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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 11 "Maple Lab 7" }}{PARA 0 "
" 0 "" {TEXT -1 136 "Today we will have fun with parametric and polar \+
equations.  Here's how we plot polar curves in Maple.  First load up t
he plots package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(p
lots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "To plot the polar equat
ion r = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 7 " sin (3" }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 ") as " }{XPPEDIT 18 
0 "theta;" "6#%&thetaG" }{TEXT -1 17 " varies from -10 " }{XPPEDIT 18 
0 "Pi;" "6#%#PiG" }{TEXT -1 8 "  to 10 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG
" }{TEXT -1 8 " , type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "
polarplot(t*sin(3*t),t=-10*Pi..10*Pi);" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 30 "In Maple, we use t instead of " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 136 " .  Also, as a remi
nder on how to graph parametric equations: to plot x = t + 2 sin (2t) \+
, y = t + 2 cos (5t) from t = -10 to 10,  type:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "plot([t+2*sin(2*t),t+2*cos(5*t),t=-10..10]);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Welp, that's it for the intro.
  Problems are a'comin'.  Note that problems 1 and 3 deal with PARAMET
RIC equations, while problem 2 deals with POLAR equations.  " }{TEXT 
266 40 "PLEASE SAVE YOUR WORKSHEET PERIODICALLY." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "1) Consider the fo
llowing parametric equations: x = sin(t) , y = sin(t+sin(t)). " }
{TEXT 267 65 " Find the equation of the tangent line to this curve whe
n t = 0. " }{TEXT -1 34 "Do all relevant commands on Maple!" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "2) a) " }{TEXT 
260 24 "Graph the polar equation" }{TEXT -1 5 " r = " }{XPPEDIT 18 0 "
theta;" "6#%&thetaG" }{TEXT -1 2 "  " }{TEXT 261 3 "as " }{XPPEDIT 18 
0 "theta;" "6#%&thetaG" }{TEXT 265 21 " ranges from  0 to 30" }{TEXT 
-1 3 ".  " }{TEXT 262 62 "Briefly and concisely explain why the plot a
ppears as it does." }}{PARA 0 "" 0 "" {TEXT -1 81 "    b) Investigate \+
the family of curves defined by the polar equation  r = sin(n " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 35 "), where n is a posi
tive integer.  " }{TEXT 263 57 "How does it seem that the number of lo
ops in the graph of" }{TEXT -1 12 "  r = sin(n " }{XPPEDIT 18 0 "theta
;" "6#%&thetaG" }{TEXT -1 3 ")  " }{TEXT 264 30 "is related to the val
ue of n? " }{TEXT -1 394 " (Hint: to try and see the relationship, jus
t try graphing it for some values of n and notice a pattern.  I DO NOT
 WANT ANY OF THESE GRAPHS PRINTED OUT TO BE HANDED IN!!!  NONE!! DO YO
U HEAR ME!!!  If you want, you could start a new worksheet just to foo
l around with the graphs.  At any rate, once you see what the pattern \+
is, just explain it to me.  You don't have to tell me WHY it happens.)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "3) Th
e following parametric equations define a curve known as Cormu's spira
l:" }}{PARA 257 "" 0 "" {TEXT -1 11 "x = C(t) = " }{XPPEDIT 18 0 "int(
cos(Pi*u^2/2),u = 0 .. t);" "6#-%$intG6$-%$cosG6#*(%#PiG\"\"\"*$%\"uG
\"\"#F+\"\"#!\"\"/F-;\"\"!%\"tG" }{TEXT -1 21 "    ,     y = S(t) = " 
}{XPPEDIT 18 0 "int(sin(Pi*u^2/2),u = 0 .. t);" "6#-%$intG6$-%$sinG6#*
(%#PiG\"\"\"*$%\"uG\"\"#F+\"\"#!\"\"/F-;\"\"!%\"tG" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 257 85 "Define the functions C(t) and S(t) as abo
ve.  Then graph this curve from t = -5 to 5." }{TEXT -1 60 "  It seems
 it might be converging to two different points.  " }{TEXT 258 23 "To \+
check this out, find" }{TEXT -1 3 "   " }{XPPEDIT 18 0 "limit(C(t),t =
 infinity);" "6#-%&limitG6$-%\"CG6#%\"tG/F)%)infinityG" }{TEXT -1 4 " \+
   " }{TEXT 256 3 "and" }{TEXT -1 5 "     " }{XPPEDIT 18 0 "limit(S(t)
,t = infinity);" "6#-%&limitG6$-%\"SG6#%\"tG/F)%)infinityG" }{TEXT -1 
93 "  .  So we see that indeed as t gets larger, the curve gets closer
 and closer to the point ( " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"\"\"\"\"
\"#!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"\"\"\"\"
\"#!\"\"" }{TEXT -1 99 " ) (upper right).  In the same fashion, as t g
ets more negative the curve approaches the point ( - " }{XPPEDIT 18 0 
"1/2;" "6#*&\"\"\"\"\"\"\"\"#!\"\"" }{TEXT -1 6 " , -  " }{XPPEDIT 18 
0 "1/2;" "6#*&\"\"\"\"\"\"\"\"#!\"\"" }{TEXT -1 152 " ) (lower left). \+
 Let a be some positive number.  From the text, we know that the equat
ion for the length of this curve from t = 0 to t = a is given by:" }}
{PARA 258 "" 0 "" {XPPEDIT 18 0 "int(sqrt(diff(C(t),t)^2+diff(S(t),t)^
2),t = 0 .. a);" "6#-%$intG6$-%%sqrtG6#,&*$-%%diffG6$-%\"CG6#%\"tGF1\"
\"#\"\"\"*$-F,6$-%\"SG6#F1F1\"\"#F3/F1;\"\"!%\"aG" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 259 40 "Now evaluate the above formula in Maple \+
" }{TEXT -1 112 "(depending on how you choose to do this, you may need
 to use a \"simplify\" command to get a nice looking answer)." }{TEXT 
268 1 " " }{TEXT -1 313 " You should find that your answer is in fact \+
a!  This means that the length of the curve from t = 0 to t = a is a. \+
 For example, the length of the curve from t = 0 to t = 8 billion is e
qual to 8 billion.  So even though the curve is bounded and converging
 to a point, it is \"infinitely long\" (in both directions)!" }}}}
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