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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 25 "Graphing Polar Equations \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 214 "The command polarplo
t can be used to graph an equation in polar coordinates.  The animatec
urve command can be used to visualize the manner in which the graph is
 drawn.  These commands can only be used after typing " }{TEXT 257 11 
"with(plots)" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with
(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "r:=2*sin(t):" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "polarplot(r,t=0..Pi,scaling=constr
ained,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "ani
matecurve([r,t,t=0..Pi],coords=polar,thickness=2,frames=50,scaling=con
strained);#Click on the plot and use the toolbar buttons to animate." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 
0 "" {TEXT 258 5 "Plot " }{XPPEDIT 257 0 "r = 2-2*sin(2*theta);" "6#/%
\"rG,&\"\"#\"\"\"*&F&F'-%$sinG6#*&F&F'%&thetaGF'F'!\"\"" }{TEXT 259 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=2-2*sin(2*t):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "polarplot(f,t=0..2*Pi,scalin
g=constrained,thickness=2);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "animatecurve([f,t,t=0..2*Pi]
,coords=polar,thickness=2,frames=50,scaling=constrained);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 55 "The following set of commands creates a g
raph called a " }{TEXT 260 9 "cardiod  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "g:=1+sin(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "polarplot(g,t=0..2*Pi,scaling=constrained,thickness=2);" }}
{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 85 "animatecurve([g,t,t=0..2*Pi],coords=polar,thickness=2,frames=50,
scaling=constrained);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 54 "The following set of commands creates a g
raph called a" }{TEXT 261 17 " four-leaved rose" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "r:=cos(2*t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
66 "polarplot(r,t=0..2*Pi,scaling=constrained,thickness=2,color=blue);
" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 85 "animatecurve([r,t,t=0..2*Pi],coords=polar,thickness=2
,frames=50,scaling=constrained);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The followi
ng family of graphs based on r = 1 + c sin(t) are called " }{TEXT 262 
9 "limacons " }{TEXT -1 120 "after a French word for snail.  Notice th
e shape of the graphs.  Feel free to try other values of c to see what
 happens." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(p
lots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "a:=1+3*sin(t):pol
arplot(a,t=0..2*Pi,scaling=constrained,thickness=2,color=blue);" }}
{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 81 "b:=1+2*sin(t):polarplot(b,t=0..2*Pi,scaling=constrained,thicknes
s=2,color=green);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 83 "c:=1+.5*sin(t):polarplot(c,t=0..2*Pi,scalin
g=constrained,thickness=2,color=orange);" }}{PARA 13 "" 1 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "d:=1-.5*sin(t):polar
plot(d,t=0..2*Pi,scaling=constrained,thickness=2,color=red);" }}{PARA 
13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "
e:=1-2*sin(t):polarplot(e,t=0..2*Pi,scaling=constrained,thickness=2,co
lor=aquamarine);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 82 "f:=1-3*sin(t):polarplot(f,t=0..2*Pi,scaling=
constrained,thickness=2,color=violet);" }}{PARA 13 "" 1 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 26 "A lemniscate has the form " }{XPPEDIT 18 0 "r^2 = a^
2*cos(2*t);" "6#/*$)%\"rG\"\"#\"\"\"*&)%\"aGF'F(-%$cosG6#*&F'F(%\"tGF(
F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Maple has a tough time wit
h this plot, but the following will give you an idea of what hte graph
 looks like." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:wit
h(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p:=(cos(2*t))^
(1/2):q:=-(cos(2*t))^(1/2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "polarplot(\{p,q\},t=0..20*Pi,thickness=2,color=aquamarine);" }}
{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Here are a few other plo
ts" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "polarplot(sin(9*t/5),
t=0..10*Pi,scaling=constrained, thickness=2);" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "polarplot(s
in(t)+(sin(5*t/2))^3,t=0..4*Pi,scaling=constrained, thickness=2);" }}
{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 76 "polarplot((sin(4*t))^2+cos(4*t),t=0..2*Pi,scaling=constrained, t
hickness=2);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 66 "polarplot((5/sqrt(t)),t=0..6*Pi,scaling=constrai
ned, thickness=2);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 70 "polarplot(1/(16+4*cos(t)),t=0..2*Pi,scali
ng=constrained, thickness=2);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "45" 0 }
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