{VERSION 3 0 "IBM INTEL NT" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 255 255 0 0 0 0 2 0 0 0 
0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "Highlight" -1 256 "" 0 0 0 255 0 1 0 1 0 0 0 0 0 0 0 }
{CSTYLE "Define" -1 257 "Times" 1 12 0 0 0 1 1 1 2 0 0 2 0 0 0 }
{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 
0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 
0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 
0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 
0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 
0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 
0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "
" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 116 114 97 0 0 1 0 0 0 0 0 0 0 }3 
0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 256 "" 0 "" {TEXT -1 17 "Parametric Curves" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 50 "We begin with a simple plot of a parametric curve." }}
{PARA 0 "" 0 "" {TEXT -1 28 "Its parametric equations are" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 73 "x:=t->cos(t)+cos(7*t)/2+sin(17*t)/3;\ny:=t->
sin(t)+sin(7*t)/2+cos(17*t)/3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 
"To plot a parametric curve we can use the " }{TEXT 261 4 "plot" }
{TEXT -1 20 " command as before. " }}{PARA 0 "" 0 "" {TEXT -1 84 "Howe
ver, the syntax is slightly different from the one used for plotting f
unctions. " }}{PARA 0 "" 0 "" {TEXT -1 85 "We now need to specify the \+
coordinate functions along with a range for the parameter " }{XPPEDIT 
18 0 "t" "6#%\"tG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "(whi
ch we take as " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiG
F'" }{TEXT -1 51 " in this example) all enclosed in square brackets. \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot([x(t), y(t), t=0..2*Pi]);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Next we illustrate the genera
tion of a " }{TEXT 262 7 "cycloid" }{TEXT -1 136 ". \nFrom class we kn
ow that a cycloid is obtained by tracking the position of a fixed poin
t on the circumference of a circle with radius " }{XPPEDIT 18 0 "r" "6
#%\"rG" }{TEXT -1 32 " rolling along a straight line. " }}{PARA 0 "" 
0 "" {TEXT -1 28 "The parametric equations are" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 39 "x:=t->r*(t-sin(t));\ny:=t->r*(1-cos(t));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "We will generate the picture in se
veral steps." }}{PARA 0 "" 0 "" {TEXT -1 31 "First we use the Maple co
mmand " }{TEXT 259 7 "animate" }{TEXT -1 48 " to create an animation o
f a circle with radius " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 18 " m
oving along the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 16 "-axis fro
m 0 to " }{XPPEDIT 18 0 "4*Pi" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 11 "(we choose " }{XPPEDIT 18 0 "4*Pi" "6
#*&\"\"%\"\"\"%#PiGF%" }{TEXT -1 41 " to see two full arches of the cy
cloid). " }}{PARA 0 "" 0 "" {TEXT -1 23 "Animate is part of the " }
{TEXT 260 5 "plots" }{TEXT -1 38 " package which has to be loaded firs
t." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 80 "To have a specific value for the radius r
 of the circle for the plot we set r=1." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 5 "r:=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Note that the " }
{TEXT 271 7 "animate" }{TEXT -1 31 " command requires as parameters" }
}{PARA 0 "" 0 "" {TEXT -1 25 "   - a curve to be drawn," }}{PARA 0 "" 
0 "" {TEXT -1 24 "   - a second parameter " }{XPPEDIT 18 0 "d" "6#%\"d
G" }{TEXT -1 6 " (the " }{TEXT 272 15 "frame parameter" }{TEXT -1 84 "
) which will simulate the movement of the circle from left to right in
 our example. " }}{PARA 0 "" 0 "" {TEXT -1 60 "   - the definition of \+
the number of frames to be generated." }}{PARA 0 "" 0 "" {TEXT -1 56 "
One can think of animate as a command which generates a " }{TEXT 263 
5 "movie" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 119 "The result \+
of animate is a sequence of frames, which when viewed successively at \+
a high enough speed look like a movie." }}{PARA 0 "" 0 "" {TEXT -1 20 
"Here is the circle: " }}{PARA 0 "" 0 "" {TEXT -1 33 "(To start the an
imation you must " }{TEXT 273 17 "click on the plot" }{TEXT -1 10 " an
d then " }}{PARA 0 "" 0 "" {TEXT -1 71 "   - either use the VCR button
s which appear at the top of the screen, " }}{PARA 0 "" 0 "" {TEXT -1 
77 "   - select Play from the Animation pull down menu at the top of t
he screen, " }}{PARA 0 "" 0 "" {TEXT -1 80 "   - or click on the plot \+
with the right mouse button and choose animate-play.) " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 151 "animate( [-r*sin(t) + d, -r*cos(t) + r, t=0..
2*Pi], d=0..4*r*Pi, frames=30, color=magenta, scaling=constrained, axe
s=frame, view=[-Pi/2..9*Pi/2,0..2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 75 "Let's take a closer look at how animate works, and what its par
ameters are." }}{PARA 0 "" 0 "" {TEXT -1 44 "First of all, you can dis
regard the options " }{TEXT 264 5 "color" }{TEXT -1 2 ", " }{TEXT 265 
7 "scaling" }{TEXT -1 2 ", " }{TEXT 266 4 "axes" }{TEXT -1 6 ", and " 
}{TEXT 267 4 "view" }{TEXT -1 38 ". They are for cosmetic purposes onl
y." }}{PARA 0 "" 0 "" {TEXT -1 27 "As noted above, the option " }
{TEXT 268 6 "frames" }{TEXT -1 52 " tells Maple how many individual fr
ames to produce. " }}{PARA 0 "" 0 "" {TEXT -1 95 "Let's use a trimmed \+
down version of the above command to make things a little more transpa
rent:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "animate( [-r*sin(t) + d, -
r*cos(t) + r, t=0..2*Pi], d=0..4*r*Pi, frames=3);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 125 "If you animated the previous plot you probably no
ticed that the animation was not very smooth (there weren't enough fra
mes). " }}{PARA 0 "" 0 "" {TEXT -1 40 "But that should not bother us r
ight now." }}{PARA 0 "" 0 "" {TEXT -1 60 "Let's try to understand the \+
other two parameters remaining: " }}{PARA 0 "" 0 "" {TEXT -1 9 "   - t
he " }{TEXT 269 6 "curve " }}{PARA 0 "" 0 "" {TEXT -1 9 "   - and " }
{TEXT 270 15 "frame parameter" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d" "6#%
\"dG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "Notice that the f
irst argument in animate is a curve " }}{PARA 0 "" 0 "" {TEXT -1 69 "(
here a parametric curve, although other objects are also possible), " 
}}{PARA 0 "" 0 "" {TEXT -1 64 "and it is generated (for any fixed valu
e of the frame parameter " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 40 "
) in our example by the curve parameter " }{XPPEDIT 18 0 "t" "6#%\"tG
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "So, for " }{XPPEDIT 
18 0 "d=0" "6#/%\"dG\"\"!" }{TEXT -1 101 ", say, the above command doe
s the same as the following plot statement, i.e. it just draws a circl
e: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "f1:=plot( [-r*sin(t) + 0, -r
*cos(t) + r, t=0..2*Pi] ):\ndisplay(f1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 37 "Now we can take another value, e.g.  " }{XPPEDIT 18 0 "d=
2*Pi" "6#/%\"dG*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 43 ", and we get the sa
me circle - but shifted " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF
%" }{TEXT -1 20 " units to the right." }}{PARA 0 "" 0 "" {TEXT -1 75 "
This is the second frame generated in our simplified animate command a
bove." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f2:=plot( [-r*sin(t) + 2*P
i, -r*cos(t) + r, t=0..2*Pi] ):\ndisplay(f2);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 43 "Finally, we can produce a third plot, with " }
{XPPEDIT 18 0 "d=4*Pi" "6#/%\"dG*&\"\"%\"\"\"%#PiGF'" }{TEXT -1 1 ":" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f3:=plot( [-r*sin(t) + 4*Pi, -r*c
os(t) + r, t=0..2*Pi] ):\ndisplay(f3);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 87 "And all that animate does, is it takes these three plots \+
and displays them in sequence." }}{PARA 0 "" 0 "" {TEXT -1 21 "The fol
lowing use of " }{TEXT 274 7 "display" }{TEXT -1 29 " accomplishes the
 same thing:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "display(\{f1,f2,f3
\}, insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Now we r
eturn to the cycloid." }}{PARA 0 "" 0 "" {TEXT -1 121 "We save our ani
mation of the rolling circle in the variable a1 so that we can later d
isplay it together with the cycloid." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 157 "a1 := animate( [-r*sin(t) + d, -r*cos(t) + r, t=0..2*Pi], d=0..
4*r*Pi, frames=30, color=magenta, scaling=constrained, axes=frame, vie
w=[-Pi/2..9*Pi/2,0..2]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "Next
 we create an animation of the moving point on the circumference of th
e circle by creating a little blue tickmark which rotates along with t
he circle. " }}{PARA 0 "" 0 "" {TEXT -1 24 "We save this part in a2." 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "a2 := animate( [-r*t*sin(d) + d,
 -r*t*cos(d)+r, t=0.9..1.1], d=0..4*r*Pi, frames=30, color=blue, scali
ng=constrained, axes=frame, view=[-Pi/2..9*Pi/2,0..2]):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 74 "Finally, we create the animation for the \+
cycloid curve, and save it in a3." }}{PARA 0 "" 0 "" {TEXT -1 42 "Note
 how the frame parameter is used here:" }}{PARA 0 "" 0 "" {TEXT -1 36 
"The graph of the cycloid is given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 65 "plot([x(t),  y(t), t=0..4*Pi], scaling=constrained, color=green)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "In order to simulate the cur
ve being drawn we use the frame parameter " }{XPPEDIT 18 0 "d" "6#%\"d
G" }{TEXT -1 50 " to determine how much of the curve we want drawn." }
}{PARA 0 "" 0 "" {TEXT -1 44 "So, in the statement to follow the range
 of " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "
[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 11 ", with \n   " }{XPPEDIT 18 0 "d
=0" "6#/%\"dG\"\"!" }{TEXT -1 42 " corresponding to \"nothing is drawn
\", \n   " }{XPPEDIT 18 0 "d=1" "6#/%\"dG\"\"\"" }{TEXT -1 36 " \"the \+
entire curve is drawn\",\n    0<" }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT 
-1 38 "<1 \"a fraction of the curve is drawn\"." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 137 "a3 := animate( [x(d*t),  y(d*t), t=0..4*Pi], d=0..1,
 frames=30, color=green, scaling=constrained, axes=frame, view=[-Pi/2.
.9*Pi/2,0..2]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Now we are re
ady to display all three animations (the rolling circle, the tickmark,
 and the curve) together. " }}{PARA 0 "" 0 "" {TEXT -1 197 "Note: you \+
can modify the speed of the animation by clicking on the plot and then
 \n   - either using the VCR buttons which appear at the top of the sc
reen (>> makes the animation faster, << slower)," }}{PARA 0 "" 0 "" 
{TEXT -1 181 "   - selecting Faster/Slower from the Animation pull dow
n menu at the top of the screen, \n   - or clicking on the plot with t
he right mouse button and choose animate-faster/slower. " }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "display(\{a1,a2,a3\});" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 40 "The following creates an animation of a " }{TEXT 
276 11 "hypocycloid" }{TEXT -1 107 " (the curve obtained by following \+
the motion of a fixed point on the circumference of a circle with radi
us " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 23 " which rolls along the
 " }{TEXT 278 6 "inside" }{TEXT -1 31 " of another circle with radius \+
" }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "b" "6
#%\"bG" }{TEXT -1 2 "):" }}{PARA 0 "" 0 "" {TEXT -1 38 "The general pa
rametric equations are: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "x:=t->(
a-b)*cos(t)+b*cos((a-b)*t/b);\ny:=t->(a-b)*sin(t)-b*sin((a-b)*t/b);\n
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "For the following plot we cho
ose " }{XPPEDIT 18 0 "a=4" "6#/%\"aG\"\"%" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "b=1" "6#/%\"bG\"\"\"" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 546 "a:=4: b:=1:\na1:=animate([(a-b)*cos(d)+b*cos(t), (a-
b)*sin(d)+b*sin(t), t=0..2*Pi], d=0..2*Pi, frames=30, color=magenta, v
iew=[-a..a,-a..a]): # animation of the rolling circle\na2:=plot([a*cos
(t), a*sin(t), t=0..2*Pi], color=red): # plot of the outer circle\na3:
=animate([x(d*t), y(d*t), t=0..2*Pi], d=0..1, frames=30, color=green, \+
view=[-a..a,-a..a]): # animation of the hypocycloid\na4:=animate([t*x(
d), t*y(d), t=.98..1.02], d=0..2*Pi, frames=30, color=blue, view=[-a..
a,-a..a]): # animation of the tickmark\ndisplay(\{a1,a2,a3,a4\});\na:=
'a': b:='b':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "An " }{TEXT 277 
10 "epicycloid" }{TEXT -1 107 " is a curve obtained by following the m
otion of a fixed point on the circumference of a circle with radius " 
}{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 23 " which rolls along the " }
{TEXT 279 7 "outside" }{TEXT -1 31 " of another circle with radius " }
{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "b" "6#%
\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Its general para
metric equations are given by: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "
x:=t->(a+b)*cos(t)-b*cos((a+b)*t/b);\ny:=t->(a+b)*sin(t)-b*sin((a+b)*t
/b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "And here is the animation
 of an epicycloid with " }{XPPEDIT 18 0 "a=4" "6#/%\"aG\"\"%" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "b=1" "6#/%\"bG\"\"\"" }{TEXT -1 1 ":" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 601 "a:=4: b:=1:\na1:=animate([(a+b)*co
s(d)+b*cos(t), (a+b)*sin(d)+b*sin(t), t=0..2*Pi], d=0..2*Pi, frames=30
, color=magenta, view=[-(a+2*b)..(a+2*b),-(a+2*b)..(a+2*b)]): # animat
ion of the rolling circle\na2:=plot([a*cos(t), a*sin(t), t=0..2*Pi], c
olor=red): # plot of the inner circle\na3:=animate([x(d*t), y(d*t), t=
0..2*Pi], d=0..1, frames=30, color=green, view=[-(a+2*b)..(a+2*b),-(a+
2*b)..(a+2*b)]): # animation of the epicycloid\na4:=animate([t*x(d), t
*y(d), t=.98..1.02], d=0..2*Pi, frames=30, color=blue, view=[-(a+2*b).
.(a+2*b),-(a+2*b)..(a+2*b)]): # animation of the tickmark\ndisplay(\{a
1,a2,a3,a4\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT 258 11 "Assignment:" }}{PARA 0 "" 0 "" {TEXT -1 
54 "1. a) Create a plot of the following parametric curve:" }}{PARA 0 
"" 0 "" {TEXT -1 15 "               " }{XPPEDIT 18 0 "x(t) = 2*cos(t) \+
+ cos(2*t)" "6#/-%\"xG6#%\"tG,&*&\"\"#\"\"\"-%$cosG6#F'F+F+-F-6#*&\"\"
#F+F'F+F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 15 "            \+
   " }{XPPEDIT 18 0 "y(t) = 2*sin(t) - sin(2*t)" "6#/-%\"yG6#%\"tG,&*&
\"\"#\"\"\"-%$sinG6#F'F+F+-F-6#*&\"\"#F+F'F+!\"\"" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 15 "               " }{XPPEDIT 18 0 "t" "6#%
\"tG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"
\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "     b) Wh
at happens if you replace 2 by -2?" }}{PARA 0 "" 0 "" {TEXT -1 43 "   \+
  c) What happens if you replace 2 by 3?" }}{PARA 0 "" 0 "" {TEXT -1 
60 "     d) What happens as you increase the value even further?" }}
{PARA 0 "" 0 "" {TEXT -1 44 "2. What is the difference between the cur
ves" }}{PARA 0 "" 0 "" {TEXT -1 14 "              " }{XPPEDIT 18 0 "x(
t) = 6*cos(t) + 5*cos(3*t)" "6#/-%\"xG6#%\"tG,&*&\"\"'\"\"\"-%$cosG6#F
'F+F+*&\"\"&F+-F-6#*&\"\"$F+F'F+F+F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "
" {TEXT -1 14 "              " }{XPPEDIT 18 0 "y(t) = 6*sin(t)-5*sin(3
*t)" "6#/-%\"yG6#%\"tG,&*&\"\"'\"\"\"-%$sinG6#F'F+F+*&\"\"&F+-F-6#*&\"
\"$F+F'F+F+!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 "     \+
         " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }{XPPEDIT 
18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 10 "      and " }}{PARA 0 "" 0 "" {TEXT -1 
14 "              " }{XPPEDIT 18 0 "x(t) = 6*cos(2*t)+5*cos(6*t)" "6#/
-%\"xG6#%\"tG,&*&\"\"'\"\"\"-%$cosG6#*&\"\"#F+F'F+F+F+*&\"\"&F+-F-6#*&
\"\"'F+F'F+F+F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 "      \+
        " }{XPPEDIT 18 0 "y(t) = 6*sin(2*t)-5*sin(6*t)" "6#/-%\"yG6#%
\"tG,&*&\"\"'\"\"\"-%$sinG6#*&\"\"#F+F'F+F+F+*&\"\"&F+-F-6#*&\"\"'F+F'
F+F+!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 "            \+
  " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0
,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 3 "3
. " }{TEXT 275 16 "Lissajous curves" }{TEXT -1 98 " are the type of cu
rves that appear on oscilloscopes in physics or electronics. They are \+
given by " }}{PARA 0 "" 0 "" {TEXT -1 11 "           " }{XPPEDIT 18 0 
"x(t) = cos(a*t)" "6#/-%\"xG6#%\"tG-%$cosG6#*&%\"aG\"\"\"F'F-" }{TEXT 
-1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 11 "           " }{XPPEDIT 18 0 "y
(t) = sin(b*t)" "6#/-%\"yG6#%\"tG-%$sinG6#*&%\"bG\"\"\"F'F-" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "     a) Plot the case " }
{XPPEDIT 18 0 "a=3, b=5" "6$/%\"aG\"\"$/%\"bG\"\"&" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0,2*Pi
]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 25 "     b) What happens for " }{XPPEDIT 18 0 "a=b" "6#/%\"aG
%\"bG" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 74 "     c) Create a
n animation that shows how the curve in a) is being drawn." }}{PARA 0 
"" 0 "" {TEXT -1 6 "4. Let" }}{PARA 0 "" 0 "" {TEXT -1 24 "           \+
             " }{XPPEDIT 18 0 "x(t) = a*cos(t) - b*cos(p*t)" "6#/-%\"x
G6#%\"tG,&*&%\"aG\"\"\"-%$cosG6#F'F+F+*&%\"bGF+-F-6#*&%\"pGF+F'F+F+!\"
\"" }}{PARA 0 "" 0 "" {TEXT -1 24 "                        " }
{XPPEDIT 18 0 "y(t) = c*sin(t) -d*sin(q*t)" "6#/-%\"yG6#%\"tG,&*&%\"cG
\"\"\"-%$sinG6#F'F+F+*&%\"dGF+-F-6#*&%\"qGF+F'F+F+!\"\"" }}{PARA 0 "" 
0 "" {TEXT -1 49 "     a) Plot the \"slinky curve\" corresponding to \+
" }{XPPEDIT 18 0 "a=16, b=5, c=12, d=3, p=47/3, q=44/3" "6(/%\"aG\"#;/
%\"bG\"\"&/%\"cG\"#7/%\"dG\"\"$/%\"pG*&\"#Z\"\"\"\"\"$!\"\"/%\"qG*&\"#
WF3\"\"$F5" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 
-1 4 " in " }{XPPEDIT 18 0 "[0,12*Pi]" "6#7$\"\"!*&\"#7\"\"\"%#PiGF'" 
}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 98 "     b) Play with the p
arameters to create an even prettier (more interesting) curve. Be crea
tive!" }}{PARA 0 "" 0 "" {TEXT -1 34 "5.  Consider the parametric curv
e " }{XPPEDIT 18 0 "x=sin(2*t)" "6#/%\"xG-%$sinG6#*&\"\"#\"\"\"%\"tGF*
" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=cos(t^2)" "6#/%\"yG-%$cosG6#*$%\"
tG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " i
n " }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 39 ".\n     a) \+
Plot the curve.\n     b) Find " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"
\"%#dxG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "d^2y/dx^2" "6#*(%\"d
G\"\"#%\"yG\"\"\"*$%#dxG\"\"#!\"\"" }{TEXT -1 14 " at the point " }
{XPPEDIT 18 0 "t[0]=Pi/2" "6#/&%\"tG6#\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }
{TEXT -1 64 ".\n     c) Find an equation for the tangent line to the c
urve at " }{XPPEDIT 18 0 "t[0]=Pi/2" "6#/&%\"tG6#\"\"!*&%#PiG\"\"\"\"
\"#!\"\"" }{TEXT -1 152 ". \n     d) Plot the curve together with the \+
tangent line.\n     e) Find the length of the curve. You will have to \+
numerically evaluate the integral with " }{TEXT 280 6 "evalf." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "52 0 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 }