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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 58 "Parametric Curves and Vector-val
ued Functions in the Plane" }}{PARA 19 "" 0 "" {TEXT -1 10 "Bill Farr \+
" }}{PARA 19 "" 0 "" {TEXT -1 13 "MA 1024 B '99" }}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 66 "Introduction - comands for curve computations in the
 CalcP package" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(CalcP
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "There are several command \+
in the CalcP package that are useful for computations with curves. The
 ones you will probably need for this lab are listed below. " }}}
{EXCHG {PARA 15 "" 0 "" {TEXT -1 87 "VPlot - Plots vector valued funct
ions or parametric curves in two and three dimensions." }}{PARA 15 "" 
0 "" {TEXT -1 65 "VDiff - For computing the derivative of a vector-val
ued function." }}{PARA 15 "" 0 "" {TEXT -1 57 "ParamPlot - An animated
 plot of a vector valued function." }}{PARA 15 "" 0 "" {TEXT -1 36 "Sp
eed - Calculates the speed, ds/dt." }}{PARA 15 "" 0 "" {TEXT -1 46 "ta
nvect - Computes the unit tangent vector, T." }}{PARA 15 "" 0 "" 
{TEXT -1 36 "Curvature - Computes the curvature, " }{XPPEDIT 18 0 "kap
pa" "I&kappaG6\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 48 "norm
alvect - Computes the unit normal vector, N." }}{PARA 15 "" 0 "" 
{TEXT -1 48 "TanAccel - Computes the tangential acceleration." }}
{PARA 15 "" 0 "" {TEXT -1 46 "NormAccel - Computess the normal acceler
ation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 70 "Defining parametric curves and vector val
ued functions simply in Maple" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "
The easiest way to define a \"vector valued\" function or a parametric
 curve is to use the Maple list notation with square brackets []." }
{TEXT -1 150 " Strictly speaking, this does not define something that \+
Maple recognizes as a vector, but it will work with all of the command
s you need for this lab." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := t \+
-> [2*cos(t),2*sin(t)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "You ca
n evaluate this function at any value of t in the usual way." }{TEXT 
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(0);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 92 "This is how to access a single component. You woul
d use f(t)[2] to get the second component." }{TEXT -1 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "f(t)[1];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Plotting an
d animating" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The VPlot command i
s a simple command for plotting." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "VPlot(f(t),t=0..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 22 "The ParamPlot command " }{TEXT -1 41 "can be used to anim
ate parametric plots. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 281 "When you execute the command, at first you will jus
t see a set of axes. Click anywhere on the plot, and the controls will
 become visible in the tool bar (just below the menu shorcuts bar). Th
e controls are a lot like those on a VCR. If you have problems, ask fo
r help from your IA." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ParamPlot(f
(t),t=0..2*Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Examples of curve computation com
mands" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "The speed command comput
es the speed, ds/dt. It can either produce an expression depending on \+
t or a value at a specific time. See the examples below." }{TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "This
 function corresponded to uniform circular motion, so the speed is con
stant." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Speed(f(t),t);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "This is an example where the speed
 isn't constant." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "
g := t -> [t+2,t^3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Spe
ed(g(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Speed(g(t),t
=1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The tanvect command compu
tes the unit tangent vector T." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "tanvect(f(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "tanvect(g(t),t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
148 "This form of the command lets you compute the tangent vector at a
 specific value of t. This works for the rest of the commands shown be
low, as well." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "tan
vect(g(t),t=1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The Curvature \+
command computes the curvature." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 104 "This example was uniform circular motion
 on a circle of radius 2, so you expect the curvature to be 1/2." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Curvature(f(t),t);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 163 "By definition, the curvature is always n
on-negative. The signum function appearing below is 1 if t is greater \+
than or equal to zero, and -1 if t is less than zero." }{TEXT -1 0 "" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Curvature(g(t),t);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 80 "Since the output of the Curvature command
 is an expression, it is easy  to plot." }{TEXT -1 0 "" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "plot(Curvature(g(t),t),t=-2..2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 55 "The normalvect command computes the unit \+
normal vector." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "no
rmalvect(f(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "normal
vect(g(t),t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The TanAccel and
 NormAccel co" }{TEXT -1 72 "mmands compute the tangential and normal \+
components of the acceleration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 86 "The first example was uniform circular mo
tion, so the tangential acceleration is zero." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "TanAccel(f(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "NormAccel(f(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "TanAccel(g(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "NormAccel(g(t),t);" }}}{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 "" {MPLTEXT 1 0 0 "" }}}
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