{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 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 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 "Normal" -1 256 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 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 257 68 "Calculus Exploration 12A: Visu alizing Motion on a Parametric Curve " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 256 426 "In each exercise below, view the pa rametric curve movie and write a description of the particle motion al ong the curve, including: (1) the start point and end point on the cur ve, (2) the number of times the curve is traversed from start time to \+ end time, and (3) interesting aspects of the shape of each particular \+ curve. In addition, (4) check the velocity and acceleration functions \+ given by taking the derivatives yourself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 166 "Finally, after completing Exerci ses 1.1-1.12 below, create two new parametric curves yourself, using ( appropriately editing) the two sections at the end of this file." } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 29 "Parametric Curve Exercise 1.1" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in red area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1424 "restart:\nn:=1:\nx:=t->t^3-2*t: # enter here\n y:=t->t^2-t: # enter here\ntstart:=-3: # enter here\ntend:=3: # enter here\nm:=4: # enter here\nxleft:=-m: \nxri ght:=m:\nybot:=-m: \nytop:=m: \nwith(linalg):\nr:=t->[x(t),y(t)]:\n`r '`:=t->[D(x)(t),D(y)(t)]:\n`r''`:=t->[(D@D)(x)(t),(D@D)(y)(t)]:\nwith( plottools):\nk:=100:\ndt:=abs(tstart-tend)/k:\nfor i from 0 to k do\nt [i+1]:=evalf(tstart+i*dt,4):\ntstr:=convert(t[i+1],name);\nTitle[i+1]: =`Parametric Curve Exercise 1.`||n:\ntxtplot[i+1]:=plots[textplot]([0, m-.05*m,\n ` t = `||tstr],\n \+ font=[HELVETICA,DEFAULT,12],\n color= blue,align=\{RIGHT\}):\np[0]:=plot([x(t),y(t),t=tstart..tend],color=bl ue,\n titlefont=[HELVETICA,DEFAULT,14],\n title= ``||Title[i+1]):\np[i+1]:=plots[display]([p[0],txtplot[i+1],\n \+ disk([x(tstart+i*dt),y(tstart+i*dt)],.1,color=navy)]): \nend:\nplotfcns:=plots[display](\n [seq(p[i],i=1..k+1)],\n insequence=true,\n scaling=constrained,\n \+ labels=[``,``],\n view=[xleft..xright,ybot..ytop]): \nplotfcns;\n`Parameterize x & y Coordinates: `*'x'(t)=x(t),'y'(t)=y( t);\n`Position Function (Curve): `*'r(t)'*` =`*['x'(t),'y'(t)]=r(t),t =tstart..tend;\n`Velocity Function: `*'`r '`(t)'*` =`*[`x '`(t),`y '` (t)]=`r'`(t);\n`Acceleration Function: `*'`r ''`(t)'*` =`*[`x ''`(t), `y ''`(t)]=`r''`(t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Parametric Curve Exercise 1.2 " }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in red area and Press \+ [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1416 "restart:\nn:=2:\nx:=t- >t^3-1: # enter here\ny:=t->-t^2+2: # enter here\ntstart:=-3: # \+ enter here\ntend:=3: # enter here\nm:=4: # enter here\n xleft:=-m: \nxright:=m: \nybot:=-m: \nytop:=m: \nwith(linalg):\nr:=t ->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)(t)]:\n`r''`:=t->[(D@D)(x)(t),(D @D)(y)(t)]:\nwith(plottools):\nk:=100:\ndt:=abs(tstart-tend)/k:\nfor i from 0 to k do\nt[i+1]:=evalf(tstart+i*dt,4):\ntstr:=convert(t[i+1],n ame);\nTitle[i+1]:=`Parametric Curve Exercise 1.`||n:\ntxtplot[i+1]:=p lots[textplot]([0,m-.05*m,\n ` t = `||tstr], \n font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{RIGHT\}):\np[0]:=plot([x(t),y(t),t=tsta rt..tend],color=blue,\n titlefont=[HELVETICA,DEFAULT,14],\n title=``||Title[i+1]):\np[i+1]:=plots[display]([p[0],txtpl ot[i+1],\n disk([x(tstart+i*dt),y(tstart+i*dt)] ,.1,color=navy)]):\nend:\nplotfcns:=plots[display](\n [seq( p[i],i=1..k+1)],\n insequence=true,\n scaling=c onstrained,\n labels=[``,``],\n view=[xleft..xri ght,ybot..ytop]):\nplotfcns;\n`Parameterize x & y Coordinates: `*'x'( t)=x(t),'y'(t)=y(t);\n`Position Function (Curve): `*'r(t)'*` =`*['x'( t),'y'(t)]=r(t),t=tstart..tend;\n`Velocity Function: `*'`r '`(t)'*` = `*[`x '`(t),`y '`(t)]=`r'`(t);\n`Acceleration Function: `*'`r ''`(t)' *` =`*[`x ''`(t),`y ''`(t)]=`r''`(t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Parametric \+ Curve Exercise 1.3" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in r ed area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1432 "rest art:\nn:=3:\nx:=t->sin(3*t): # enter here\ny:=t->sin(4*t): # enter here\ntstart:=-4: # enter here\ntend:=4: # enter here \nm:=2: # enter here\nxleft:=-m: \nxright:=m: \nybot:=-m: \nytop:=m: \nwith(linalg):\nr:=t->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D( y)(t)]:\n`r''`:=t->[(D@D)(x)(t),(D@D)(y)(t)]:\nwith(plottools):\nk:=10 0:\ndt:=abs(tstart-tend)/k:\nfor i from 0 to k do\nt[i+1]:=evalf(tstar t+i*dt,4):\ntstr:=convert(t[i+1],name);\nTitle[i+1]:=`Parametric Curve Exercise 1.`||n:\ntxtplot[i+1]:=plots[textplot]([0,m-.05*m,\n \+ ` t = `||tstr],\n font=[HEL VETICA,DEFAULT,12],\n color=blue,align=\{RIGHT \}):\np[0]:=plot([x(t),y(t),t=tstart..tend],color=blue,\n t itlefont=[HELVETICA,DEFAULT,14],\n title=``||Title[i+1]):\n p[i+1]:=plots[display]([p[0],txtplot[i+1],\n di sk([x(tstart+i*dt),y(tstart+i*dt)],.05,color=navy)]):\nend:\nplotfcns: =plots[display](\n [seq(p[i],i=1..k+1)],\n inseq uence=true,\n scaling=constrained,\n labels=[`` ,``],\n view=[xleft..xright,ybot..ytop]):\nplotfcns;\n`Para meterize x & y Coordinates: `*'x'(t)=x(t),'y'(t)=y(t);\n`Position Fun ction (Curve): `*'r(t)'*` =`*['x'(t),'y'(t)]=r(t),t=tstart..tend;\n`V elocity Function: `*'`r '`(t)'*` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`Ac celeration Function: `*'`r ''`(t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`(t );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Parametric Curve Exercise 1.4" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in red area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1442 "restart:\nn:=4:\nx:=t->t+sin(2*t): # \+ enter here\ny:=t->t+sin(3*t): # enter here\ntstart:=-10: # en ter here\ntend:=10: # enter here\nm:=10: # ente r here\nxleft:=-m: \nxright:=m: \nybot:=-m: \nytop:=m: \nwith(linalg ):\nr:=t->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)(t)]:\n`r''`:=t->[(D@D)( x)(t),(D@D)(y)(t)]:\nwith(plottools):\nk:=100:\ndt:=abs(tstart-tend)/k :\nfor i from 0 to k do\nt[i+1]:=evalf(tstart+i*dt,4):\ntstr:=convert( t[i+1],name);\nTitle[i+1]:=`Parametric Curve Exercise 1.`||n:\ntxtplot [i+1]:=plots[textplot]([0,m-.05*m,\n ` t = ` ||tstr],\n font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{RIGHT\}):\np[0]:=plot([x(t),y(t ),t=tstart..tend],color=blue,\n titlefont=[HELVETICA,DEFAUL T,14],\n title=``||Title[i+1]):\np[i+1]:=plots[display]([p[ 0],txtplot[i+1],\n disk([x(tstart+i*dt),y(tstar t+i*dt)],.15,color=navy)]):\nend:\nplotfcns:=plots[display](\n \+ [seq(p[i],i=1..k+1)],\n insequence=true,\n \+ scaling=constrained,\n labels=[``,``],\n view=[x left..xright,ybot..ytop]):\nplotfcns;\n`Parameterize x & y Coordinates : `*'x'(t)=x(t),'y'(t)=y(t);\n`Position Function (Curve): `*'r(t)'*` =`*['x'(t),'y'(t)]=r(t),t=tstart..tend;\n`Velocity Function: `*'`r ' `(t)'*` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`Acceleration Function: `*'` r ''`(t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`(t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Par ametric Curve Exercise 1.5" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Cl ick in red area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1457 "restart:\nn:=5:\nx:=t->sin(t+sin(t)): # enter here\ny:=t->cos( t+cos(t)): # enter here\ntstart:=-10: # enter here\ntend:= 10: # enter here\nm:=2: # enter here\nxl eft:=-m: \nxright:=m: \nybot:=-m: \nytop:=m: \nwith(linalg):\nr:=t-> [x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)(t)]:\n`r''`:=t->[(D@D)(x)(t),(D@D )(y)(t)]:\nwith(plottools):\nk:=100:\ndt:=abs(tstart-tend)/k:\nfor i f rom 0 to k do\nt[i+1]:=evalf(tstart+i*dt,4):\ntstr:=convert(t[i+1],nam e);\nTitle[i+1]:=`Parametric Curve Exercise 1.`||n:\ntxtplot[i+1]:=plo ts[textplot]([0,m-.05*m,\n ` t = `||tstr],\n font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{RIGHT\}):\np[0]:=plot([x(t),y(t),t=tstart ..tend],color=blue,\n titlefont=[HELVETICA,DEFAULT,14],\n \+ title=``||Title[i+1]):\np[i+1]:=plots[display]([p[0],txtplot [i+1],\n disk([x(tstart+i*dt),y(tstart+i*dt)],. 05,color=navy)]):\nend:\nplotfcns:=plots[display](\n [seq(p [i],i=1..k+1)],\n insequence=true,\n scaling=co nstrained,\n labels=[``,``],\n view=[xleft..xrig ht,ybot..ytop]):\nplotfcns;\n`Parameterize x & y Coordinates: `*'x'(t )=x(t),'y'(t)=y(t);\n`Position Function (Curve): `*'r(t)'*` =`*['x'(t ),'y'(t)]=r(t),t=tstart..tend;\n`Velocity Function: `*'`r '`(t)'*` =` *[`x '`(t),`y '`(t)]=`r'`(t);\n`Acceleration Function: `*'`r ''`(t)'* ` =`*[`x ''`(t),`y ''`(t)]=`r''`(t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Parametric \+ Curve Exercise 1.6" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in r ed area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1477 "rest art:\nn:=6:\nx:=t->cos(t): # enter here\ny:=t->sin(t+sin(5* t)): # enter here\ntstart:=-10: # enter here\ntend:=10: \+ # enter here\nm:=2: # enter here\nxl eft:=-m: \nxright:=m: \nybot:=-m: \nytop:=m: \nwith(linalg):\nr:=t-> [x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)(t)]:\n`r''`:=t->[(D@D)(x)(t),(D@D )(y)(t)]:\nwith(plottools):\nk:=100:\ndt:=(abs(tstart-tend)/k)+.000001 :\nfor i from 0 to k do\nt[i+1]:=evalf(tstart+i*dt,4):\ntstr:=convert( t[i+1],name);\nTitle[i+1]:=`Parametric Curve Exercise 1.`||n:\ntxtplot [i+1]:=plots[textplot]([0,m-.05*m,\n ` t = ` ||tstr],\n font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{RIGHT\}):\np[0]:=plot([x(t),y(t ),t=tstart..tend],color=blue,\n titlefont=[HELVETICA,DEFAUL T,14],\n title=``||Title[i+1]):\np[i+1]:=plots[display]([p[ 0],txtplot[i+1],\n disk([x(tstart+i*dt),y(tstar t+i*dt)],.05,color=navy)]):\nend:\nplotfcns:=plots[display](\n \+ [seq(p[i],i=1..k+1)],\n insequence=true,\n \+ scaling=constrained,\n labels=[``,``],\n view=[x left..xright,ybot..ytop]):\nplotfcns;\n`Parameterize x & y Coordinates : `*'x'(t)=x(t),'y'(t)=y(t);\n`Position Function (Curve): `*'r(t)'*` =`*['x'(t),'y'(t)]=r(t),t=tstart..tend;\n`Velocity Function: `*'`r ' `(t)'*` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`Acceleration Function: `*'` r ''`(t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`(t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Par ametric Curve Exercise 1.7" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Cl ick in red area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1491 "restart:\nn:=7:\nwith(linalg):\nx:=t->t+2*sin(2*t): # enter here\ny:=t->t+2*cos(5*t): # enter here\ntstart:=-10: \+ # enter here\ntend:=10: # enter here\nm:=12: \+ # enter here\nxleft:=-m: \nxright:=m: \nybot:=-m: \nytop:=m: \nwith(linalg):\nr:=t->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)(t)]:\n`r ''`:=t->[(D@D)(x)(t),(D@D)(y)(t)]:\nwith(plottools):\nk:=100:\ndt:=abs (tstart-tend)/k + .000001:\nfor i from 0 to k do\nt[i+1]:=evalf(tstart +i*dt,4):\ntstr:=convert(t[i+1],name);\nTitle[i+1]:=`Parametric Curve \+ Exercise 1.`||n:\ntxtplot[i+1]:=plots[textplot]([0,m-.05*m,\n \+ ` t = `||tstr],\n font=[HELV ETICA,DEFAULT,12],\n color=blue,align=\{RIGHT \}):\np[0]:=plot([x(t),y(t),t=tstart..tend],color=blue,\n t itlefont=[HELVETICA,DEFAULT,14],\n title=``||Title[i+1]):\n p[i+1]:=plots[display]([p[0],txtplot[i+1],\n di sk([x(tstart+i*dt),y(tstart+i*dt)],.15,color=navy)]):\nend:\nplotfcns: =plots[display](\n [seq(p[i],i=1..k+1)],\n inseq uence=true,\n scaling=constrained,\n labels=[`` ,``],\n view=[xleft..xright,ybot..ytop]):\nplotfcns;\n`Para meterize x & y Coordinates: `*'x'(t)=x(t),'y'(t)=y(t);\n`Position Fun ction (Curve): `*'r(t)'*` =`*['x'(t),'y'(t)]=r(t),t=tstart..tend;\n`V elocity Function: `*'`r '`(t)'*` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`Ac celeration Function: `*'`r ''`(t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`(t );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Parametric Curve Exercise 1.8" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in red area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1506 "restart:\nn:=8:\nx:=t->cos(t)-cos(80*t) *sin(t): # enter here\ny:=t->2*sin(t)-sin(80*t): # enter here \ntstart:=-4: # enter here\ntend:=4: \+ # enter here\nm:=4: # enter her e\nxleft:=-m: \nxright:=m: \nybot:=-m: \nytop:=m: \nwith(linalg):\nr :=t->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)(t)]:\n`r''`:=t->[(D@D)(x)(t) ,(D@D)(y)(t)]:\nwith(plottools):\nk:=100:\ndt:=abs(tstart-tend)/k:\nfo r i from 0 to k do\nt[i+1]:=evalf(tstart+i*dt,4):\ntstr:=convert(t[i+1 ],name);\nTitle[i+1]:=`Parametric Curve Exercise 1.`||n:\ntxtplot[i+1] :=plots[textplot]([0,m-.05*m,\n ` t = `||tst r],\n font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{RIGHT\}):\np[0]:=plot([x(t),y(t),t=t start..tend],color=blue,\n titlefont=[HELVETICA,DEFAULT,14] ,\n title=``||Title[i+1]):\np[i+1]:=plots[display]([p[0],tx tplot[i+1],\n disk([x(tstart+i*dt),y(tstart+i*d t)],.1,color=navy)]):\nend:\nplotfcns:=plots[display](\n [s eq(p[i],i=1..k+1)],\n insequence=true,\n scalin g=constrained,\n labels=[``,``],\n view=[xleft.. xright,ybot..ytop]):\nplotfcns;\n`Parameterize x & y Coordinates: `*' x'(t)=x(t),'y'(t)=y(t);\n`Position Function (Curve): `*'r(t)'*` =`*[' x'(t),'y'(t)]=r(t),t=tstart..tend;\n`Velocity Function: `*'`r '`(t)'* ` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`Acceleration Function: `*'`r ''`( t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`(t);\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Parametri c Curve Exercise 1.9" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in red area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1431 "re start:\nn:=9:\nx:=t->t*cos(t): # enter here\ny:=t->t*sin(t): # ent er here\ntstart:=0: # enter here\ntend:=20: # enter her e\nm:=20: # enter here\nxleft:=-m: \nxright:=m: \nybot:=-m : \nytop:=m: \nwith(linalg):\nr:=t->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D (y)(t)]:\n`r''`:=t->[(D@D)(x)(t),(D@D)(y)(t)]:\nwith(plottools):\nk:=1 00:\ndt:=abs(tstart-tend)/k:\nfor i from 0 to k do\nt[i+1]:=evalf(tsta rt+i*dt,4):\ntstr:=convert(t[i+1],name);\nTitle[i+1]:=`Parametric Curv e Exercise 1.`||n:\ntxtplot[i+1]:=plots[textplot]([0,m-.05*m,\n \+ ` t = `||tstr],\n font=[HE LVETICA,DEFAULT,12],\n color=blue,align=\{RIGH T\}):\np[0]:=plot([x(t),y(t),t=tstart..tend],color=blue,\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=``||Title[i+1]): \np[i+1]:=plots[display]([p[0],txtplot[i+1],\n \+ disk([x(tstart+i*dt),y(tstart+i*dt)],.5,color=navy)]):\nend:\nplotfcns :=plots[display](\n [seq(p[i],i=1..k+1)],\n inse quence=true,\n scaling=constrained,\n labels=[` `,``],\n view=[xleft..xright,ybot..ytop]):\nplotfcns;\n`Par ameterize x & y Coordinates: `*'x'(t)=x(t),'y'(t)=y(t);\n`Position Fu nction (Curve): `*'r(t)'*` =`*['x'(t),'y'(t)]=r(t),t=tstart..tend;\n` Velocity Function: `*'`r '`(t)'*` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`A cceleration Function: `*'`r ''`(t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`( t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Parametric Curve Exercise 1.10" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in red area and Press [Enter]" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 1431 "restart:\nn:=10:\nx:=t->t*sin(t): # enter here\ny:=t->t*cos(t): # enter here\ntstart:=0: # e nter here\ntend:=20: # enter here\nm:=20: # enter h ere\nxleft:=-m: \nxright:=m: \nybot:=-m: \nytop:=m: \nwith(linalg): \nr:=t->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)(t)]:\n`r''`:=t->[(D@D)(x) (t),(D@D)(y)(t)]:\nwith(plottools):\nk:=100:\ndt:=abs(tstart-tend)/k: \nfor i from 0 to k do\nt[i+1]:=evalf(tstart+i*dt,4):\ntstr:=convert(t [i+1],name);\nTitle[i+1]:=`Parametric Curve Exercise 1.`||n:\ntxtplot[ i+1]:=plots[textplot]([0,m-.2*m,\n ` t = `|| tstr],\n font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{RIGHT\}):\np[0]:=plot([x(t),y(t), t=tstart..tend],color=blue,\n titlefont=[HELVETICA,DEFAULT, 14],\n title=``||Title[i+1]):\np[i+1]:=plots[display]([p[0] ,txtplot[i+1],\n disk([x(tstart+i*dt),y(tstart+ i*dt)],.5,color=navy)]):\nend:\nplotfcns:=plots[display](\n \+ [seq(p[i],i=1..k+1)],\n insequence=true,\n sca ling=constrained,\n labels=[``,``],\n view=[xlef t..xright,ybot..ytop]):\nplotfcns;\n`Parameterize x & y Coordinates: \+ `*'x'(t)=x(t),'y'(t)=y(t);\n`Position Function (Curve): `*'r(t)'*` =` *['x'(t),'y'(t)]=r(t),t=tstart..tend;\n`Velocity Function: `*'`r '`(t )'*` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`Acceleration Function: `*'`r ' '`(t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`(t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Parame tric Curve Exercise 1.11" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Clic k in red area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1503 "restart:\nn:=11:\nx:=t->exp(-t/10)*cos(t): # enter here\n y:=t->exp(-t/10)*sin(t): # enter here\ntstart:=0: \+ # enter here\ntend:=50: # enter here\nm: =1: # enter here\nxleft:=-m: \nxright:=m: \+ \nybot:=-m: \nytop:=m: \nwith(linalg):\nr:=t->[x(t),y(t)]:\n`r'`:=t-> [D(x)(t),D(y)(t)]:\n`r''`:=t->[(D@D)(x)(t),(D@D)(y)(t)]:\nwith(plottoo ls):\nk:=100:\ndt:=abs(tstart-tend)/k:\nfor i from 0 to k do\nt[i+1]:= evalf(tstart+i*dt,4):\ntstr:=convert(t[i+1],name);\nTitle[i+1]:=`Param etric Curve Exercise 1.`||n:\ntxtplot[i+1]:=plots[textplot]([0,m-.05*m ,\n ` t = `||tstr],\n \+ font=[HELVETICA,DEFAULT,12],\n color=blue,al ign=\{RIGHT\}):\np[0]:=plot([x(t),y(t),t=tstart..tend],color=blue,\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=``||Tit le[i+1]):\np[i+1]:=plots[display]([p[0],txtplot[i+1],\n \+ disk([x(tstart+i*dt),y(tstart+i*dt)],.02,color=navy)]):\nend: \nplotfcns:=plots[display](\n [seq(p[i],i=1..k+1)],\n \+ insequence=true,\n scaling=constrained,\n \+ labels=[``,``],\n view=[xleft..xright,ybot..ytop]):\nplotf cns;\n`Parameterize x & y Coordinates: `*'x'(t)=x(t),'y'(t)=y(t);\n`P osition Function (Curve): `*'r(t)'*` =`*['x'(t),'y'(t)]=r(t),t=tstart ..tend;\n`Velocity Function: `*'`r '`(t)'*` =`*[`x '`(t),`y '`(t)]=`r '`(t);\n`Acceleration Function: `*'`r ''`(t)'*` =`*[`x ''`(t),`y ''`( t)]=`r''`(t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Parametric Curve Exercise 1.12" } }{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in red area and Press [En ter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1502 "restart:\nn:=12:\nx:=t->e xp(-t/10)*sin(t): # enter here\ny:=t->exp(-t/10)*cos(t): \+ # enter here\ntstart:=0: # enter here\ntend:=50: # enter here\nm:=1: # enter here\nxleft:=-m: \nxright:=m: \nybot:=-m: \nytop:=m: \nwith(l inalg):\nr:=t->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)(t)]:\n`r''`:=t->[( D@D)(x)(t),(D@D)(y)(t)]:\nwith(plottools):\nk:=100:\ndt:=abs(tstart-te nd)/k:\nfor i from 0 to k do\nt[i+1]:=evalf(tstart+i*dt,4):\ntstr:=con vert(t[i+1],name);\nTitle[i+1]:=`Parametric Curve Exercise 1.`||n:\ntx tplot[i+1]:=plots[textplot]([0,m-.2*m,\n ` t = `||tstr],\n font=[HELVETICA,DEFAULT,12],\n \+ color=blue,align=\{RIGHT\}):\np[0]:=plot([x(t) ,y(t),t=tstart..tend],color=blue,\n titlefont=[HELVETICA,DE FAULT,14],\n title=``||Title[i+1]):\np[i+1]:=plots[display] ([p[0],txtplot[i+1],\n disk([x(tstart+i*dt),y(t start+i*dt)],.02,color=navy)]):\nend:\nplotfcns:=plots[display](\n \+ [seq(p[i],i=1..k+1)],\n insequence=true,\n \+ scaling=constrained,\n labels=[``,``],\n vie w=[xleft..xright,ybot..ytop]):\nplotfcns;\n`Parameterize x & y Coordin ates: `*'x'(t)=x(t),'y'(t)=y(t);\n`Position Function (Curve): `*'r(t )'*` =`*['x'(t),'y'(t)]=r(t),t=tstart..tend;\n`Velocity Function: `*' `r '`(t)'*` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`Acceleration Function: \+ `*'`r ''`(t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`(t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "My Parametric Curve 1" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click \+ in red area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1416 " restart:\nn:=1:\nx:=t->t^3-2*t: # enter here\ny:=t->t^2-t: # ent er here\ntstart:=-3: # enter here\ntend:=3: # enter here \nm:=4: # enter here\nxleft:=-m: \nxright:=m:\nybot:=-m: \n ytop:=m: \nwith(linalg):\nr:=t->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)( t)]:\n`r''`:=t->[(D@D)(x)(t),(D@D)(y)(t)]:\nwith(plottools):\nk:=100: \ndt:=abs(tstart-tend)/k:\nfor i from 0 to k do\nt[i+1]:=evalf(tstart+ i*dt,4):\ntstr:=convert(t[i+1],name);\nTitle[i+1]:=`My Parametric Curv e `||n:\ntxtplot[i+1]:=plots[textplot]([0,m-.05*m,\n \+ ` t = `||tstr],\n font=[HELVETICA,DEF AULT,12],\n color=blue,align=\{RIGHT\}):\np[0] :=plot([x(t),y(t),t=tstart..tend],color=blue,\n titlefont=[ HELVETICA,DEFAULT,14],\n title=``||Title[i+1]):\np[i+1]:=pl ots[display]([p[0],txtplot[i+1],\n disk([x(tsta rt+i*dt),y(tstart+i*dt)],.1,color=navy)]):\nend:\nplotfcns:=plots[disp lay](\n [seq(p[i],i=1..k+1)],\n insequence=true, \n scaling=constrained,\n labels=[``,``],\n \+ view=[xleft..xright,ybot..ytop]):\nplotfcns;\n`Parameterize x \+ & y Coordinates: `*'x'(t)=x(t),'y'(t)=y(t);\n`Position Function (Curv e): `*'r(t)'*` =`*['x'(t),'y'(t)]=r(t),t=tstart..tend;\n`Velocity Fun ction: `*'`r '`(t)'*` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`Acceleration \+ Function: `*'`r ''`(t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`(t);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 21 "My Parametric Curve 2" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 35 "Click in red area and Press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1407 "restart:\nn:=2:\nx:=t->t^3-1: # enter here\ny:=t- >-t^2+2: # enter here\ntstart:=-3: # enter here\ntend:=3: # \+ enter here\nm:=4: # enter here\nxleft:=-m: \nxright:=m: \nyb ot:=-m: \nytop:=m: \nwith(linalg):\nr:=t->[x(t),y(t)]:\n`r'`:=t->[D(x )(t),D(y)(t)]:\n`r''`:=t->[(D@D)(x)(t),(D@D)(y)(t)]:\nwith(plottools): \nk:=100:\ndt:=abs(tstart-tend)/k:\nfor i from 0 to k do\nt[i+1]:=eval f(tstart+i*dt,4):\ntstr:=convert(t[i+1],name);\nTitle[i+1]:=`My Parame tric Curve `||n:\ntxtplot[i+1]:=plots[textplot]([0,m-.05*m,\n \+ ` t = `||tstr],\n font=[HELV ETICA,DEFAULT,12],\n color=blue,align=\{RIGHT \}):\np[0]:=plot([x(t),y(t),t=tstart..tend],color=blue,\n t itlefont=[HELVETICA,DEFAULT,14],\n title=``||Title[i+1]):\n p[i+1]:=plots[display]([p[0],txtplot[i+1],\n di sk([x(tstart+i*dt),y(tstart+i*dt)],.1,color=navy)]):\nend:\nplotfcns:= plots[display](\n [seq(p[i],i=1..k+1)],\n insequ ence=true,\n scaling=constrained,\n labels=[``, ``],\n view=[xleft..xright,ybot..ytop]):\nplotfcns;\n`Param eterize x & y Coordinates: `*'x'(t)=x(t),'y'(t)=y(t);\n`Position Func tion (Curve): `*'r(t)'*` =`*['x'(t),'y'(t)]=r(t),t=tstart..tend;\n`Ve locity Function: `*'`r '`(t)'*` =`*[`x '`(t),`y '`(t)]=`r'`(t);\n`Acc eleration Function: `*'`r ''`(t)'*` =`*[`x ''`(t),`y ''`(t)]=`r''`(t) ;" }}}{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 28 "Filename: ExploreCalc12A.mws" }}{PARA 0 "" 0 "" {TEXT -1 36 "Co pyright 2005, All Rights Reserved." }}{PARA 0 "" 0 "" {TEXT -1 44 "Per mission 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 Departme nt-MICDS" }}{PARA 0 "" 0 "" {TEXT -1 45 "E-mail: pais@micds.org or pai s@kinetigram.com" }}{PARA 0 "" 0 "" {TEXT -1 33 "URL: http://kinetigr am.com/micds" }}{PARA 0 "" 0 "" {TEXT -1 37 "_________________________ ____________" }}}{MARK "23 0" 24 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }