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"" 1 18 0 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 "" 0 300 1 {CSTYLE "" -1 -1 "" 1 18 0 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 "" 0 301 1 {CSTYLE "" -1 -1 "" 1 18 0 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 "" 0 302 1 {CSTYLE "" -1 -1 "" 1 18 0 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 "" 0 303 1 {CSTYLE "" -1 -1 "" 1 18 0 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 }} {SECT 0 {EXCHG {PARA 287 "" 0 "" {TEXT -1 0 "" }}{PARA 288 "" 0 "" {TEXT 258 21 "Curves in Three Space" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 93 "We begin this se ction by plotting the parametric equation of a curve. We begin with t he case" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 40 " x = f(t) y = g(t)." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 148 "That is: x is a function of time and y is a function of \+ time. In this case, time or t is the independent variable. It is oft en convenient to write" }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 38 " \+ x = x(t) y = y(t)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 " " 0 "" {TEXT -1 124 "Using Maple we can easily graph parametric equati ons. Begin by using the help menu and \"Topic Search\" to look up th e word " }}{PARA 266 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 4 "plot" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 77 "You should find an example of a parametric equation simil ar to the following:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot([cos (t),sin(t), t=0..2*Pi], title=Example1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 94 "We have obtained a circ le. Why? If it does not look like a circle, correct the aspect ratio ." }}{PARA 271 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 62 " Consider the following case: Can the aspect ratio be changed?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([cos(t), 2*sin(t), t=0. .2*Pi], scaling=constrained, title=Example2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT -1 68 "Note that the scalin g attribute in the above graph forces the graph " }}{PARA 292 "" 0 "" {TEXT -1 44 "to treat the x and y axes distances equally." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT -1 79 "What kind of gra ph is obtained in Example 2? Prove that your answer is correct." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT -1 65 "A major problem with this graphing technique is that it is often " }}{PARA 289 "" 0 "" {TEXT -1 65 "helpful to see the graph unfold as a function of time. In other " }}{PARA 290 "" 0 "" {TEXT -1 60 "words we need t o animate the parametric graph. We begin by " }}{PARA 291 "" 0 "" {TEXT -1 45 "loading a Maple library by the name of plots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wit h(plots);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 " " {TEXT -1 55 "Next we will need to graph several plot structures and \+ " }}{PARA 293 "" 0 "" {TEXT -1 50 "save them. To do this we shall wri te a \"do\" loop." }}{PARA 280 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " nframes \+ := 20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "for i from 1 to n frames do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 " para[i] := spacecur ve([cos(t), 2*sin(t),t], t=0..0.2*i*Pi, scaling=constrained, title=Exa mple3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT -1 162 "Note that the u pper limit for each curve is determined by the index variable - i -. \+ The first graph goes from t=0..Pi/5.0 and the last graph goes from t=0 ..2*Pi." }}{PARA 282 "" 0 "" {TEXT -1 0 "" }}{PARA 283 "" 0 "" {TEXT -1 209 "We can animate this graph by using a display command. We will plot the sequence of graphs and use the attribute insequence to assur e that they are displayed in the same sequence in which they were calc ulated." }}{PARA 284 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display(seq(para[k], k=1..nframes), insequence=true); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 285 "" 0 " " {TEXT -1 324 "As you can see a parametric equation is often construc ted with a partical motion in mind. This need not be the case. Param etric equations are useful in physics for discribing the motion of obj ects. They are also used in mathematics for solving differential equa tions and graphing surfaces to name just a few applications." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 275 " " 0 "" {TEXT -1 51 "Construct a similar example to Example 3 above wit h" }}{PARA 276 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT -1 57 " x(t) = cos(t) y(t) = -2*sin(t) z(t) = t." }}{PARA 286 "" 0 "" {TEXT -1 0 "" }}{PARA 299 "" 0 "" {TEXT -1 70 "Recall that the Riemann Sum for the 2-dimensional parametric curve is:" }}{PARA 300 "" 0 "" {TEXT -1 0 "" }}{PARA 301 "" 0 "" {TEXT -1 42 " Sum(sqr t(xprime^2 + yprime^2) delta t)" }}{PARA 302 "" 0 "" {TEXT -1 0 "" }} {PARA 303 "" 0 "" {TEXT -1 60 "We can realize this as a maple command \+ in the following way:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "a := 0.0; b := 2*Pi; n := 10: delta_t := (b - a)/n: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f := cos:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "fprime := D(f):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "g := sin:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gprime := D(g): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Sum(sqrt(fprime(i*delta _t)^2 + gprime(i*delta_t)^2)*delta_t, i=0..n);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "ApproxArcL := evalf(%);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "ActualArcL := evalf(2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "percenterr := evalf((ApproxArcL - ActualArc L)*100/ActualArcL);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 57 "As You can see the percent error is about 10 w hen n=10. " }}{PARA 0 "" 0 "" {TEXT 257 42 "Find the percent error wh en n=100, n=1000." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 294 "" 0 " " {TEXT -1 50 "Rewrite the formula for a three dimensional curve." }} {PARA 295 "" 0 "" {TEXT -1 0 "" }}{PARA 296 "" 0 "" {TEXT -1 45 "Take \+ x(t) = cos(t), y(t) = sin(t), z(t) = t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a := 0.0; b := 2*Pi; n := 1000: del ta_t := (b - a)/n:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f := cos:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fprime := D(f):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " g := sin:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gprime := D(g):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "h := t -> t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "hprime := D(h):" }}}{EXCHG {PARA 297 "" 0 "" {TEXT -1 0 "" }}{PARA 298 "" 0 "" {TEXT -1 35 "Fill in the correct summation below" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Sum( sqrt( 1 )*delta_t, i=0..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ApproxArcL := evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ActualArcL := evalf(2*Pi*sqrt(2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "percenterr := evalf((ApproxArcL - ActualArcL)*100/Act ualArcL);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}}{MARK "0 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }