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{SECT 0 {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 259 0 "" }{TEXT 260 18 
"EXPLORING LIMACONS" }{TEXT 257 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }{TEXT 264 19 "Calculus
 II Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT 
261 1 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 0 "" }
{TEXT 266 11 "Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 29 "a) Plot curv
es in polar form." }}{PARA 0 "" 0 "" {TEXT -1 66 "b) Explore the effec
ts of a parameter on the shape of the Limacon." }}{PARA 0 "" 0 "" 
{TEXT -1 34 "c) Construct a series of limacons." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "Using Maple allows the s
tudent to quickly and efficiently vary the parameter for a family of L
imacons and visualize the results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "
" }{TEXT 267 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 19 " a) P
lot the curve " }{XPPEDIT 18 0 "r = cos(n*theta);" "6#/%\"rG-%$cosG6#*
&%\"nG\"\"\"%&thetaGF*" }{TEXT -1 23 " for n from 1 to 4 and " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 29 " from 0 to 2 Pi, usi
ng a loop" }}{PARA 0 "" 0 "" {TEXT -1 91 " b) Explain, using complete \+
sentences, the relationship between the value of the parameter " }
{TEXT 291 1 "n" }{TEXT -1 50 " and the number of leaves in the       c
urve cos(n" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 2 ")." }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 68 "To plot
 a curve in polar coordinates use the following Maple syntax:" }}
{PARA 0 "" 0 "" {TEXT -1 90 "plot( [r(t), t, t = a..b], x = c..d, y = \+
e..f,  coords = polar, scaling=constrained);     " }}{PARA 0 "" 0 "" 
{TEXT -1 83 "Note: constrained scaling means the scaling on the x- and
 y-axis are in a 1:1 ratio" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT 257 1 "T" }{TEXT -1 3 "o p" }{TEXT 258 70 "lot a set
 of polar curves using a loop use the following Maple syntax:" }}
{PARA 0 "" 0 "" {TEXT -1 25 "for n from 1 to k by 1 do" }}{PARA 0 "" 
0 "" {TEXT -1 85 "plot( [r(t), t, t = a..b], x = c..d, y = e..f,  coor
ds = polar, scaling=constrained);" }}{PARA 0 "" 0 "" {TEXT -1 3 "od;" 
}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 268 0 "" }{TEXT 269 0 "" }
{TEXT 270 0 "" }{TEXT -1 1 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "
" }{TEXT 271 0 "" }{TEXT 272 0 "" }{TEXT 273 0 "" }{TEXT 274 9 "Soluti
on:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "for n from 2 by 1 to
 8 do\nplot([cos(n*theta), theta, theta=0..2*Pi],x=-2..2,y=-2..2,coord
s=polar, scaling=constrained)\nod;\n" }}{PARA 13 "" 1 "" {GLPLOT2D 
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G\"e\\$z0%p%F=7$$\"3PoBe'fXRi*F/$!3)H6UOTQfG$F=7$$\"3OM!zsm[yy*F/$!3m1
W)eP**f]#F=7$$\"3Kvl1ZI^0**F/$!39)o**y!\\a!p\"F=7$$\"3#*Hwnm@\"o%**F/$
!3[-G[%)39t7F=7$$\"3w&GS6g[j(**F/$!35SeP!z]D^)F27$$\"3e*3piH&3%***F/$!
3]11J,XwjUF27$F($\"36YKhSr8/#)!#F-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%(SC
ALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!\"#F*$\"\"#F*Fcat" 1 2 0 1 10 0 2 
9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "_______________________
___________________________________________" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 275 0 "" }{TEXT 276 
10 "ASSIGNMENT" }{TEXT 277 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 278 0 "" 
}{TEXT 279 0 "" }{TEXT 280 10 "Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 281 43 "a) Using a loop, plot the set of limacons  " }
{XPPEDIT 293 0 "r = 1+k*sin(theta);" "6#/%\"rG,&\"\"\"F&*&%\"kGF&-%$si
nG6#%&thetaGF&F&" }{TEXT 292 31 "   for k from 0.2  by 0.2  to 2" }}
{PARA 0 "" 0 "" {TEXT -1 77 "b) Decribe in words what is happening to \+
the curve as the value of k changes." }}{PARA 0 "" 0 "" {TEXT -1 62 "c
) For what value of k is the curve a cardioid (heart-shaped)?" }}
{PARA 0 "" 0 "" {TEXT -1 62 "d) For what values of 1/k does the limaco
n have an inner loop?" }}{PARA 0 "" 0 "" {TEXT -1 62 "e) For what valu
es of 1/k does the limacon have no inner loop?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 0 "" }{TEXT 283 0 "" }{TEXT 284 10 "Problem 2:" }{TEXT 
285 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The general polar form of a li
macon is   " }{XPPEDIT 18 0 "r = a+b*sin(theta);" "6#/%\"rG,&%\"aG\"\"
\"*&%\"bGF'-%$sinG6#%&thetaGF'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 28 "a) Find a set of values for " }{TEXT 286 1 "a" }{TEXT -1 
5 " and " }{TEXT 287 2 "b " }{TEXT -1 36 "such that the Limacon is a c
ardioid." }}{PARA 0 "" 0 "" {TEXT -1 28 "b) Find a set of values for \+
" }{TEXT 288 1 "a" }{TEXT -1 5 " and " }{TEXT 289 2 "b " }{TEXT -1 40 
"such that the Limacon has no inner loop." }}{PARA 0 "" 0 "" {TEXT -1 
28 "c) Find a set of values for " }{TEXT 256 1 "a" }{TEXT 294 1 " " }
{TEXT -1 4 "and " }{TEXT 257 1 "b" }{TEXT 290 1 " " }{TEXT -1 40 "such
 that the Limacon has an inner loop." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "______
____________________________________________________________" }}{PARA 
0 "" 0 "" {TEXT -1 94 "MSIP Grant #P120A80089-98: \"Three Urban Calcul
us Reform Programs: Adopting the Best\" 1998-2001" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{MARK "8 1 7 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }