{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 274 56 "Calculus Exploration 14: Taylo r Polynomials - The Movie " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 85 "Analytic and visual exploration of the Taylor polynomials for various nice functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Taylor Polynomials Basic E xample: exp(x) " }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 40 "Click in the r ed area and press [Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "restart:\nf:=x->exp(x):\nt:=0:\nn:=10:\nf(x)=taylor( f(x),x=t,n+1);\nP[0]:=unapply(f(0),x):\nprint(` P`[0](x)=P[0](x));\nfo r k from 1 to n do\nP[k]:=unapply(P[k-1](x)+(D@@k)(f)(0)*x^k/k!,x):\np rint(` P`[k](x)=P[k](x));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 23 "Taylor Polynomial Mov ie" }{TEXT -1 1 "\n" }{TEXT 258 173 "Click in the red area and press [ Enter].\nThen click on the plot to see VCR type controls for the movie .\nYou should slow the movie down to about 3 fps using the [<<] contro l." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 907 "xleft:=-10:\nxright:=10:\nyb ot:=-10:\nytop:=10:\npp[0]:=plot(f(x),x=xleft..xright,y=ybot..ytop,\n \+ color=blue,labels=[``,``],\n titlefont =[HELVETICA,DEFAULT,14],\n title=`Original Function`): \nList:=[]:\nfor i from 0 to n do\nF[i]:=P[i](x):\nTitle[i]:=`Original Function and Taylor Polynomial P`||i:\np[i]:=plot(F[i],x=xleft..xrig ht,\n y=ybot..ytop,\n color=red):\np[i]: =plots[display]([pp[0],p[i]],\n titlefont=[HELVETICA,DEF AULT,14],\n title=Title[i]):\n#combp[i]:=plots[display]( [seq(p[j],j=1..i)]):\nList:=[op(List),` P`[i](x)=F[i]]:\nod:\nTaylorPo lynomialMovie:=plots[display](\n [pp[0],seq(p[i],i=0..n)],\n insequence=true,\n labels=[``,``],\n t ickmarks=[5,9]\n #view=[x_left..x_right,y_vertex-20..y_verte x+20]\n ):\nTaylorPolynomialMovie;\n'f'(x)=f(x);\nop(List); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 41 "Taylor Polynomials Basic Example: sin(x) " }} {EXCHG {PARA 0 "" 0 "" {TEXT 259 40 "Click in the red area and press [ Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "restart: \nf:=x->sin(x):\nt:=0:\nn:=20:\nf(x)=taylor(f(x),x=t,n+1);\nP[0]:=unap ply(f(0),x):\nprint(` P`[0](x)=P[0](x));\nfor k from 1 to n do\nP[k]:= unapply(P[k-1](x)+(D@@k)(f)(0)*x^k/k!,x):\nprint(` P`[k](x)=P[k](x)); \nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 23 "Taylor Polynomial Movie" }{TEXT -1 1 "\n" } {TEXT 261 173 "Click in the red area and press [Enter].\nThen click on the plot to see VCR type controls for the movie.\nYou should slow the movie down to about 3 fps using the [<<] control." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 907 "xleft:=-10:\nxright:=10:\nybot:=-10:\nytop:=10:\np p[0]:=plot(f(x),x=xleft..xright,y=ybot..ytop,\n color= blue,labels=[``,``],\n titlefont=[HELVETICA,DEFAULT,14 ],\n title=`Original Function`):\nList:=[]:\nfor i fro m 0 to n do\nF[i]:=P[i](x):\nTitle[i]:=`Original Function and Taylor P olynomial P`||i:\np[i]:=plot(F[i],x=xleft..xright,\n y =ybot..ytop,\n color=red):\np[i]:=plots[display]([pp[0] ,p[i]],\n titlefont=[HELVETICA,DEFAULT,14],\n \+ title=Title[i]):\n#combp[i]:=plots[display]([seq(p[j],j=1..i)]):\n List:=[op(List),` P`[i](x)=F[i]]:\nod:\nTaylorPolynomialMovie:=plots[d isplay](\n [pp[0],seq(p[i],i=0..n)],\n insequence =true,\n labels=[``,``],\n tickmarks=[5,9]\n \+ #view=[x_left..x_right,y_vertex-20..y_vertex+20]\n ):\n TaylorPolynomialMovie;\n'f'(x)=f(x);\nop(List);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Tayl or Polynomials Basic Example: cos(x) " }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 40 "Click in the red area and press [Enter]." }{TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "restart:\nf:=x->cos(x):\nt:=0:\nn: =20:\nf(x)=taylor(f(x),x=t,n+1);\nP[0]:=unapply(f(0),x):\nprint(` P`[0 ](x)=P[0](x));\nfor k from 1 to n do\nP[k]:=unapply(P[k-1](x)+(D@@k)(f )(0)*x^k/k!,x):\nprint(` P`[k](x)=P[k](x));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 23 "Tayl or Polynomial Movie" }{TEXT -1 1 "\n" }{TEXT 264 173 "Click in the red area and press [Enter].\nThen click on the plot to see VCR type contr ols for the movie.\nYou should slow the movie down to about 3 fps usin g the [<<] control." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 907 "xleft:=-10: \nxright:=10:\nybot:=-10:\nytop:=10:\npp[0]:=plot(f(x),x=xleft..xright ,y=ybot..ytop,\n color=blue,labels=[``,``],\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=`Ori ginal Function`):\nList:=[]:\nfor i from 0 to n do\nF[i]:=P[i](x):\nTi tle[i]:=`Original Function and Taylor Polynomial P`||i:\np[i]:=plot(F [i],x=xleft..xright,\n y=ybot..ytop,\n c olor=red):\np[i]:=plots[display]([pp[0],p[i]],\n titlefo nt=[HELVETICA,DEFAULT,14],\n title=Title[i]):\n#combp[i] :=plots[display]([seq(p[j],j=1..i)]):\nList:=[op(List),` P`[i](x)=F[i] ]:\nod:\nTaylorPolynomialMovie:=plots[display](\n [pp[0],seq (p[i],i=0..n)],\n insequence=true,\n labels=[``,` `],\n tickmarks=[5,9]\n #view=[x_left..x_right,y_v ertex-20..y_vertex+20]\n ):\nTaylorPolynomialMovie;\n'f'(x)= f(x);\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 42 "Taylor Polynomials Basic Example: sinh(x) " }}{EXCHG {PARA 0 "" 0 "" {TEXT 265 40 "Click in the red are a and press [Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "restart:\nf:=x->sinh(x):\nt:=0:\nn:=10:\nf(x)=taylor(f(x),x=t,n+1 );\nP[0]:=unapply(f(0),x):\nprint(` P`[0](x)=P[0](x));\nfor k from 1 t o n do\nP[k]:=unapply(P[k-1](x)+(D@@k)(f)(0)*x^k/k!,x):\nprint(` P`[k] (x)=P[k](x));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 266 23 "Taylor Polynomial Movie" }{TEXT -1 1 "\n" }{TEXT 267 173 "Click in the red area and press [Enter].\nTh en click on the plot to see VCR type controls for the movie.\nYou shou ld slow the movie down to about 3 fps using the [<<] control." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 908 "xleft:=-10:\nxright:=10:\nybot:=-1 0:\nytop:=10:\npp[0]:=plot(f(x),x=xleft..xright,y=ybot..ytop,\n \+ color=blue,labels=[``,``],\n titlefont=[HELV ETICA,DEFAULT,14],\n title=`Original Function`):\nList :=[]:\nfor i from 0 to n do\nF[i]:=P[i](x):\nTitle[i]:=`Original Funct ion and Taylor Polynomial P`||i:\np[i]:=plot(F[i],x=xleft..xright,\n \+ y=ybot..ytop,\n color=red):\np[i]:=plots [display]([pp[0],p[i]],\n titlefont=[HELVETICA,DEFAULT,1 4],\n title=Title[i]):\n\n#combp[i]:=plots[display]([seq (p[j],j=1..i)]):\nList:=[op(List),` P`[i](x)=F[i]]:\nod:\nTaylorPolyno mialMovie:=plots[display](\n [pp[0],seq(p[i],i=0..n)],\n \+ insequence=true,\n labels=[``,``],\n tickm arks=[5,9]\n #view=[x_left..x_right,y_vertex-20..y_vertex+20 ]\n ):\nTaylorPolynomialMovie;\n'f'(x)=f(x);\nop(List);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 42 "Taylor Polynomials Basic Example: cosh(x) " }}{EXCHG {PARA 0 "" 0 "" {TEXT 268 40 "Click in the red area and press [Enter]. " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "restart:\nf:=x- >cosh(x):\nt:=0:\nn:=10:\nf(x)=taylor(f(x),x=t,n+1);\nP[0]:=unapply(f( 0),x):\nprint(` P`[0](x)=P[0](x));\nfor k from 1 to n do\nP[k]:=unappl y(P[k-1](x)+(D@@k)(f)(0)*x^k/k!,x):\nprint(` P`[k](x)=P[k](x));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 269 23 "Taylor Polynomial Movie" }{TEXT -1 1 "\n" }{TEXT 270 173 "Click in the red area and press [Enter].\nThen click on the plot \+ to see VCR type controls for the movie.\nYou should slow the movie dow n to about 3 fps using the [<<] control." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 907 "xleft:=-10:\nxright:=10:\nybot:=-10:\nytop:=10:\npp[ 0]:=plot(f(x),x=xleft..xright,y=ybot..ytop,\n color=bl ue,labels=[``,``],\n titlefont=[HELVETICA,DEFAULT,14], \n title=`Original Function`):\nList:=[]:\nfor i from \+ 0 to n do\nF[i]:=P[i](x):\nTitle[i]:=`Original Function and Taylor Pol ynomial P`||i:\np[i]:=plot(F[i],x=xleft..xright,\n y=y bot..ytop,\n color=red):\np[i]:=plots[display]([pp[0],p [i]],\n titlefont=[HELVETICA,DEFAULT,14],\n \+ title=Title[i]):\n#combp[i]:=plots[display]([seq(p[j],j=1..i)]):\nLi st:=[op(List),` P`[i](x)=F[i]]:\nod:\nTaylorPolynomialMovie:=plots[dis play](\n [pp[0],seq(p[i],i=0..n)],\n insequence=t rue,\n labels=[``,``],\n tickmarks=[5,9]\n \+ #view=[x_left..x_right,y_vertex-20..y_vertex+20]\n ):\nTa ylorPolynomialMovie;\n'f'(x)=f(x);\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Taylor Polynomials: Enter Your Function " }}{EXCHG {PARA 0 "" 0 "" {TEXT 271 40 "Click in the red area and press [Enter]." }{TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "restart:\nf:=x->x^2*exp(-x):\nt:=0 :\nn:=20:\nf(x)=taylor(f(x),x=t,n+1);\nP[0]:=unapply(f(0),x):\nprint(` P`[0](x)=P[0](x));\nfor k from 1 to n do\nP[k]:=unapply(P[k-1](x)+(D@ @k)(f)(0)*x^k/k!,x):\nprint(` P`[k](x)=P[k](x));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 23 "Taylor Polynomial Movie" }{TEXT -1 1 "\n" }{TEXT 273 173 "Clic k in the red area and press [Enter].\nThen click on the plot to see VC R type controls for the movie.\nYou should slow the movie down to abou t 3 fps using the [<<] control." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 904 "xleft:=-1:\nxright:=10:\nybot:=-2:\nytop:=2:\npp[0]:=plot(f(x),x=xlef t..xright,y=ybot..ytop,\n color=blue,labels=[``,``],\n titlefont=[HELVETICA,DEFAULT,14],\n t itle=`Original Function`):\nList:=[]:\nfor i from 0 to n do\nF[i]:=P[i ](x):\nTitle[i]:=`Original Function and Taylor Polynomial P`||i:\np[i ]:=plot(F[i],x=xleft..xright,\n y=ybot..ytop,\n \+ color=red):\np[i]:=plots[display]([pp[0],p[i]],\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=Title[i]):\n #combp[i]:=plots[display]([seq(p[j],j=1..i)]):\nList:=[op(List),` P`[i ](x)=F[i]]:\nod:\nTaylorPolynomialMovie:=plots[display](\n [ pp[0],seq(p[i],i=0..n)],\n insequence=true,\n lab els=[``,``],\n tickmarks=[5,9]\n #view=[x_left..x_ right,y_vertex-20..y_vertex+20]\n ):\nTaylorPolynomialMovie; \n'f'(x)=f(x);\nop(List);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 27 "Filename: ExploreCalc14.mws" }}{PARA 0 " " 0 "" {TEXT -1 36 "Copyright 2006, All Rights Reserved." }}{PARA 0 " " 0 "" {TEXT -1 44 "Permission is granted to use and modify for " }} {PARA 0 "" 0 "" {TEXT -1 37 "academic and non-commercial purposes." }} {PARA 0 "" 0 "" {TEXT -1 13 "Dr. John Pais" }}{PARA 0 "" 0 "" {TEXT -1 28 "Mathematics Department-MICDS" }}{PARA 0 "" 0 "" {TEXT -1 45 "E- mail: pais@micds.org or pais@kinetigram.com" }}{PARA 0 "" 0 "" {TEXT -1 33 "URL: http://kinetigram.com/micds" }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}}{MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }