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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 258 
"" 0 "" {TEXT -1 25 "Lesson 27:  Polar Graphs\n" }}}{PARA 0 "" 0 "" 
{TEXT -1 83 "_________________________________________________________
__________________________" }}{PARA 4 "" 0 "" {TEXT -1 23 "A. Cardioid
s & Limacons" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________
__________________________________________________________" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "We're going to lo
ok at a variety of cardioids, which are graph of the form" }}{PARA 0 "
" 0 "" {TEXT -1 30 "                     y = a +- " }{XPPEDIT 18 0 "b*
 sin(theta)" "6#*&%\"bG\"\"\"-%$sinG6#%&thetaGF%" }{TEXT -1 11 " or y=
a +- " }{XPPEDIT 18 0 "b* cos(theta)" "6#*&%\"bG\"\"\"-%$cosG6#%&theta
GF%" }}{PARA 0 "" 0 "" {TEXT -1 97 "and see how the relationship among
 the components effects the graph.\n\n          COMPARING a AND b" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "In parti
cular, there are three cases : |a| = |b|. |a| > |b|, and |a| < |b|. Ea
ch of these cases creates a distinctive version of the limacon." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "When |a| \+
= |b|, the graph passes through the origin." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "This shape is known a cardioid
, or heart shaped curve. Note the reference circles of radius 1 and 2.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning,
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,CONSTRAINEDG-%%VIEWG6$%(DEFAULTGFeco" 1 2 0 1 10 0 2 9 1 4 1 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "To
 see all of these varieties in one glance, execute the next block of c
ommands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
100 "This shape is known a cardioid, or heart shaped curve. Note the r
eference circles of radius 1 and 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "display( polarplot( 8 + 8*
cos(theta) , theta = 0..2*Pi, scaling = constrained, color = green, th
ickness = 3), polarplot(\{8 + a*cos(theta) $ a = 9..15\}, theta = 0..2
*Pi, color = blue), polarplot(\{ 8 + a*cos(theta) $ a = 1..7\}, theta \+
= 0..2*Pi, color = red));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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\"z0Q]r]\"FfvFg^x-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$%(DEFAULTGFhf^l
" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "C
urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C
urve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve \+
15" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 23 
"CHOICE OF TRIG FUNCTION" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 168 "There are four variations iin the format : sine, \+
cosine, -sine, and -cosine. How does the choice of one of these effect
 the graph? Lets take a look at all four at once!" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Can you decide which gra
ph belongs to which? Think about what values of theta make the sine an
d cosine maxima!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 124 "polarplot(\{ 8 + 7*sin(theta), 8 + 7*cos(thet
a), 8 - 7*sin(theta), 8 -7*cos(theta)\}, theta = 0..2*Pi, scaling = co
nstrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-
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FAULTGF^[z" 1 2 0 1 10 0 2 6 1 4 1 1.000000 45.000000 45.000000 0 0 "C
urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________________
_________________________________________________" }}{PARA 4 "" 0 "" 
{TEXT -1 18 "B. The Rose Garden" }}{PARA 0 "" 0 "" {TEXT -1 83 "______
______________________________________________________________________
_______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 127 "We're going to look polar functions
 of the form f = a sin(n ) and r = a cos(n ) which are sometimes calle
d multi-petaled roses." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 26 "EVEN AND ODD NUMBER PETALS" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The first distinction to be mad
e is between when n is an even or odd number." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "When n is an odd number, \+
the resulting rose has exactly n petals" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polarplot( \{9, 9*sin(5
* theta)\}, theta = 0..2*Pi, scaling = constrained);" }}{PARA 13 "" 1 
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rve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 122 "Try creating some other roses on your own with different
 numbers of petals to verify that the even/odd relationship holds." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "What abou
t a single-petaled rose?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 63 "Do you recognize the inner shaped of the \"single \+
petaled rose\"?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 70 "polarplot( \{9, 9*sin(theta)\}, theta = 0..2*P
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