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{SECT 0 {PARA 256 "" 0 "" {TEXT 258 37 "Moving up a dimension...Space \+
Curves!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
326 "MAPLE can produce parametric plots of curves in three dimensions \+
quite easily. The appropriate command is \"spacecurve\" and it requir
es three equations: x(t), y(t), and z(t), which describe the x-, y-, a
nd z-components, respectively, of a point on the curve at time t. To \+
illustrate, let's add the parametric equation z(t) = " }{XPPEDIT 18 0
"0.25*t" "6#*&-%&FloatG6$\"#D!\"#\"\"\"%\"tGF)" }{TEXT -1 74 " to the \+
parameterization of the unit circle: x(t) = cos(t), y(t) = sin(t)." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "with(plots):\nspacecurve([co
s(t),sin(t),0.25*t],t=0..6*Pi,thickness=3,scaling=constrained);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 66 "Was the output surprising to you? It shouldn't be too su
rprising!" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 "Exercise 1 " }}
{PARA 0 "" 0 "" {TEXT -1 132 "Suppose that a tiny bug is spiraling aro
und a pole stuck into the ground. The bug's motion is described by th
e parameterization: \{" }{TEXT 257 1 "x" }{TEXT -1 37 "(t), y(t), z(t
)\} = \{cos(t), sin(t), " }{XPPEDIT 18 0 "0.25*t" "6#*&-%&FloatG6$\"#
D!\"#\"\"\"%\"tGF)" }{TEXT -1 135 "\}, where t in measured in seconds,
and x, y, and z in centimeters. Assume also that the bug starts out \+
at time t = 0 on the ground. " }}{PARA 0 "" 0 "" {TEXT -1 52 "a.) Wh
at is the height of the bug after 1.5 minutes?" }}{PARA 0 "" 0 ""
{TEXT -1 71 "b.) How many turns around the pole will the bug make afte
r 1.5 minutes?" }}{PARA 0 "" 0 "" {TEXT -1 209 "c.) Find the parameter
ization that describes the motion of a bug that makes exactly 3 turns \+
around the pole in 1 minute, ending up at a height of 10 cm above the \+
ground. Then produce a plot of the bug's path." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Exer
cise 2" }}{PARA 0 "" 0 "" {TEXT -1 175 "Generalize the ideas presented
above about parameterizations of lines in the plane to produce the pa
rameterization of the line through the two points (1, 2, 3) and (7, 9,
15)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA
0 "" 0 "" {TEXT -1 82 "Feel free to experiment with more space curves.
Here is a cool one...it's a knot!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "knot:= [ -10*cos(t) - 2*c
os(5*t) + 15*sin(2*t),\n -15*cos(2*t) + 10*sin(t) - 2*sin(5*t)
, 10*cos(3*t), t= 0..2*Pi]:\nspacecurve(knot,thickness=4,numpoints=100
);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "7 0 0"
9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }