C\303\241lculo IILi\303\247\303\243o 22: Aproxima\303\247\303\243o de Fun\303\247\303\265es Gen\303\251ricas por S\303\251rie de Taylor Na se\303\247\303\243o anterior, estudamos como aproximar as fun\303\247\303\265es seno e coseno de forma arbitrariamente precisa, em torno do ponto x = 0, utilizando polin\303\264mios. Neste folha de trabalho, verificaremos que o mesmo procedimento pode ser usado para aproximar outras fun\303\247\303\265es. Al\303\251m disso, veremos como gerar aproxima\303\247\303\265es em torno de outros pontos.

restart; Recordemos o m\303\251todo utilizado: deve-se encontrar uma aproxima\303\247\303\243o para a fun\303\247\303\243o em torno de 0. Em seguida, subtrai-se a aproxima\303\247\303\243o da fun\303\247\303\243o, e observa-se com que velocidade esta diferen\303\247a tende a 0 quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIwRidGPkYr. Para isto, deve-se dividir o resultado desta diferen\303\247a por pot\303\252ncias de x. f := x -> sqrt(1 + x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUklc3FydEdGJTYjLCYiIiJGODkkRjhGKEYoRig3I0Yu

plot(f, -1..1);

6*-%'CURVESG6$7dw7$$!#5!""$""!!""7$$!1)[(\**y%[*)*!#;$"2U(QZ#)pVD5!#<7$$!1xa4>cN.)*!#;$"1%)\&>Z*H-9!#;7$$!1BX!4)HY+(*!#;$"1">s2y82t"!#;7$$!1)e<NI))of*!#;$"2vn>1<kx+#!#<7$$!1iC\)*e!Q\*!#;$"20r)pxd()\A!#<7$$!0%yc8`B)R*!#:$"2)e`:\#*3`C!#<7$$!19Fa3vF*H*!#;$"2N0[%Rj6ZE!#<7$$!1D^-0_$p>*!#;$"1\^+*yRQ$G!#;7$$!1lGd96#\4*!#;$"2W`d/!HX3I!#<7$$!1'3<M3()**)*)!#;$"2a%\!z.q!yJ!#<7$$!1"He;8hv*))!#;$"0s1q[*H?L!#:7$$!1D]+,9^$z)!#;$"2v^2]ZaMZ$!#<7$$!1/3;KW.*o)!#;$"2ZUf'\es?O!#<7$$!1Ryc8:N)e)!#;$"1$*=,(3">dP!#;7$$!1[%*)yx@p\)!#;$"0(3;x_&p(Q!#:7$$!1sW*)yH?)Q)!#;$"/fa%>>Z,%!#97$$!1(Qxa*Q5'H)!#;$"1)f$QUw#y7%!#;7$$!0<Mowq*)=)!#:$"0dzG"*=cD%!#:7$$!1v[(\zRT4)!#;$"2;R9W)ehlV!#<7$$!1CZ%*)=&4!*z!#;$"2v7FO>'>$[%!#<7$$!0*zf>,-"*y!#:$"1z&Qe4jBf%!#;7$$!1f>Ryck(y(!#;$"1hSitqc.Z!#;7$$!1$yc8n:Fp(!#;$"2FUDt&*>M![!#<7$$!1a3<M3K!f(!#;$"16fj-$[)3\!#;7$$!1)RzePhR[(!#;$"1d&fD)H,;]!#;7$$!1pQxa^P"R(!#;$"1)*Q(z')pu5&!#;7$$!1)[(\*\z8H(!#;$"1/Sk#*3W/_!#;7$$!1w],.U2)=(!#;$"1;1Zn%fFI&!#;7$$!1:JiC,,(3(!#;$"1'>?z77sR&!#;7$$!1/3;KaA*)p!#;$"1vYE\D0([&!#;7$$!1v],.Al!)o!#;$"1%HW;<7^e&!#;7$$!0>QwK%4$y'!#:$"1T*H.@x<n&!#;7$$!13=Os/$*ym!#;$"1U$yx8sGw&!#;7$$!1V%)oPDa%e'!#;$"0J8'G:>We!#:7$$!1\)pRf^8['!#;$"1MusC'>=$f!#;7$$!1X*)ydpD%Q'!#;$"1***H.*[58g!#;7$$!0;Kk3lFG'!#:$"1Q(4D38p4'!#;7$$!1yd:J#QN='!#;$"1$HW7$=vxh!#;7$$!1Rxa4plzg!#;$"0em:Kk7E'!#:7$$!1c7D]ogzf!#;$"1_s6CqlSj!#;7$$!1^.297Hxe!#;$"1x3%QLK3U'!#;7$$!1v],.G#ex&!#;$"1AeyVqO*\'!#;7$$!1$yc8(Qe#o&!#;$"1fu^yZqql!#;7$$!1T$oO`@dd&!#;$"1e"*z()\_^m!#;7$$!1BY#\eU,[&!#;$"1"35O[))Hs'!#;7$$!1hAX!\N#y`!#;$"1/RIegN)z'!#;7$$!0&**)zR(p!G&!#:$"1Rv;X!=(po!#;7$$!1NqS")QRs^!#;$"1&z%pmw4[p!#;7$$!0&)pR*4hy]!#:$"1v'y`Ah_,(!#;7$$!2vW*)ydm@(\!#<$"1n[f*p@24(!#;7$$!117C[S:v[!#;$"1#*4()QH!)er!#;7$$!2NnMpeP!pZ!#<$"11%R)e#RDB(!#;7$$!1zc8FqZxY!#;$"1)*\@JBc&H(!#;7$$!1Hf=P-?tX!#;$"1))=i)e"omt!#;7$$!2/:Ig+uCZ%!#<$"1()G"pMLZV(!#;7$$!28U%)od8=P%!#<$"1URK")R7-v!#;7$$!2C['HfM_rU!#<$"1=pI*)*p'ov!#;7$$!1KjE`g<vT!#;$"11e.CM0Kw!#;7$$!1%*)yd&>-rS!#;$"0_()Hu&)**p(!#:7$$!2%=Pu[bvrR!#<$"1;]i,q<kx!#;7$$!21C['HBGnQ!#<$"1vcw2q;Jy!#;7$$!2NrU&3*3Fx$!#<$"1Etn0(=8*y!#;7$$!1Z$pQd(=oO!#;$"10(yKkos&z!#;7$$!2$=Pu[V2oN!#<$"1xE%*4(G*>!)!#;7$$!005?![?oM!#:$"1/+;QA&>3)!#;7$$!1MoOt1*QO$!#;$"1%)=iW=BY")!#;7$$!2'=Qw_N"yE$!#<$"1o$)o<-*\?)!#;7$$!2&zf>RuTpJ!#<$"1A1!)\Qtk#)!#;7$$!1a4>QkxgI!#;$"1xhw9-?I$)!#;7$$!1kFb54TiH!#;$"1^:qmR.*Q)!#;7$$!29MoOt8='G!#<$"1<&ovwy([%)!#;7$$!1**)zf**3&fF!#;$"1NB7wu54&)!#;7$$!1`3<M5ZlE!#;$"1;6Ick=k&)!#;7$$!28Ig?hH_c#!#<$"1*y!4g9^A')!#;7$$!2V&3<M7vlC!#<$"1$GR%\]+!o)!#;7$$!2fGd9HS$fB!#<$"1CKAAn3T()!#;7$$!2d5@UCPaE#!#<$"11')z%HOYz)!#;7$$!2v_06i#4d@!#<$"1:5<r!>g&))!#;7$$!2Yze<&*>$f?!#<$"1eb*zM\5"*)!#;7$$!22:Ig+zD'>!#<$"1`'\eZm^'*)!#;7$$!1x_069be=!#;$"11S26w*H-*!#;7$$!1v],.I5a<!#;$"1lcd_@p!3*!#;7$$!2wU&3</2f;!#<$"1%ev7)*oG8*!#;7$$!2L!3;K+:e:!#<$"1j."H0az=*!#;7$$!2'*******fh'f9!#<$"143xK^RT#*!#;7$$!2E^-05**GN"!#<$"1&pq@#*y*)H*!#;7$$!1#Qw_&)>0E"!#;$"0fEmn=&[$*!#:7$$!2>MoO$*=W:"!#<$"1vm6J\40%*!#;7$$!2hBZ%*[yY0"!#<$"1n\[#)4(zX*!#;7$$!0Hd9H9!f&*!#;$"1^p5nl/5&*!#;7$$!1Nxa4zXq&)!#<$"1>^mJ'z=c*!#;7$$!1rV([(H>iv!#<$"0"GPG#eWh*!#:7$$!0)e<N!\M\'!#;$"1([+$R&y)p'*!#;7$$!1Uu[(\:N^&!#<$"1>ZdgeT?(*!#;7$$!1l;Lm_>YX!#<$"1x4^,j/q(*!#;7$$!2%*['Hfe97N!#=$"1J&zZ7BG#)*!#;7$$!2X=Pu[Z.[#!#=$"1&="eAR?v)*!#;7$$!2jQze<ZUb"!#=$"1Y!=5J$)>#**!#;7$$!1tJkGd9sX!#=$"1oK!338r(**!#;7$$"1R#f=P9tg%!#=$"2ati)45I-5!#;7$$"2C/5?SsV`"!#=$"2bu2plUw+"!#;7$$"1JtY$pe!fD!#<$"1A:mkWr75!#:7$$"1Lu[(\")RZ$!#<$"2rxnah@s,"!#;7$$".(Rzy!H]%!#9$"1Xr"yjmA-"!#:7$$"1GjE`YlQb!#<$"2/!R))z+KF5!#;7$$"1%e<Nqy%pl!#<$"2pxhD%\KK5!#;7$$"13Qw_X=Dv!#<$"2c+_7^Vp."!#;7$$"1"4:IgiZ^)!#<$"1_=:$>/</"!#:7$$"1b4>Qc=Q&*!#<$"1Hb&*=\gY5!#:7$$"2fNrUlKe0"!#<$"2C0`l>n90"!#;7$$"2[8Faom2;"!#<$"2%*y9%)HXk0"!#;7$$"2#Hf=PE>`7!#<$"2.%QYj1"31"!#;7$$"2a>RyOUsN"!#<$"2'>EfMOql5!#;7$$"1=MoO$><Y"!#;$"1,:a'>%fq5!#:7$$"2GQw_D-Cc"!#<$"2\RLL6'Gv5!#;7$$"2Fxa4*>$Ql"!#<$"2LW/i:H&z5!#;7$$"2$[(\**y]Dw"!#<$"2=>#pRHb%3"!#;7$$"2P$oOt)\Y&=!#<$"2tpjwg!z)3"!#;7$$"2805?+$yh>!#<$"2n3MYO*p$4"!#;7$$"2zMpQ(Rhc?!#<$"2<^Kp8E!)4"!#;7$$"1(\**)z&e1;#!#;$"2,ln7w`F5"!#;7$$"2<BY#\OtfA!#<$"2yl&>:oB26!#;7$$"2<E_/43JO#!#<$"2-?3I9&*=6"!#;7$$"2$Qu[(4Q!eC!#<$"1'piy"e:;6!#:7$$"1nLnMHVgD!#;$"20p\IPL27"!#;7$$"1D['HR#zmE!#;$"2oylKXoa7"!#;7$$"2ONqShy$fF!#<$"2h9LZ>u&H6!#;7$$"1LnMpUPfG!#;$"1)=)y5=*R8"!#:7$$"1X"HeczE'H!#;$"2&R3^Rw`Q6!#;7$$"2j5@UkVP1$!#<$"2b[\0NnH9"!#;7$$"1<MoO$G:;$!#;$"27xkL/Ps9"!#;7$$"2m9Hec,,F$!#<$"1_,jo#f>:"!#:7$$"0.17WfwO$!#:$"2LtMq%f=c6!#;7$$"2DT#['HB=Z$!#<$"2e3_#Q=og6!#;7$$"1yd:J7@mN!#;$"2D6OlzSZ;"!#;7$$"0Pu[<-%pO!#:$"1rZ8'=i"p6!#:7$$"2xFb5"o\mP!#<$"2L6#[brIt6!#;7$$"017Co))z'Q!#:$"1i]ZZUix6!#:7$$"1U%)oPb@nR!#;$"2GB;#[(H=="!#;7$$"2O['Hfo4rS!#<$"2G_'fUl@'="!#;7$$"1lHf=p9rT!#;$"2y6sZEE/>"!#;7$$"1oQxaDYtU!#;$"218AF(fr%>"!#;7$$"1Z"He'4$\P%!#;$"2ky+>+b*)>"!#;7$$"1Nu[(*)p"oW!#;$"2z9'oeq$G?"!#;7$$"2w(e<NA.vX!#<$"2<!)3=)4F27!#;7$$"2'*f>R=61n%!#<$"2&\qe%)HA67!#;7$$"2&f>Ry#=Dx%!#<$"1)4T%*[Aa@"!#:7$$"1sU&3Pc+([!#;$"0a()eSG%>7!#97$$"1&=Pu))f$y\!#;$"2$)eJ')4hQA"!#;7$$"1rV([xU@2&!#;$"2%G]1bloF7!#;7$$"1tZ&4>(ey^!#;$"1>9Y!49?B"!#:7$$"1:Ig?(*fv_!#;$"0K"Hr\%fB"!#97$$"1Y&4>=;<Q&!#;$"2z:)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Como 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, a primeira aproxima\303\247\303\243o deve ser a fun\303\247\303\243o constante 1 (polin\303\264mio de grau 0): p0 := x -> 1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMEdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCIiIkYoRihGKDcjRi4=

limit(f(x) - p0(x), x=0);

IiIh

Como esperado, esta \303\251 uma boa aproxima\303\247\303\243o, j\303\241 que a diferen\303\247a tende a 0 quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIwRidGPkYr . Mas com que velocidade esta diferen\303\247a tende a 0? limit((f(x) - p0(x))/x, x=0);

IyIiIiIiIw==

Este limite indica que 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 \303\251 dado por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y4USdub3JtYWxGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRy8lKWJldmVsbGVkR1EmZmFsc2VGJw== quando x \303\251 pequeno. Portanto, a aproxima\303\247\303\243o seguinte \303\251 dada por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiUtRiw2JVEjcDBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInhGJ0Y2RjlGQC1GPTYtUSIrRidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRlxvLUkmbWZyYWNHRiQ2KEZZLUYjNiMtSSNtbkdGJDYkUSIyRidGQC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXHAvJSliZXZlbGxlZEdGREYr : p1 := x -> 1 + (1/2)*x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmOSQjIiIiIiIjRjdGN0YoRihGKDcjRi4=

Antes de darmos prosseguimento aos c\303\241lculos, exibiremos o gr\303\241fico da fun\303\247\303\243o original, f, juntamente com a aproxima\303\247\303\243o linear, p1: plot({f,p1}, -1..1);

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Nota-se que (o gr\303\241fico de) a aproxima\303\247\303\243o linear \303\251 simplesmente a reta tangente ao gr\303\241fico de f, em x = 0. limit((f(x) - p1(x))/x, x=0);

IiIh

limit((f(x) - p1(x))/x^2, x=0);

IyEiIiIiKQ==

Estes limites indicam uma melhora na aproxima\303\247\303\243o, e tamb\303\251m informam qual deve ser o pr\303\263ximo termo. O processo pode ser repetido quanto se queira. Abaixo seguem mais alguns passos. p2 := x -> 1 + (1/2)*x - (1/8)*x^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwoIiIiRjU5JCNGNSIiIyokRjZGOCMhIiIiIilGKEYoRig3I0Yu

limit((f(x) - p2(x))/x^3, x=0);

IyIiIiIjOw==

p3 := x -> p2(x) + (1/16)*x^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcDJHRig2IzkkIiIiKiRGOCIiJCNGOSIjO0YoRihGKDcjRi4=

limit((f(x) - p3(x))/x^4, x=0);

IyEiJiIkRyI=

p4 := x -> p3(x) - (5/128)*x^4;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcDNHRig2IzkkIiIiKiRGOCIiJSMhIiYiJEciRihGKEYoNyNGLg==

p4(x);

LCwiIiJGI0kieEc2IiNGIyIiIyokRiRGJyMhIiIiIikqJEYkIiIkI0YjIiM7KiRGJCIiJSMhIiYiJEci

Qu\303\243o eficaz o m\303\251todo se mostrou? plot({f,p2,p4}, -1..1);

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N\303\243o h\303\241 nada em especial na fun\303\247\303\243o f escolhida: o m\303\251todo \303\251 eficaz com qualquer fun\303\247\303\243o razoavelmente bem comportada. Apresentamos a seguir uma importante fun\303\247\303\243o da estat\303\255stica (usada para descrever a distribui\303\247\303\243o normal). g := x -> exp(-x^2/2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJnR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUkkZXhwR0YlNiMsJCokOSQiIiMjISIiRjpGKEYoRig3I0Yu

plot(g, -4..4, 0..2);

6*-%'CURVESG6$7dw7$$!#S!""$"2&=^-ziiaL!#?7$$!1&**)zf"Rz&R!#:$"2Zw$4!fmd'R!#?7$$!1">QwCU8#R!#:$"1;?di#>3e%!#>7$$!14=O#>&=!)Q!#:$"1K.hngcy`!#>7$$!1NqS@`vQQ!#:$"1*=@n!z:6j!#>7$$!1&)pRfB_(z$!#:$"11J5HjA(Q(!#>7$$!1OrUDTHfP!#:$"1LM*)G4;N&)!#>7$$!2b3<M+6(>P!#;$"1P)3W%)Qo*)*!#>7$$!005?3u(yO!#9$"2By()f`+::"!#>7$$!1Y"HeWozj$!#:$"2(QhsUF*oL"!#>7$$!2V$oOL[*ff$!#;$"2hS/m!Q2c:!#>7$$!2mJjEXC!fN!#;$"2&)[Jy&Q6w<!#>7$$!0,-/c/u^$!#9$"2d#>mIP!z0#!#>7$$!2:KkGx8cZ$!#;$"1PSE&f)o"Q#!#=7$$!2c8FagS`V$!#;$"2$fRhQ>IPF!#>7$$!1zd:6(o()R$!#:$"1h%oPqq;5$!#=7$$!1*yd:>"GbL!#:$"1[D(=)RK#f$!#=7$$!2[&4>e:W=L!#;$"1rch"y8A1%!#=7$$!1oOt1$)evK!#:$"0'yJ]%z'yY!#<7$$!0&**)z"flPK!#9$"1R!zgSeQH&!#=7$$!2%*)ydv!Qg>$!#;$"1ik-uEA_g!#=7$$!1'>Ry/3k:$!#:$"1W0`8?1ko!#=7$$!2Oyc8Fe]6$!#;$"1GmH(*eM9y!#=7$$!2MrU&oi3xI!#;$"1L6BVV@*y)!#=7$$!2;MoOLGh.$!#;$"1sGjc-Sh**!#=7$$!1f<N]Xe$*H!#:$"2wo`")=kC8"!#=7$$!2xa4>1]l&H!#;$"2eKmkisVE"!#=7$$!1&**)z*z^l"H!#:$"2$fA!pUb>U"!#=7$$!2..17oH_(G!#;$"2Wx?7@7Fg"!#=7$$!1Y#\)\S![$G!#:$"2(*pU^9)y)z"!#=7$$!2:KkG<!p&z#!#;$"1[2OZ"z"3?!#<7$$!0.17)3E_F!#9$"2Ci<u:%GlA!#=7$$!1w_0JxB8F!#:$"2x+,R71-_#!#=7$$!2LsW*)=s:n#!#;$"2=$\')\=R>G!#=7$$!1x`2:q"Qj#!#:$"2e?$G"['Q;J!#=7$$!2%RzeP1a#f#!#;$"1Y>O'=E8Z$!#<7$$!1yb6$y-Pb#!#:$"2C<PNddh$Q!#=7$$!1kGdMg58D!#:$"2%e>xG8q^U!#=7$$!29JiCH:MZ#!#;$"28o0VTpRp%!#=7$$!2d4>Qwi=V#!#;$"2bNY\xtv>&!#=7$$!2A]+,uU=R#!#;$"1M:,WgECd!#<7$$!219Gc[;4N#!#;$"1"=A%Gzm2j!#<7$$!0.177H.J#!#9$"1Yzj.iYLp!#<7$$!2LrU&[N.tA!#;$"1AUr!RG@b(!#<7$$!2jLnMh)GIA!#;$"11+u[#G^J)!#<7$$!2$\)pR.d?>#!#;$"1%=!3)>G'[!*!#<7$$!2W!4='>%H^@!#;$"1N$Rx:qh))*!#<7$$!0)f>f*yA6#!#9$"2\RoVmeV2"!#<7$$!19Gc_v&*o?!#:$"2k0f<8/i<"!#<7$$!0%ze(RW9.#!#9$"2oHK++N-F"!#<7$$!1zd:Jm')))>!#:$"2w6Z!H)RPQ"!#<7$$!2C['H>;1]>!#;$"0(Q&oEQO\"!#:7$$!2%pQxM]h2>!#;$"2.5W`zr5i"!#<7$$!2<Fa3"3*4(=!#;$"2#Q>jXC@P<!#<7$$!2<Pu[4!GH=!#;$"2pO3*=Cew=!#<7$$!2-17Cg*)*)y"!#;$"28`>Q&oZ=?!#<7$$!2&oPvIas[<!#;$"27`Tx2zu;#!#<7$$!1$f=PQ4'3<!#:$"29VSPDAJK#!#<7$$!2F`18Uq+n"!#;$"2A[7/@-%zC!#<7$$!2wb6By3%G;!#;$"1$*>%>4^dl#!#;7$$!2u[(\>Aq)e"!#;$"11Fv/'34$G!#;7$$!2iHf=$H"pa"!#;$"1oV$)QOfAI!#;7$$!2a3<Mc$34:!#;$"2wS"=FvX-K!#<7$$!2)QxaH]Fn9!#;$"2juqn6R!3M!#<7$$!2t[(\R(HsU"!#;$"1jcme**Q6O!#;7$$!0-/3#>G(Q"!#9$"2$)>HMzD-#Q!#<7$$!2OtY$pibX8!#;$"2&)>iPzjV/%!#<7$$!2u_06UDrI"!#;$"1<4_h"feD%!#;7$$!2=Ryc(pwn7!#;$"1zsit(oqZ%!#;7$$!2<Qw_d5VA"!#;$"1)z%)oFyhs%!#;7$$!2c5@UOk\="!#;$"0A(y/Ieb\!#:7$$!2lLnM\DZ9"!#;$"1z5))\5R$>&!#;7$$!2'f>R)f.Q5"!#;$"1dFAM^!zV&!#;7$$!28MoOT)=m5!#;$"1UbJV(HWm&!#;7$$!207C[%=4E5!#;$"1j:ALs32f!#;7$$!1<MoO\+j)*!#;$"1a>[(3$R[h!#;7$$!1V"He;htV*!#;$"1/fc7$*>1k!#;7$$!1BW)o(*[<1*!#;$"1<2-UynKm!#;7$$!06AW[q$G')!#:$"1hh="Rh=*o!#;7$$!1y^.2)zsB)!#;$"1K[&paWH7(!#;7$$!1.17CgJ]y!#;$"1AZ&*[T9[t!#;7$$!136AWc?Mu!#;$"1(4_X-sbe(!#;7$$!/.17?T;q!#9$"1\m4()z/=y!#;7$$!0rT$o;GOm!#:$"1yui*ptN-)!#;7$$!18KkG,gKi!#;$"1%pI"*[8ZB)!#;7$$!1)******RY'Qe!#;$"1@0k28'GV)!#;7$$!005?S'f6a!#:$"1<5=>*))yj)!#;7$$!1Gb5@%z?/&!#;$"1;*e"*4Vj!))!#;7$$!2xOtYtvwh%!#<$"1@&y@!*>())*)!#;7$$!2V%*)ydRr=U!#<$"1A$38#zc["*!#;7$$!1;He;dgBQ!#;$"1v-^t03&H*!#;7$$!1%4>Q;$=GM!#;$"1>@/C(4$H%*!#;7$$!2%[(\**=x[-$!#<$"1AUwgJ"Gb*!#;7$$!1_.29'ztf#!#;$"1z/VInIo'*!#;7$$!1x\**)>1a?#!#;$"1CsaPCuf(*!#;7$$!1mE`1"y%==!#;$"1(>7PG;g$)*!#;7$$!2df=PMe[S"!#<$"0-tW&R!=!**!#:7$$!1Q([(\**Q@**!#<$"12-)y#R!4&**!#;7$$!1Xv^.())p@'!#<$"1'\/u=$p!)**!#;7$$!2$psX"He)G=!#=$"14n]yxK)***!#;7$$"2cpV([d#H%=!#=$"18i_c>I)***!#;7$$"0<S!3'*[Ph!#;$"1-mBOL=")**!#;7$$"2C$pQxMiB5!#<$"1C!*RYnuZ**!#;7$$"1t\**)f#f*Q"!#;$"1l-c.i"R!**!#;7$$"/)e<:j6!=!#9$"1NCVv!*4R)*!#;7$$"28`18'=Y:A!#<$"1jDwVLdd(*!#;7$$"2O.29["zFE!#<$"1/`6jzig'*!#;7$$"2M_06#Q25I!#<$"1*RC7C"3d&*!#;7$$"2l.17/0fS$!#<$"1$Q;)\0\O%*!#;7$$"1#Qw_Du_"Q!#;$"16&o(z)Q!)H*!#;7$$"2NU&3<1LBU!#<$"1#**y:J&yY"*!#;7$$"1R&3<umIk%!#;$"1")*[J4e"y*)!#;7$$"1<Pu[0x7]!#;$"1`b1['G$>))!#;7$$"1#yc8Zp*Ga!#;$"1">66re(H')!#;7$$"1sOtYt(o%e!#;$"1EZQ'z1)G%)!#;7$$"1Jb5@!4'\i!#;$"0^'pEm(fA)!#:7$$"1"4>Q'zK:m!#;$"1np4x4sM!)!#;7$$"1$**)zfJ?]q!#;$"10#e+S*[*z(!#;7$$"1NtY$\*f=u!#;$"1CR#QhoVf(!#;7$$"10-/3?8Zy!#;$"1E!3Pr!)*\t!#;7$$"1"Rxa*eXE#)!#;$"1**pu!*RHHr!#;7$$"1))zf>VjU')!#;$"10ugSvP$)o!#;7$$"1F\)pfM*Q!*!#;$"1=Kk1qQYm!#;7$$"1Z!4=OKCX*!#;$"1LaGQo3(R'!#;7$$"1`(\**Q_@$)*!#;$"1#e]'=<5nh!#;7$$"2nMpQ<tT-"!#;$"1FDVpsr=f!#;7$$"0$f=dprm5!#9$"1b&RGzP7m&!#;7$$"299GcW^P5"!#;$"1N^]w"=#Qa!#;7$$"2JpQxq\P9"!#;$"1$[&))oG>*>&!#;7$$"1e;LE=2&="!#:$"1:7]y>&\&\!#;7$$"2DW)odu\D7!#;$"2;qp0F7$>Z!#<7$$"2nOtYL6YE"!#;$"1u8v2L*\\%!#;7$$"2'e;LE1/38!#;$"1^&e?>p2D%!#;7$$"17C[wP1Z8!#:$"1FxW!zjh.%!#;7$$"1lHf=$H()Q"!#:$"1z.Ey'eD"Q!#;7$$"27JiC\%[E9!#;$"2cY?D@K_h$!#<7$$"1[(\*p3wn9!#:$"2u0C275cS$!#<7$$"16@UC()f1:!#:$"2l2"4<)yW@$!#<7$$"1C['HZ&>Z:!#:$"1'[0Dts7-$!#;7$$"2pPv]@')oe"!#;$"1$=ML9$3RG!#;7$$"2Mf=PuQ%G;!#;$"1=)f8_@cl#!#;7$$"1'=Puwe%o;!#:$"2M(H,p93'[#!#<7$$"2sa4>-&Q4<!#;$"2B*oxZ[/?B!#<7$$"2(e;L'Qs*\<!#;$"2jz/y<cF;#!#<7$$"1u\**ezE(y"!#:$"2k&)Q,N+Z-#!#<7$$"1^.2%*G,I=!#:$"1rip^!oS(=!#;7$$"2)RyctWCo=!#;$"0QdqFbhu"!#:7$$"2Qyc8J2!4>!#;$"2Km67\qnh"!#<7$$"2)3<M[D-[>!#;$"1ZZzZhe*\"!#;7$$"1u[(\&RM"*>!#:$"2xX&p$fMpP"!#<7$$"2$[(\*4r&)G?!#;$"1poQpX#pF"!#;7$$"2$4>Qw[Vr?!#;$"2ooYhN(=q6!#<7$$"117C)))R-6#!#:$"/2mcP**y5!#97$$"2%=Q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Quando x = 0, 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 ser\303\241 a primeira aproxima\303\247\303\243o. q0 := x -> 1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMEdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCIiIkYoRihGKDcjRi4=

limit(g(x) - q0(x), x=0);

IiIh

limit((g(x) - q0(x))/x, x=0);

IiIh

limit((g(x) - q0(x))/x^2, x=0);

IyEiIiIiIw==

Estes limites mostram que a constante 1 \303\251 uma aproxima\303\247\303\243o linear, como se verifica no gr\303\241fico de g. A aproxima\303\247\303\243o quadr\303\241tica adiciona ainda um novo termo. q2 := x -> q0(x) - (1/2)*x^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcTBHRig2IzkkIiIiKiRGOCIiIyMhIiJGO0YoRihGKDcjRi4=

q2(x);

LCYiIiJGIyokSSJ4RzYiIiIjIyEiIkYn

Na verdade, as \303\272nicas pot\303\252ncias de x que aparecem nas aproxima\303\247\303\265es de g em torno de x = 0 s\303\243o pares. (p\303\264r que?) Abaixo, beneficiaremo-nos deste fato a fim de evitar c\303\241lculos desnecess\303\241rios. limit((g(x) - q2(x))/x^4, x=0);

IyIiIiIiKQ==

q4 := x -> q2(x) + (1/8)*x^4;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcTJHRig2IzkkIiIiKiRGOCIiJSNGOSIiKUYoRihGKDcjRi4=

limit((g(x) - q4(x))/x^6, x=0);

IyEiIiIjWw==

q6 := x -> q4(x) - (1/48)*x^6; q6(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNkdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcTRHRig2IzkkIiIiKiRGOCIiJyMhIiIiI1tGKEYoRig3I0Yu

LCoiIiJGIyokSSJ4RzYiIiIjIyEiIkYnKiRGJSIiJSNGIyIiKSokRiUiIicjRikiI1s=

plot({g,q6}, -4..4, -2..2);

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restart; Como se pode notar nos exemplos acima, tem-se boas aproxima\303\247\303\265es em torno de x = 0, mas n\303\243o em outros pontos quaisquer. Suponhamos que deseja-se aproximar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbXNxcnRHRiQ2Iy1GIzYlLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIitGJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkwtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjVRJ2l0YWxpY0Yn em torno de x = 3, por exemplo. Obviamente, em x = 3 esta express\303\243o \303\251 exatamente igual a 2. Neste caso, esta seria certamente a primeira aproxima\303\247\303\243o. f := x -> sqrt(1 + x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUklc3FydEdGJTYjLCYiIiJGODkkRjhGKEYoRig3I0Yu

p0 := x -> 2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMEdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCIiI0YoRihGKDcjRi4=

Qual deve ser a pr\303\263xima aproxima\303\247\303\243o? Deseja-se saber a velocidade com que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiUtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEieEYnRjZGOUZALUY9Ni1RKCZtaW51cztGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGXG8tSSNtbkdGJDYkUSIyRidGQEYr tende a 0 quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIzRidGPkYr. Para isso, \303\251 preciso comparar esta diferen\303\247a com outras express\303\265es mais simples, que aproximam-se de 0. Quando aproxim\303\241vamos em torno de x = 0, compar\303\241vamos a diferen\303\247a com pot\303\252ncias de x. Agora que se trata de x = 3, as pot\303\252ncias de x n\303\243o mais aproximam-se de 0 --- deve-se usar pot\303\252ncias de (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GKw==)! limit(f(x) - p0(x), x=3);

IiIh

limit((f(x) - p0(x))/x, x=3);

IiIh

limit((f(x) - p0(x))/(x-3), x=3);

IyIiIiIiJQ==

p1 := x -> p0(x) + (1/4)*(x-3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwoLUkjcDBHRig2IzkkIiIiRjgjRjkiIiUjISIkRjtGOUYoRihGKDcjRi4=

Lamentavelmente, o Maple insiste em simplificar algumas das aproxima\303\247\303\265es polinomiais; veremos mais adiante como for\303\247\303\241-lo a escrev\303\252-las como pot\303\252ncias de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GKw== . Apesar disso, observa-se que a aproxima\303\247\303\243o linear mais uma vez \303\251 a reta tangente: plot({f,p1}, -1..5);

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Continuando o processo, ainda \303\251 poss\303\255vel gerar aproxima\303\247\303\265es de ordem superior. limit((f(x) - p1(x))/(x-3)^2, x=3);

IyEiIiIjaw==

p2 := x -> p1(x) - (1/64)*(x-3)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcDFHRig2IzkkIiIiKiQsJkY4RjkhIiRGOSIiIyMhIiIiI2tGKEYoRig3I0Yu

limit((f(x) - p2(x))/(x-3)^3, x=3);

IyIiIiIkNyY=

p3 := x -> p2(x) + (1/512)*(x-3)^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcDJHRig2IzkkIiIiKiQsJkY4RjkhIiRGOSIiJCNGOSIkNyZGKEYoRig3I0Yu

limit((f(x) - p3(x))/(x-3)^4, x=3);

IyEiJiImJVE7

p4 := x -> p3(x) - (5/16384)*(x-3)^4; p4(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcDNHRig2IzkkIiIiKiQsJkY4RjkhIiRGOSIiJSMhIiYiJiVRO0YoRihGKDcjRi4=

LCwjIiImIiIlIiIiSSJ4RzYiI0YmRiUqJCwmRidGJiEiJEYmIiIjIyEiIiIjayokRisiIiQjRiYiJDcmKiRGK0YlIyEiJiImJVE7

plot({f,p4}, -1..5);

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Apresentamos a seguir aproxima\303\247\303\265es para outra fun\303\247\303\243o, em torno de 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. Embora os coeficientes n\303\243o sejam muito simples, pode-se verificar nos gr\303\241ficos que o m\303\251todo ainda \303\251 eficiente. (Para aproxima\303\247\303\265es decimais deve-se utilizar evalf. Por\303\251m, \303\251 preciso calcular "aproxima\303\247\303\265es exatas" inicialmente, e utilizar evalf somente no final do procedimento.) g := x -> exp(-x^2/2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJnR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUkkZXhwR0YlNiMsJCokOSQiIiMjISIiRjpGKEYoRig3I0Yu

q0 := x -> g(-2); q0(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMEdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKC1JImdHRig2IyEiI0YoRihGKDcjRi4=

LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMhIiM=

limit(g(x) - q0(x), x=-2);

IiIh

l1 := limit((g(x) - q0(x))/(x + 2), x=-2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsMUdGKCwkLUkkZXhwR0YlNiMhIiMiIiM3I0Yu

Note que as aproxima\303\247\303\265es encontram-se em pot\303\252ncias de (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiK0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GKw==) = (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiK0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GKw==) . q1 := x -> q0(x) + l1*(x+2); q1(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcTBHRig2IzkkIiIiKiZJI2wxR0YoRjksJkY4RjkiIiNGOUY5RjlGKEYoRig3I0Yu

LCYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyEiIyIiIiomRiNGKywmSSJ4R0YoRisiIiNGK0YrRi8=

Como sempre, a aproxima\303\247\303\243o linear \303\251 dada por uma reta tangente: plot({g,q1}, -4..0);

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limit((g(x) - q1(x))/(x+2), x=-2);

IiIh

l2 := limit((g(x) - q1(x))/(x+2)^2, x=-2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsMkdGKCwkLUkkZXhwR0YlNiMhIiMjIiIkIiIjNyNGLg==

q2 := x -> q1(x) + l2*(x+2)^2; q2(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcTFHRig2IzkkIiIiKiZJI2wyR0YoRjksJkY4RjkiIiNGOUY9RjlGKEYoRig3I0Yu

LCgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyEiIyIiIiomRiNGKywmSSJ4R0YoRisiIiNGK0YrRi8qJkYjRitGLUYvIyIiJEYv

l3 := limit((g(x) - q2(x))/(x+2)^3, x=-2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsM0dGKCwkLUkkZXhwR0YlNiMhIiMjIiIiIiIkNyNGLg==

q3 := x -> q2(x) + l3*(x+2)^3; q3(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcTJHRig2IzkkIiIiKiZJI2wzR0YoRjksJkY4RjkiIiNGOSIiJEY5RihGKEYoNyNGLg==

LCotSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyEiIyIiIiomRiNGKywmSSJ4R0YoRisiIiNGK0YrRi8qJkYjRitGLUYvIyIiJEYvKiZGI0YrRi1GMiNGK0Yy

l4 := limit((g(x) - q3(x))/(x+2)^4, x=-2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsNEdGKCwkLUkkZXhwR0YlNiMhIiMjISImIiNDNyNGLg==

q4 := x -> q3(x) + l4*(x+2)^4; q4(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKCwmLUkjcTNHRig2IzkkIiIiKiZJI2w0R0YoRjksJkY4RjkiIiNGOSIiJUY5RihGKEYoNyNGLg==

LCwtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyEiIyIiIiomRiNGKywmSSJ4R0YoRisiIiNGK0YrRi8qJkYjRitGLUYvIyIiJEYvKiZGI0YrRi1GMiNGK0YyKiZGI0YrRi0iIiUjISImIiND

plot({g,q4}, -4..0);

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Visto como gerar aproxima\303\247\303\265es polinomiais para qualquer fun\303\247\303\243o em qualquer ponto, chegou a hora de apresentar o comando do Maple que realiza tais c\303\241lculos. Os polin\303\264mios que vem sendo usados at\303\251 ent\303\243o s\303\243o chamados polin\303\264mios de Taylor, e podem ser gerados atrav\303\251s do comando taylor. Para utilizar este comando, deve-se informar a express\303\243o a ser expandida, o ponto em torno do qual deseja-se expandir e o ordem da expans\303\243o desejada. (O Maple possui regras sobre a quantidade de termos na expans\303\243o. Contudo, nem sempre a resposta obtida \303\251 a resposta desejada. Por\303\251m, \303\251 poss\303\255vel acrescentar mais termos na expans\303\243o.) O \303\272ltimo par\303\242metro pode ser omitido; neste caso, o Maple utiliza a vari\303\241vel padr\303\243o Order para determinar o n\303\272mero de termos. f := x -> sqrt(1 + x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUklc3FydEdGJTYjLCYiIiJGODkkRjhGKEYoRig3I0Yu

taylor(f(x), x=0, 6);

KzFJInhHNiIiIiIiIiEjRiUiIiNGJSMhIiIiIilGKCNGJSIjOyIiJCMhIiYiJEciIiIlIyIiKCIkYyMiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiJw==

taylor(f(x), x=0, 12);

Kz1JInhHNiIiIiIiIiEjRiUiIiNGJSMhIiIiIilGKCNGJSIjOyIiJCMhIiYiJEciIiIlIyIiKCIkYyMiIiYjISNAIiVDNSIiJyMiI0wiJVs/RjQjISRIJSImb0YkRisjIiQ6KCImT2InIiIqIyElSkMiJ1dARSIjNSMiJSo+JSInKUdDJiIjNi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM3

taylor(f(x), x=0);

KzFJInhHNiIiIiIiIiEjRiUiIiNGJSMhIiIiIilGKCNGJSIjOyIiJCMhIiYiJEciIiIlIyIiKCIkYyMiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiJw==

taylor(f(x), x=3);

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g := x -> exp(-x^2/2);

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taylor(g(x), x=0);

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taylor(g(x), x=-2);

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Note que estas s\303\243o as mesmas s\303\251ries obtidas anteriormente. Apresentamos tamb\303\251m outras s\303\251ries familiares a seguir. taylor(sin(x), x=0, 10);

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taylor(cos(x), x=0, 10);

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taylor(exp(x), x=0, 10);

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