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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT -1 19 "An Introduction to " }
{TEXT 256 5 "Maple" }{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 71 "Grap
hing Implicit Functions, Parametric Equations, and Polar Equations\n" 
}{TEXT 274 44 "by Bob Bradshaw, Ohlone College, Fremont, CA" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "\nStart by loading the plotting su
broutines" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plots);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 
"" {TEXT 288 18 "Implicit Functions" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
289 0 "" }{TEXT -1 145 "An implicit function is one in which you do no
t solve the equation for y. The implicitplot command will allow you to
 graph this type of equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "implicitplot(x^2+y^2=1,x=-2..2,y=-2..2,thickness=3);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 262 "\nTo make the ellipse look like a circle
, I want to set the scaling option so that scaling = constrained. Sinc
e I want this to be true for all the graphs that follow, I set the opt
ion in a separate statement now, rather than retypring the command wit
h every plot." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "setoptions(scaling
=constrained);\nimplicitplot(x^2+y^2=1,x=-2..2,y=-2..2,thickness=3);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "\nUnfortunely, implicitplot can
 give an ugly graph." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "implicitplo
t(x^3*y+x^2*y^3=4,x=-10..10,y=-10..10);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 100 "\nIncreasing the number of points helps but the graph is
 incorrect since this equation is a function!" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "implicitplot(x^3*y+x^2*y^3=4,x=-10..10,y=-10..10,nump
oints=1000);\n" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 290 20 "Parametric E
quations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 106 "Parametric equations describe the horizontal and vertica
l positions of a point on a curve by writing both " }{TEXT 258 1 "x" }
{TEXT -1 5 " and " }{TEXT 259 1 "y" }{TEXT -1 30 " in terms of a third
 variable " }{TEXT 260 1 "t" }{TEXT -1 122 ", which often represents t
ime. This material is usually covered in Calculus II. To graph paramet
ric equations, define the " }{TEXT 261 1 "x" }{TEXT -1 66 " and y valu
es and then use the plot command with the format plot([" }{TEXT 262 
16 "x,y,t=start..end" }{TEXT -1 87 "]). You can use any of the options
 that are used with the graphs of standard functions." }{MPLTEXT 1 0 
1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x:=cos(t);\ny:=sin
(t);\nplot([x,y,t=0..2*Pi]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "horiz:=t-2*sin(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vert:
=1-2*cos(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot([horiz,vert,t=
0..8*Pi]);\n" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 291 17 "Polar Coordina
tes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 134 "A polar coordinate system plots coordinates based on the dista
nce of a point from the origin (the radius) and the angle formed by th
e " }{TEXT 257 1 "x" }{TEXT -1 54 " -axis and the line between the ori
gin and the point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 104 "There are three different commands for plotting in pol
ar coordinates. I suggest that you use polarplot([" }{TEXT 263 31 "rad
ius, angle, angle=start..end" }{TEXT -1 24 "], scaling=constrained)." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "To grap
h the curve " }{TEXT 275 1 "r" }{TEXT -1 9 " = cos (2" }{TEXT 276 1 "q
" }{TEXT -1 7 ") , use" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "polarplot
([cos(2*theta),theta,theta=0..2*Pi]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "A second method is to use
 the standard plot command but to forcce " }{TEXT 277 5 "Maple" }
{TEXT -1 26 " to use polar coordinates." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 38 "plot(cos(2*t),t=0..4*Pi,coords=polar);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 245 "The disadvantage of usinghe polarplot or standard p
lot commands is thatmost standard polar functions are centered around \+
the origin. It requires some messy algebra to move the above graph so \+
that it is centered around another point, say (2, 3).\n" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "\nTo avoid this dif
ficulty, we can use parametric equations. The relationship between a p
olar equation " }{TEXT 278 1 "r" }{TEXT -1 34 " and parametric equatio
ns is that " }{TEXT 279 1 "x" }{TEXT -1 3 " = " }{TEXT 280 1 "r" }
{TEXT -1 5 " cos(" }{TEXT 284 1 "q" }{TEXT -1 6 ") and " }{TEXT 281 1 
"y" }{TEXT -1 3 " = " }{TEXT 282 1 "r" }{TEXT -1 5 " sin(" }{TEXT 283 
1 "q" }{TEXT -1 80 "). Consider the following. Noptice that the graph \+
is now centered around (2, 3)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 88 "r1:=cos(2*theta);\nx1:=2+r1*cos(theta);\ny1:=3+r1*sin(theta);
\nplot([x1,y1,theta=0..2*Pi]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 
"We can now use this method to combine multpile polar plots." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 612 "r1:=2*cos(2*theta);\nx1:=-2
+r1*cos(theta);\ny1:=3+r1*sin(theta);\nfirstcurve:=plot([x1,y1,theta=0
..2*Pi],color=orange,thickness=3):\n\nr2:=2*cos(3*theta);\nx2:=2+r2*co
s(theta);\ny2:=3+r2*sin(theta);\nsecondcurve:=plot([x2,y2,theta=0..2*P
i],color=green,thickness=4):\n\nr3:=2*cos(6*theta);\nx3:=-2+r3*cos(the
ta);\ny3:=-3+r3*sin(theta);\nthirdcurve:=plot([x3,y3,theta=0..2*Pi],co
lor=blue,thickness=1):\n\nr4:=1.5*cos(5*theta);\nx4:=2+r4*cos(theta);
\ny4:=-3+r4*sin(theta);\nfourthcurve:=plot([x4,y4,theta=0..2*Pi],color
=magenta,thickness=2):\n\ndisplay([firstcurve,secondcurve,thirdcurve,f
ourthcurve],scaling=constrained,axes=boxed);\n\n" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 29 "\nOur look at polar graphs in " }{TEXT 264 5 "Mapl
e" }{TEXT -1 54 " would be incomplete without looking at the following
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "S := 100/(100+(t-Pi/2)^8): \n
R := S*(2-sin(7*t)-cos(30*t)/2):\nplot([R,t,t=-Pi/2..3/2*Pi],coords=po
lar,numpoints=2000,axes=NONE);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "You Try It!" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "(1)  Graph the set of parametric e
quations " }}{PARA 0 "" 0 "" {TEXT 265 1 "x" }{TEXT -1 8 " = 5cos(" }
{TEXT 266 1 "t" }{TEXT -1 12 ") + 14cos(15" }{TEXT 267 1 "t" }{TEXT 
-1 12 "), y = 5sin(" }{TEXT 271 1 "t" }{TEXT -1 12 ") + 14sin(15" }
{TEXT 272 1 "t" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 30 "(2)  Graph the polar function " }{TEXT 
268 1 "r" }{TEXT -1 3 " = " }{TEXT 269 1 "t" }{TEXT -1 11 " where 0 < \+
" }{TEXT 270 1 "t" }{TEXT -1 7 " < 6*Pi" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 33 "(3)  Graph the implicit function " }
{XPPEDIT 18 0 "abs(x)^(2/3)+abs(y)^(2/3) = 1;" "6#/,&)-%$absG6#%\"xG*&
\"\"#\"\"\"\"\"$!\"\"F,)-F'6#%\"yG*&F+F,F-F.F,F," }{TEXT -1 22 " on th
e interval -1 < " }{TEXT 285 1 "x" }{TEXT -1 11 " < 1, -1 < " }{TEXT 
286 1 "y" }{TEXT -1 0 "" }{TEXT 287 1 " " }{TEXT -1 64 "< 1. Use 1000 \+
points and abs( ) for the absolute value function." }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 96 "(4)  Create a single graph containing a plot of an implic
it, a parametric, and a polar equation." }}}}}{MARK "6" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }