es ne#ativo su apertura es K:acia abajoL= in e!bar#o' la e+presión en for!a de ecuación no fue viable :asta el desarrollo desarrollo de la #eo!etría analítica' la analo#ía #eo!"trica !encionada en la ecuación anterior %a e+istía
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G-#/+0( :E01#0+) &' %# 5#-.$(%# 0() 4-*+0' ') '% (-+9')
/ale tener sie!pre presente ;ue la parábola va a tener su apertura :acia donde está el foco por lo ;ue si el foco tiene coordenadas coordenadas ne#ativas ne#ativas es posible posible de ;ue pueda abrir :acia
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[ FP ] =√ ( x − p ) + y 2
PNQ) P0Q donde
2
¿ X + P
[ PA PA ] =[ X −(− P ) ] =¿ ¿
√ ( X − P )
2
2
+ Y
)P&Q
-( ) *& M por lo ;ue es la lon#itud del vertice al foco %
del vertice a la directri< 4a cuerda focal @lado recto B es la cuerda perpendicular al eje de la parabola ;ue pasa por el foco '% su lon#itud es P* Q i se necesita #raficar una parabola es suficiente con conocer tres aspectos del vertice 'el valor de % su eje por ende 3
i ˃ A i ˂ A
la parabola se abre por la derec:a la parabola abre la i<;uierda
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i ˂ A
;uiere decir ;ue la parabola abre :acia abajo
P#-#$(%# 0() V'-*+0' /1'-# &' O-+9')
4a arábola arábola Hori
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%( .%S S( ) *+p *:p %( .%S S( *:p *+p ) A i D ) .S' E ) *p' N ) S( *:p' por lo tanto' la fór!ula es3 %( D% E+ N ) A arábola /ertical de /"rtice / @:'SB fuera del Ori#en' eje de si!etría paralelo al eje -' % cu%o Noco está a una distancia 5 del /"rtice' en donde aplicando la PF? PM ? Ecuación de la Directri< % elevando al cuadrado % si!plificando se obtiene lo si#uiente3 @& :B( P% @S pBQ( ) P% @S pBQ( @& :B( %( .%@S pB @S pB( ) %( .%@S pB @S pB( @& :B( %( .%S .%p S( .Sp p( ) %( .%S .%p S( .Sp p(
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&( D+ E% N ) A @4e:!ann' 29=p'262B
T#)9')*' T# )9')*' &' 5#-.$(%#
L# *#)9')*' $+'0# '% .)91%( ')*-' '% /(0(, '% 51)*( &' *#)9')0+# > 1 5-(>'00+) 4a tan#ente en las parábolas !uestra una propiedad ;ue proporciona su tra
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ara ;ue se produ
G-#/+0( : E5#%' &' 5#-.$(%# ') '% 51)*( 84
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or !edio de estos datos se constru%e la parábola de e!pal!e' pero se debe tener en cuenta ;ue si las dos curvas poseen i#ual o si!ilar curvatura en el punto de unión' la prolon#ación entre estas es !ás perceptible= percep tible=
T#)9')*' T# )9')*' &' %# P#-.$(%# P#-.$( %# &'&' '% 51)*( +)*'-+(En el punto , se #ráfica la circunferencia de radio F % v"rtice en ' la cual biseca a la recta directri< ;ue en la parábola :ace de circunferencia focal de radio infinito' en los puntos 2 % .= e puede denotar en la #ráfica ;ue las !ediatrices de los se#!entos 2N % .N
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Grafico 8:Tangente a la Parábola paralela a una determinada dirección
A5%+0#0+()' &' %# P#-.$(%#
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Ta!bi"n se pueden percibir for!as parabólicas e!itidas por un :a< lu!inosos de for!a cónica pro%ectado sobre una superficie plana % blanca ' están presente en la ar;uitectura ' donde se pueden apreciar apreciar en los #randes #randes arcos for!ados for!ados en !onu!entos !onu!entos anti#uos anti#uos en el viejo !undo' !undo' pero se encuentran 'co!o 'co!o se dijo al principio en diversas estructuras estructuras co!o puentes'la !ontaña rusa 'en el deporte' un telescopio' un radar ':asta en el si!ple :ec:o de saltar una cuerda se for!a una parábola 'por eso es ;ue está presente en el diario vivir del ser :u!ano =
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Esta investi#aci investi#ación ón per!itió per!itió :acer un enfo;ue #eneral no solo a nivel !ate!átic !ate!áticaa sino
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B+$%+(9-#/6# Alamar Penadés & Roig Salas. 2008!. Matemáticas básicas; . "alen#ia$ %P"
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