{VERSION 4 0 "IBM INTEL LINUX22" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 2 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 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 18 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 25 "Curve Examples with Maple" }{TEXT 264 0 "" }} {PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wi th(plots);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot( [cos(2* t),sin(2*t), t=0..Pi],-3..3,-3..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "That shows the obvious thing: a circle traversed once." }} {PARA 0 "" 0 "" {TEXT -1 73 "Now let's see a tangent line at some poin t, say t=0. At t=0, the curve" }}{PARA 0 "" 0 "" {TEXT -1 57 "is at the point (1,0), as this little computation shows:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "[cos(0),sin(0)]; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The velocity vector at any ti me t is given by the derivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "vel := [diff(cos(t),t), diff(sin(t ),t) ];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Thus the velocity ve ctor at time t=0 is given by substituting t=0 into" }}{PARA 0 " " 0 "" {TEXT -1 16 "this expression:" }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(t=0,vel);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Let's get a numerical value:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf( % ); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Now we can find the tangent line, since we have a point on it and \+ a vector" }}{PARA 0 "" 0 "" {TEXT -1 66 "giving direction. The t angent line has vector equation " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 " X = (1,0) + t(0,1)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "In scalar form, the line has equations x=1, y=t. Now let's plot" }}{PARA 0 "" 0 "" {TEXT -1 45 "both the tangent line and the curve together." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot( \{ [cos(t),sin(t),t=0..2*Pi], [1,t,t=0..1]\} );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Let's do something similar with a space curve. F irst, here's the curve" }}{PARA 0 "" 0 "" {TEXT -1 7 "itself:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "spacecurve( [cos(t),sin(t),t, t=0..10] , \+ color=black,axes=framed );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 84 "Does the curve look reasonable? If so, let's think about drawing a tangent line, " }}{PARA 0 "" 0 "" {TEXT -1 29 "say at the point t=Pi/2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs( t=Pi/2, [cos(t),sin(t),t] ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "vel := diff( [cos(t),sin(t),t], t ) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Note that diff worked as you'd th ink --- it differentiated each of the parts" }}{PARA 0 "" 0 "" {TEXT -1 11 "separately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs( t=Pi/2, vel); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Now we have the necessary information to write \+ down the tangent line. It has" }}{PARA 0 "" 0 "" {TEXT -1 66 "the fo rm X = (0,1,Pi/2) + t (-1,0,1). In scalar form, that's" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 " x=-t, y=1, z=Pi/2+t" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Let's try it out graphically:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "spa cecurve(\{[cos(t),sin(t),t,t=0..2*Pi],[-t,1,Pi/2+t,t=-1..1]\}, axes = \+ framed,color=black );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "Looks plausible. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Do these:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "1. (2-d probl em) Let the curve be the unit circle, as above. Use" }}{PARA 0 "" 0 "" {TEXT -1 78 " Maple to draw both the curve and the tangent l ine at the ``north pole.''" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 69 "2. (2-d proble m) Let the curve be the unit circle, as above. Use" }}{PARA 0 "" 0 "" {TEXT -1 74 " Maple to draw both the curve and the tangent lin e at t=1. What's " }}{PARA 0 "" 0 "" {TEXT -1 24 " the speed at t=1?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "3. ( 2-d problem) Use Maple to dra w both the curve and the tangent line at t=0 for" }}{PARA 0 "" 0 " " {TEXT -1 18 " the curve " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 60 " X(t) = ( cos(t), sin(2*t) ) \+ , t = 0 .. 2*Pi ." }}{PARA 0 "" 0 "" {TEXT -1 30 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 30 " What's the speed at t=0?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 " 4. ( 3-d problem) Use Maple to draw both the curve and the tangent l ine at t=0 for" }}{PARA 0 "" 0 "" {TEXT -1 18 " the curve " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 " \+ X(t) = ( cos(t), sin(2*t) , t) , t = 0 .. 2*Pi ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 " (This curve \+ is ALMOST, but not quite, like the one illustrated above.)" }}{PARA 0 "" 0 "" {TEXT -1 25 " " }}{PARA 0 "" 0 "" {TEXT -1 30 " What's the speed at t=0?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "5. ( 3-d problem) Use Maple t o draw both the curve and the tangent line at t=1 for" }}{PARA 0 " " 0 "" {TEXT -1 18 " the curve " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 63 " X(t) = ( cos(t), sin(2*t ) , t) , t = 0 .. 2*Pi ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 " (This curve is ALMOST, but not quite, like \+ the one illustrated above.)" }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 30 " What's the speed at t=0?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1 1 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }