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UNIVERSIDADE ANHANGUERA DE SO PAULOUNIAN - CAMPUS ABC.

INTEGRANTES DO GRUPO

Daniel SilvioJoo Vane! "a Silva

Manoel Pe#$ana%ellin$on "e Li&a 'on#e(a

ATPS ) *+ e ,+ ETAPA.

Cal(lo ,

SO BERNARDO DO CAMPO,*/

1


2/19

Daniel Silvio 01,//21,3Joao Vane! "a Silva 11,3243410Manoel T. P. Pe#$ana ,255,

%ellin$on "e Li&a 'on#e(a 4,521311

3semestre Engenharia Eltrica

ATPS Calculo 2

A$ivi"a"e P!a$i(a S6e!vi#iona"a a #e!

a6!e#en$a"a 7 "i#(i6lina "e Cal(lo ,

no (!#o "e Enen8a!ia El9$!i(a: "a Unive!#i"a"e

An8ane!a "e So Palo Unian Ca&6# ABC:

ao P!o;e##o!< Palo Jo!"=.

SO BERNARDO DO CAMPO,*/.

Sumrio

2


3/19

In$!o">o????????????????????????????????????????????????????????????5

E$a6a *

Pa##o *????????????????????????????????????????????????????????/

Pa##o ,????????????????????????????????????????????????????????0

Pa##o 2 ???????????????????????????????????????????????????????4

Pa##o 5????????????????????????????????????????????????????????*

E$a6a ,

Pa##o *????????????????????????????????????????????????????????**

Pa##o ,????????????????????????????????????????????????????????*5

Pa##o 2????????????????????????????????????????????????????????*3

Pa##o 5????????????????????????????????????????????????????????*0

Con(l#o????????????????????????????????????????????????????????????*0

Re;e!@n(ia# Bilio!=;i(a#???????????????????????????????????????????????*1

Introduo

3


4/19

A ATPS 9 &a o6o!$ni"a"e ni(a e i&6o!$an$e: e& e o# "e#a;io# ai

6!o6o#$o# #(a& 6!o&ove! e& $o"o# n#: alno#: o #en#o "e !e#6on#aili"a"e

in"ivi"al e (ole$ivo.

E##a a$ivi"a"e #e!= i&6o!$an$e 6a!a 6o"e!&o# ve!i;i(a! a a6li(a>o "a "e!iva"a

in#e!i"a e& (on(ei$o# =#i(o# "a ;#i(a e $a&9& e& no##o (o$i"iano. U&a o#e!va>o

&ai# a6!o;n"a"a #o!e o (on(ei$o "e "e!iva>o e & ol8a! &ai# a&6lo #o!e a

(on#$an$e "e Ele!: e 9 &i$o #a"a: &a# e &i$a# veFe# ;i(a o(l$a no (=l(lo

&a$e&=$i(o. RealiFa!e&o# o# 6a##o# a #ei!.

ETAPA

4


5/19

PASS!

#ai"o e 8= "ive!#a# &anei!a# "e "e#(!eve! o !a6i"a&en$e alo #e &ovevelo(i"a"e &9"ia e velo(i"a"e e#(ala! &9"ia: a&a# &e"i"a# #o!e & in$e!valo "e$e&6o $

Po"e&o# "e;ini! "e ;o!&a !=6i"a o (on(ei$o "e velo(i"a"e in#$an$nea (o&o avelo(i"a"e "e (o& e & &vel e 6e!(o!!e &a $!aKe$!ia n& "e$e!&ina"o in#$an$eva!ian"o #a 6o#i>o. Se a 6o#i>o "e & oKe$o &"a (o& o $e&6o: ele e#$= ani&a"o"e velo(i"a"e. Se ele e#$= e& !e6o#o: #a velo(i"a"e 9 nla.O#e!van"o a velo(i"a"e e#(ala! &9"ia ela #e&6!e ;aF !e;e!@n(ia a "oi# in#$an$e# "e $ine&6o 6o! i##o: ;ala&o# e& &9"ia. No en$an$o: a velo(i"a"e na al $e&o# &aio!in$e!e##e 9 a velo(i"a"e n& "e$e!&ina"o in#$an$e "e $e&6o. Tal velo(i"a"e 9"eno&ina"a velo(i"a"e in#$an$nea: e 6a!a "e;ini!&o# a velo(i"a"e in#$an$nea:"eve&o# !e(o!!e! a & (on(ei$o &a$e&=$i(o (on8e(i"o (o&o li&i$e.

O#e!ve&o# e a velo(i"a"e &9"ia 9 "e;ini"a $o&an"o-#e "oi# in#$an$e# "e$e&6o. Pa!a "e;ini-la n& "e$e!&ina"o in#$an$e: a#$a $o&a!&o# in$e!valo# "e$e&6o (a"a veF &eno!e#. De##a ;o!&a e#$a&o# a##e!an"o e: (a"a veF &ai#: noei#$a "i;e!en>a en$!e & 6on$o "eno&ina"o $, e $*. Po!$an$o: e#$a!e&o# ;alan"o: ao$o&a!&o# o li&i$e no al $en"e a Fe!o: "e & # in#$an$e "e $e&6o. De;ini&o#:

6o!$an$o: a velo(i"a"e in#$an$nea no in#$an$e $* a$!av9# "o 6!o(e##o li&i$e.

O 6!o(e##o li&i$e "e;ini"o a(i&a $e& o no&e "e "e!iva"a "a ;n>o $ ;aFen"o&en>o ao $e&6o e ele !e6!e#en$a

Na ;#i(a: o (on(ei$o "e velo(i"a"e &9"ia o velo(i"a"e e#(ala! &9"ia 9"i;e!en$e "o (on(ei$o "e velo(i"a"e in#$an$nea. A velo(i"a"e &9"ia e#$a lia"a a &in$e!valo "e $e&6o $ enan$o a velo(i"a"e in#$an$nea a & in#$an$e "e $e&6o $.Pa!a en$en"e! &el8o! e#$a "i;e!en>a va&o# e#$"a! o ee&6lo "e & &ovi&en$oni;o!&e&en$e va!ia"o. U& (a!!o 6a!$e "o !e6o#o velo(i"a"e ini(ial Fe!o e 6e!(o!!e2& e& *#. al a velo(i"a"e &9"ia "e#$e &vel no# *# "e &ovi&en$oQSae&o# e a va!ia>o "e e#6a>o "o &vel ;oi "e 2& e a va!ia>o "e $e&6o "o&vel ;oi "e *#: loo: a velo(i"a"e &9"ia 9 "a"a 6o!V& S$

V& 2& *#V& 2

A velo(i"a"e &9"ia "o &vel ;oi "e 2. I#$o no #ini;i(a e ele e#$ava #e&6!e(o& velo(i"a"e 2: K= e 6a!$e "o !e6o#o velo(i"a"e ini(ial Fe!o e ao lono "o

6e!(!#o a&en$a #a velo(i"a"e.Pa!a #ae! a velo(i"a"e in#$an$nea "o &vel no in#$an$e 1 #: #aen"o e a a(ele!a>o"o &e#&o 9 "e ,,: "eve&o# $iliFa! a ea>o aaioV V a.$

S#$i$in"o o# valo!e# ;o!ne(i"o#: $e&o#V V a.$

5


6/19

V , . 1V *3

Loo: a velo(i"a"e "o &vel no in#$an$e 1# 9 ial a *3 e e#$= 6o"e #e!(8a&a"a "e velo(i"a"e in#$an$nea K= e #e !e;e!e ao in#$an$e 1#.

A# ea>e# $iliFa"a# $an$o e& ;#i(a (o&o e& (al(lo #ee& a &e#&a li(a:#en"o e e& ;#i(a $iliFa&o# a "e!iva"a 6a!a "e#(!eve! a 6o#i>o "a 6a!$(la "a"o#a 6o#i>o e& !ela>o ao #e $e&6o e6!e##a"a 6o! " $"$ $$ e& e " e a"eno$a>o "a ;n>o 6o#i>o o e#6a>o e $ a "eno$a>o "a ;n>o $e&6o.

EEMPLO

Daniel Silvio RA 01,//21,3

Joo Vane! "a Silva RA 11,3243410

Manoel T. P. Pa#$ana RA ,255,

%ellin$on "e Li&a 'on#e(a RA 4,521311

Da!e&o# o# #ein$e# valo!e# 7 ea>o

a ,2 , #o&a$!ia "o l$i&o ala!i#&o e (o&6e o RA "o# in$e!an$e# "o

!6oo 8o!=!ia "a velo(i"a"e

v=v+at

v=23 t

De!ivan"o

v=d s

d t=

d

dt

v=11,5 t2

v=11,5.2 t21

v=23 t

Passo 2

Mon$a! &a $aela: #an"o #e ee&6lo a(i&a: (o& o# (=l(lo# eplote n&!=;i(o a# ;n>e# S& $# e V $# 6a!a & in$e!valo en$!e a /#: "ia e$i6o "e ;n>o vo(@ $e& e (al(la! a va!ia>o "o e#6a>o 6e!(o!!i"o e a va!ia>o "evelo(i"a"e 6a!a o in$e!valo "a"o.Cal(la! a =!ea ;o!&a"a 6ela ;n>o "a velo(i"a"e: 6a!a o in$e!valo "a"o a(i&a.

t"s# $ s"m# $ "t%s# $ &"m's# $"t% $#**./W**:/. X. X v,2. X. X#**./.*W**:/.***:/ X*.**:/X v,2.*,2 X*.,2 X#**./.,W**:/.553 X,.53 X v,2.,53 X,.53 X#**:/.2W**:/.4*2:/ X2.*2:/X v,2.234 X2.34 X

#**./.5W**:/.*3*15 X5.*15 X v,2.54, X5.4, X#**././W**:/.,/,10:/ X/.,10:/ X v,2./**/ X/.**/ X

7


8/19

'aFen"o =!ea# 6on$o a 6on$o $e&o# o !=;i(o

G!=;i(o #& $#.

0 1 2 3 4 5 6

0

50

100

150

200

250

300

350

T(s)

S(m)

G!=;i(o v $#

0 1 2 3 4 5 6

0

20

40

60

80

100

120

140

t(s)

V(m/s)

Cal(lan"o a =!ea "o !=;i(o

8


9/19

A .8, Y

A /.**/, /0/ ,10:/ &,

PASS! 3

A(ele!a>o in#$an$nea

a=dv

dt

A(ele!a>o: na ;#i(a: 9 a $aa "e va!ia>o "a velo(i"a"e "e & oKe($o.A(ele!a>o "e & oKe$o 9 o !e#l$a"o li"o "e ale! e $o"a# a# ;o!>a# e a$ao!e o oKe$o: (o&o "e#(!i$o 6o! Sen"a Lei "e NeZ$on. A ni"a"e SI 6a!a aa(ele!a>o 9 o &e$!o 6o! #en"o ao a"!a"o & # ,. A(ele!a>e# #o !an"eFa#ve$o!iai# ele# $@& &ani$"e e "i!e>o e a"i(iona! "e a(o!"o (o& a lei "o

6a!alelo!a&o. Co&o & ve$o!: a ;o!>a li"a (al(la"a 9 ial ao 6!o"$o "a &a##a"o oKe$o &a an$i"a"e e#(ala! e "a a(ele!a>o.

Po! ee&6lo: an"o & (a!!o (o&e>a a 6a!$i! "e &a 6a!ali#a>o Fe!ovelo(i"a"e !ela$iva e viaKa e& lin8a !e$a e& velo(i"a"e# (!e#(en$e#: e#$= a(ele!an"o na"i!e>o "o $!aKe$o. Se o (a!!o #e $!an#;o!&a 8= &a a(ele!a>o e& "i!e>o 7 nova

"i!e>o. Pa!a e#$e ee&6lo: 6o"e&o# (8a&a! a a(ele!a>o "o (a!!o e& ;!en$e a[a(ele!a>o linea![: e o# 6a##aei!o# "o (a!!o 6"e!a& e6e!i&en$a! (o&o ;o!>ae&6!!an"o-o# "e vol$a 6a!a o# #e# la!e#. an"o &"a! "e "i!e>o: 6o"e!a&o#(8a&a! i##o "e [a(ele!a>o no-linea![: e o# 6a##aei!o# 6o"e& e6e!i&en$a! (o&o&a ;o!>a 6a!a o la"o. Se a velo(i"a"e "o ve(lo "i&ini: i#$o 9 &a a(ele!a>o na"i!e>o o6o#$a "a "i!e>o "o ve(lo: 6o! veFe#: (8a&a"o "e "e#a(ele!a>o. \5] O#

6a##aei!o# 6o"e& e6e!i&en$a! "e#a(ele!a>o (o&o &a ;o!>a "e eleva>o 6a!a lone"o #e la!. Ma$e&a$i(a&en$e: no ei#$e &a ;!&la #e6a!a"a 6a!a a "e#a(ele!a>o:

6oi# a&o# #o &"an>a# "e velo(i"a"e. Ca"a &a "e#$a# a(ele!a>e# linea!e# no-linea!: "e "e#a(ele!a>o 6o"e #e! #en$i"a 6elo# 6a##aei!o# a$9 e a #a velo(i"a"e e"i!e>o #eKa i"@n$i(a 7 "o (a!!o.

A(ele!a>o in#$an$nea : 6o! #a veF : 9 o li&i$e "a a(ele!a>o &9"ia ao lono"e & in$e!valo in;ini$e#i&al "e $e&6o. E& $e!&o# "e (=l(lo: a(ele!a>o in#$an$nea 9o "e!iva"o "o ve($o! "e velo(i"a"e e& ;n>o "o $e&6o

Ai e e& o$!o# la!e# : #e o &ovi&en$o 9 e& lin8a !e$a : !an"eFa# ve$o!iai# 6o"ee! ##$i$"o# 6o! e#(ala!e# na# ea>e# .

Po"e #e! vi#$o e o in$e!al "a ;n>o "e a(ele!a>o a $ 9 a ;n>o "avelo(i"a"e v $ < i#$o 9 : a =!ea #o a (!va "e &a a(ele!a>o e& ;n>o "o $e&6o

&a v# $ (o!!e#6on"e 7 velo(i"a"e "o !=;i(o .

9


10/19

Da"o o ;a$o "e e a a(ele!a>o 9 "e;ini"a (o&o a "e!iva"a "a velo(i"a"e : v :e& !ela>o ao $e&6o $ e a velo(i"a"e 9 "e;ini"a (o&o a "e!iva"a "a 6o#i>o : : e&!ela>o ao $e&6o : a a(ele!a>o 6o"e #e! 6en#a"a (o&o a #en"a "e!iva"a "e (o&!e#6ei$o a $

PASS! (

G!a;i(o $# aW

0 1 2 3 4 5 6

0

100

200

300

400

500

600

700

T(s)

a(m/s2 )

10

a, $# A!ea

â ,2^W

^ *â ,2^*W ,2 *^,2 ,2,â ,2^,W 4, ,^4, *152â ,2^2W ,0 2^,0 3,*5â ,2^5W 231 5^231 *5,0/â ,2^/W /0/ /^/0/ ,10/


11/19

U#an"o a ;!&la "o (al(lo "e a!ea

A .8, Y A /./0/, *520:/ &,

ETAPA 2

PASS!

Leon8a!" Ele!

'ei$a 6o! Ja_o E&anel Han"&ann*0/3

Leon8a!" Ele! ;oi & &a$e&=$i(o #>o 6ionei!o e ;#i(o. Ele ;eF i&6o!$an$e#"e#(oe!$a# e& (a&6o# $o "ive!#o# (o&o o (=l(lo in;ini$e#i&al e $eo!ia "o# !=;i(o#.Ele $a&9& in$!o"Fi &i$o "a $e!&inoloia &a$e&=$i(a &o"e!na e no$a>o: e&e#6e(ial 6a!a a an=li#e &a$e&=$i(a: (o&o a no>o "e &a ;n>o &a$e&=$i(a. Ele$a&9& 9 (on8e(i"o 6o! #e $!aal8o e& &e(ni(a: "in&i(a "e ;li"o#: $i(a:a#$!ono&ia e $eo!ia i(al.

A (on#$an$e a6a!e(e 6ela 6!i&ei!a veF e& & a!$io *025 6elo &a$e&=$i(o#>o Leon8a!" Ele!: in$i$la"o "e P!o!e##ioni# 8a!&oni(i# O#e!va$ione# Ene#$!`& In"e 52 . Ele! #o a# no$a>e# C e O 6a!a a (on#$an$e . E& *04: o&a$e&=$i(o i$aliano Lo!enFo Ma#(8e!oni 8avia #a"o a# no$a>e# A e a 6a!a a(on#$an$e. A no$a>o a&&a a6a!e(e e& nen8& la! no# e#(!i$o# "e ale! Ele!o Ma#(8e!oni : e ;oi e#(ol8i"o e& & &o&en$o 6o#$e!io! : $alveF 6o! (a#a "a lia>o"a (on#$an$e 6a!a a ;n>o a&a . Po! ee&6lo: o &a$e&=$i(o ale&o Ca!l An$onB!e$#(8nei"e! #a"a a no$a>o a&&a e& *12/ : e A#$# "e Mo!an #o e& &liv!o 6li(a"o e& 6e>a# "e *123 a *15,.

Con#$an$e# ene!aliFa"a# "e Ele! #o "a"a# 6o!

11
http://en.wikipedia.org/wiki/Jakob_Emanuel_Handmannhttp://en.wikipedia.org/wiki/Jakob_Emanuel_Handmannhttp://en.wikipedia.org/wiki/Jakob_Emanuel_Handmann


12/19

6a!a b c b *: (o& e& (a#o e#6e(ial c *. I#$o 6o"e #e! ene!aliFa"o (o&o

6a!a aln# a "i&ini>o "a ;n>o ; 6o"e #e! . Po! ee&6lo:

Dan"o o!ie& a# (on#$an$e# S$iel$Ke# eo i"i&en#ional 9 a (on#$an$e Ma##e! - G!a&ain .Con#$an$e# Ele!- Le8&e! #o "a"a# 6elo #o&a$!io "o# inve!#o# "e n&e!o# e& &a(la##e &o"lo (o&&

A# 6!o6!ie"a"e# =#i(a# #o

E #e ("a:" en$o

12


13/19

Taela 6!o6o#$a

) limn(1+

1n )

n

* resultado

limn1

(1+11 )1

,

+limn5

(1+ 15 )5

2,48832

,limn10

(1+ 110 )10

2,5937424601

+,limn50

(1+ 150 )50

,:34*/11,4023/242145102//*/22

,,lim

n100(1+ 1100 )

100

,:051*21,45,*/,342,30*450*10/

+,,lim

n500(1+ 1500 )

500

,:0*//31/,3/*0,/4,4/4415421/0,

,,,lim

n1000(1+ 11000 )

1000

,:0*34,242,,2/14,5/021211*,*4503

+,,,lim

n5000

(1+

1

5000

)

5000

,:0*1*/**1/553125,*0*3*32

13


14/19

,,,,lim

n10000(1+ 110000 )

10000

,:0*1*5/4,31,/,,5135203353054*2*

,,,,, limn100000(1+

1100000 )

100000

,:0*1,31,20*055143312/351,55,3

,,,,,

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n1000000(1+ 11000000 )

1000000

,:0*1,31,20*055143312/351,55,3

0

0.5

1

1.5

2

2.5

3Constante de Euler

n0

Constante

Con(lo e (on;o!&e N a&en$a: o valo! 9 (on#$an$e e #e a6!oi&a "o valo! "a

(on#$an$e Ele!.

PASS! 2

S9!ie# Ha!&dni(a#

S9!ie 8a!&dni(a '#i(a

A #9!ie 8a!&dni(a #o& e!a"o! no$a# a"a# ##een$e# a6!e#en$a &a !ela>o

14


15/19

in$e!vala! (a!a($e!#$i(a e i&$=vel "e o!ie& na$!al o ;#i(a. A##i& #e $o&a!&o#

(o&o ee&6lo &a (o!"a "e & violo 3+ (o!"a ) no$a Mi G!ave: no$a!e&o# e al9&

"e vi!a! e& $o"a a #a e$en#o: $a&9& vi!a e& #a &e$a"e: e& #a $e!>a 6a!$e: e&

#a a!$a 6a!$e e in$a 6a!$e e$(.: 6!o"Fin"o #on# (a"a veF &ai# a"o#.

A vi!a>o "a (o!"a 6o"e #e! "e;ini"a (o&o (i(lo# o He!$F * (i(lo a i"a e vol$a "a

vi!a>o "a (o!"a.

A# 6!i&ei!a# no$a# "a #9!ie 8a!&dni(a 'n"a&en$al a$9 3+ no$a a 6a!$i! "o #o& e!a"o!

#o no$a# (o& #o& &ai# ;o!$e: 6o!$an$o (on#i"e!a"a# (on#onan$e#: enan$o e a 6a!$i!

"a 3 no$a #(e##iva 0+ no$a e& "ian$e a #9!ie (o&e>a a 6e!"e! a ;o!>a e a ;o!&a!

in$e!valo# (a"a veF &ai# "i##onan$e#.

S9!ie 8a!&dni(a M#i(a

'n(iona "a #ein$e &anei!a #e #6#e! &a no$a ;n"a&en$al a al #e e! &on$a!

#a e#(ala: #6on8a D ) ,32 HF: 6a!a #e a(8a! a# "e&ai# no$a# &l$i6li(a-#e a

;!e@n(ia "e D 6o! *: ,: 2: 5: /... En$!e$an$o 6a!a #e ;aFe! & $!a$a&en$o e!al: "eve-

#e 6a!$i! "o e#6e($!o (o&6le$o ao al #e e#$!$!o o# in#$!&en$o#.

S9!ie 8a!&dni(a Ma$e&=$i(a

S9!ie 8a!&dni(a 9 &a #e!ie in;ini$a. Cla##i;i(a>o an$o 7 (onve!@n(ia

^ S9!ie (onve!en$e ) ei#$e o li&i$e e 9 ;ini$o n*na

^ S9!ie "ive!en$e- ei#$e o li&i$e e 9 in;ini$o n*na

^ S9!ie o#(ilan$e- no ei#$e o li&i$e n*XanX

S9!ie 8a!&dni(a P!o!e##o Geo&9$!i(a

S9!ie in;ini$a

O no&e 8a!&dni(o 9 "evi"o 7 #e&el8an>a (o& a 6!o6o!(ionali"a"e "o# (o&6!i&en$o#

"e on"a "e &a (o!"a a vi!a! *: *,: *2: *5: .....

E#$a #9!ie "ive!e "evaa!. I##o ;oi "e&on#$!a"o na I"a"e M9"ia 6o! Ni(ole "fO!e#&e

(o&o $en"o a #9!ie #ein$e

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9 $e!&o a $e!&o &aio! e o ial 7 #9!ie

e ovia&en$e "ive!e.

PASS! 3

CRESCIMENTO POPULACIONALT8o&a# Mal$8# e& #e $!aal8o 6li(a"o e& *041 An Essay on thePrinciple of Population: a6!e#en$o & &o"elo 6a!a "e#(!eve! a 6o6la>o 6!e#en$ee& & "e$e!&ina"o a&ien$e: e& ;n>o "o $e&6o. Ele (on#i"e!o N N$ (o&o#en"o o n&e!o "e in"iv"o# e& (e!$a 6o6la>o no in#$an$e $. To&an"o a# 8i6$e#e#e o# na#(i&en$o# e a# &o!$e# naele a&ien$e e!a& 6!o6o!(ionai# 7 6o6la>o

6!e#en$e e #en"o a va!ia>o "o $e&6o (on8e(i"a en$!e o# "oi# 6e!o"o#: (on(li a#ein$e ea>o 6a!a "e#(!eve! a 6o6la>o 6!e#en$e e& & "e$e!&ina"o in#$an$e $.

(t)= ert

3=1er8

er8=

3

1

er8=3

ln(er8)=ln(3)

r=ln(3)8

r=0,137326536

Sen"o

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(48)=1e48. (0,137326536)

(48)=1e6,591648

(48)=728,99

E& 51 8o!a# 8ave!= a6!oi&a"a&en$e 0,4 v!#

PASS! (

Construir uma tabela eplote um grfico do crescimento populacional em

funo do tempo, observando o que ocorre a cada 4 horas

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0 8 1 24 32 40 48

0

100

200

300

400

500

600

700

800

Crescimento Populacional (Vrus)

t(horas)

Populao

A (a"a 5 8o!a# o (!e#(i&en$o 6o6la(ional #ee a ;o!&la

(x)=1ex. (0,137326536)

E (!e#(e a $aa "e *:02.

Concluso

O $!aal8o no# "eio &ai# a(o#$&a"o# a ve! (o&o a &a$e&=$i(a no# (e!(a no

"ia a "ia: vi&o# (o&o 6o"e&o# #=-la e& #i$a>e# (o$i"iana# (o&o (=l(lo# "e

a(ele!a>o e e "e$e!&ina>o "e e#6a>o e& ;n>o "o $e&6o. Con8e(e&o# $a&9& a

(on#$an$e Ele! e 9 #a"a la!a&en$e na vi"a e 6o"e #e! #a"a 6a!a (al(lo "e

(!e#(i&en$o 6o6la(ional.

Na"a na &a$e&=$i(a 9 6o! a(a#o: ela #ee (!i$9!io# e#6e(;i(o# e 6o"e #e!

a6li(a"o# na 6!a$i(a.

-e.er/ncias 0i1liogr.icas

18


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PLT ) ,* C=l(lo "e &a va!i=vel Deo!a8 H8e#-Halle$$ ) 2.e". ) Rio "e Janei!o

LTC ,1.

PLT ) ,4 Halli"ag: Davi": *43* ) ;n"a&en$o# "e ;#i(a v.* &e(ni(a ) Rio "e

Janei!o LTC: ,3.

La!!g C. An"!eZ# h Ronal" L. P8illi6# ,2.Mathematical Techniques for Engineers and Scientists. SPIE

P!e##. 6. *35

8$$6ZZZ.in;oe#(ola.(o&eo!a;ia(!e#(i&en$o-6o6la(ional

a(e##a"o e& &a!>o,*/

19
http://books.google.com/books?id=MwrDfvrQyWYC&pg=PA164&dq=particle+%22planar+motion%22#PPA164,M1http://www.infoescola.com/geografia/crescimento-populacional/http://books.google.com/books?id=MwrDfvrQyWYC&pg=PA164&dq=particle+%22planar+motion%22#PPA164,M1http://www.infoescola.com/geografia/crescimento-populacional/

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