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" }{TEXT -1 382 "T his lab introduces you to some of the features of Maple. Some of these features are simple graphing and numerical capabilities similar to th ose of a graphing calculator, and some are symbolic manipulations exc lusive to Computer Algebra Systems. If at anytime you are at a loss \+ as to how to do something, click the \"Help\" button on the menu bar a nd try topic or full text search." }}{PARA 257 "" 0 "" {TEXT -1 42 "Cl ick on the + symbol to open that section" }}{PARA 0 "" 0 "" {TEXT 289 78 "Evaluate all input (in red) by placing the cursor anywhere in red \+ and enter. " }{TEXT -1 1 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 " 1. Calculator, Graphing, and Numerical Features" }}{PARA 0 "" 0 "" {TEXT -1 265 "Like your graphing calculator, Maple can plot functions \+ but with greater resolution and more options. Maple can perform numer ical routines such as finding the solutions of an equation (using an a logorithm like Newton's Method) and approximating definite integrals. " }}{SECT 1 {PARA 3 "" 0 "" {TEXT 272 17 "a) Basic Commands" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 " The \">\" symbol is the Maple prompt th at lets you know Maple is ready for input. In the following pages, ev erything after the \">\" represents something you would type. " } {TEXT 266 118 "In this, and all Maple lessons, be sure to evaluate all input (in red) by pressing the enter key somewhere on the line" } {TEXT -1 460 ". If you don't do this, previous input will go unrecogn ized in current evaluations. Maple output appears in the center (in b lue). All commands must end with a semicolon or a colon. If you end t he input with a colon, the results will not be printed to the screen a s they are in the examples below. You must use \"colon equals (:=)\" \+ to assign a value or expression to a variable. The \"=\" sign alone \+ describes an equation which returns \"true\" or \"false\". " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "a := 15; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b := 9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a/b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Not ice, Maple simplifies but does give a decimal expansion. In order to \+ do this, type " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(5/3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nmmm;!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "The above command evaluates 5/3 as a floating \+ point number. evalf(5/3,n) gives the result with n units of precision. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(5/3,20);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5nmmmmmmmm;!#>" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 267 48 "When multiplying be sure to use the star symbol: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a * b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$N\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "If yo u had forgotten to use this and just typed" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "a b;" }}{PARA 8 "" 1 "" {TEXT -1 23 "missing operato r or `;`" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "you get an error mess age. " }}{PARA 0 "" 0 "" {TEXT -1 94 " To clear the value, or expressi on, assigned to a variable, you must use one of the following" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a := 'a';" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"aGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign('b');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "A # sign ma y be used for comments and everything after it is ignored by Maple:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "3*a + b; # Hi, How are you?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"aG\"\"$%\"bG\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Notice, the values assigned to a a nd b have been cleared. " }}{PARA 0 "" 0 "" {TEXT -1 62 "Finally you m ay reference previous output with the \"%\" symbol:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "2 * %;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&%\"aG\"\"'%\"bG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Here, \+ the previous output \"3 a + b\" is multiplied by 2. Note: Maple doesn 't include the multiplication symbol in the output but you must use it for input. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# end o f this section" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 273 58 "b) Defining \+ Functions, Plotting them, An important example" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Maple defines functions through the use of the \":=\" \+ and \"->\" symbols as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := x -> 3*x^3 + 7*x^2 + 2*x - 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$\"\"$\" \"\"F0*$)F/\"\"#F1\"\"(F/F4!\"&\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 146 "Here you may think of the command as \"f\" takes a var iable \"x\" and assigns it the value f(x). If you want to evaluate \" f\" at a particular number: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$X\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "You can plot \"f\" over the interval x=-2 to 2 \+ by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-3..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S 7$$!\"$\"\"!$!#HF*7$$!1nm;HU,\"*G!#:$!1+G]zd_wC!#97$$!1L$3FH'='z#F0$!1 LedER!\\9#F37$$!1nmTgBa*o#F0$!1W7&*HY\"4\"=F37$$!1nm\"H_\">#e#F0$!15:* [!GB9:F37$$!1L$3_!4NvCF0$!1n2W2\"GhD\"F37$$!1n;/wfHwBF0$!1Ybv0%R![5F37 $$!1+]PM.ttAF0$!1lpzT]/B')F07$$!1n;/,oln@F0$!1\"G^'QF!***pF07$$!1+](oW B>1#F0$!1D&)e`k7icF07$$!1LL$epjJ&>F0$!1Dkrl(eab%F07$$!1nm\"z/ot&=F0$!1 3H>J!e()y$F07$$!1++]P[_\\Pbfl'*o #F07$$!1++]i^)o`\"F0$!1T8[].5ICF07$$!1n;/^?7U9F0$!1x/f&)\\%QK#F07$$!1L L$eaR%H8F0$!1vf;E$F07$$!1)**\\7G-89)F[r$!1#H&y6/W2OF07$$!1HL$3Fp)pqF[r$!1e!G%o 6GvRF07$$!1KL3-$ff3'F[r$!1)4w>a?2I%F07$$!1nm;z%zY-&F[r$!1apShz>=YF07$$ !1kmT5!3B#RF[r$!1kXNP[d))[F07$$!1***\\iS!piHF[r$!1]pZ@g7c]F07$$!1jm;/r FE>F[r$!1]6&>)*fp9&F07$$!1&******\\2cb)!#<$!1H(fq-_<7&F07$$\"1:++DJE>> Fdt$!1tUhG],f\\F07$$\"1.+D1RU07F[r$!1x0;Jt%>l%F07$$\"1++](=S2L#F[r$!1& fgoG.c6%F07$$\"1jmm;p)=M$F[r$!1.g'*y$zyV$F07$$\"1-++v=]@WF[r$!1'=S?G0z [#F07$$\"1JLe*[$z*R&F[r$!1JU\\:'fmS\"F07$$\"11++DYKpkF[r$\"1Hvdc^=yNFd t7$$\"1lm\"H2qcZ(F[r$\"1:)R[\"pZg;F07$$\"1/+DJ5fF&)F[r$\"1yL%)*=zil$F0 7$$\"1lmmTg.c&*F[r$\"1-LOAUO@fF07$$\"1+]ilAFj5F0$\"1fR(Gi?mk)F07$$\"1M LL$)*pp;\"F0$\"1YB[/yUj6F37$$\"1ML3xe,t7F0$\"1MkY`\\!z]\"F37$$\"1n;HdO =y8F0$\"1\"f`OUA0*=F37$$\"1,++D>#[Z\"F0$\"1wq$\\g**)zAF37$$\"1nmT&G!e& e\"F0$\"1f\"=3%p$Gx#F37$$\"1MLL$)Qk%o\"F0$\"1?BAIZ'yD$F37$$\"1+]iSjE!z \"F0$\"1\"**paigH#QF37$$\"1,]P40O\"*=F0$\"1T\"QE/,@T%F37$$\"\"#F*$\"#^ F*-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$ ;F(Fgz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 } }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "and restrict the range by descr ibing y boundaries:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot (f(x),x=-3..2,y=-10..20);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"$\"\"!$!#HF*7$$!1nm;HU,\"*G!#:$!1+G ]zd_wC!#97$$!1L$3FH'='z#F0$!1LedER!\\9#F37$$!1nmTgBa*o#F0$!1W7&*HY\"4 \"=F37$$!1nm\"H_\">#e#F0$!15:*[!GB9:F37$$!1L$3_!4NvCF0$!1n2W2\"GhD\"F3 7$$!1n;/wfHwBF0$!1Ybv0%R![5F37$$!1+]PM.ttAF0$!1lpzT]/B')F07$$!1n;/,oln @F0$!1\"G^'QF!***pF07$$!1+](oWB>1#F0$!1D&)e`k7icF07$$!1LL$epjJ&>F0$!1D krl(eab%F07$$!1nm\"z/ot&=F0$!13H>J!e()y$F07$$!1++]P[_\\Pbfl'*o#F07$$!1++]i^)o`\"F0$!1T8[].5ICF07$$!1n; /^?7U9F0$!1x/f&)\\%QK#F07$$!1LL$eaR%H8F0$!1vf;E$F07$$!1)**\\7G-89)F[r$!1#H&y6/W 2OF07$$!1HL$3Fp)pqF[r$!1e!G%o6GvRF07$$!1KL3-$ff3'F[r$!1)4w>a?2I%F07$$! 1nm;z%zY-&F[r$!1apShz>=YF07$$!1kmT5!3B#RF[r$!1kXNP[d))[F07$$!1***\\iS! piHF[r$!1]pZ@g7c]F07$$!1jm;/rFE>F[r$!1]6&>)*fp9&F07$$!1&******\\2cb)!# <$!1H(fq-_<7&F07$$\"1:++DJE>>Fdt$!1tUhG],f\\F07$$\"1.+D1RU07F[r$!1x0;J t%>l%F07$$\"1++](=S2L#F[r$!1&fgoG.c6%F07$$\"1jmm;p)=M$F[r$!1.g'*y$zyV$ F07$$\"1-++v=]@WF[r$!1'=S?G0z[#F07$$\"1JLe*[$z*R&F[r$!1JU\\:'fmS\"F07$ $\"11++DYKpkF[r$\"1Hvdc^=yNFdt7$$\"1lm\"H2qcZ(F[r$\"1:)R[\"pZg;F07$$\" 1/+DJ5fF&)F[r$\"1yL%)*=zil$F07$$\"1lmmTg.c&*F[r$\"1-LOAUO@fF07$$\"1+]i lAFj5F0$\"1fR(Gi?mk)F07$$\"1MLL$)*pp;\"F0$\"1YB[/yUj6F37$$\"1ML3xe,t7F 0$\"1MkY`\\!z]\"F37$$\"1n;HdO=y8F0$\"1\"f`OUA0*=F37$$\"1,++D>#[Z\"F0$ \"1wq$\\g**)zAF37$$\"1nmT&G!e&e\"F0$\"1f\"=3%p$Gx#F37$$\"1MLL$)Qk%o\"F 0$\"1?BAIZ'yD$F37$$\"1+]iSjE!z\"F0$\"1\"**paigH#QF37$$\"1,]P40O\"*=F0$ \"1T\"QE/,@T%F37$$\"\"#F*$\"#^F*-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXE SLABELSG6$Q\"x6\"Q\"yFf[l-%%VIEWG6$;F(Fgz;$!#5F*$\"#?F*" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 448 "Suppose you want an appoximation of the x-intercept in t he above graph. You can do this by clicking on the graph and then put ting the arrow near the intercept and clicking there. A window will a ppear in the upper left giving you the coordinates of the point to whi ch the arrow is pointing. Try this for the above curve and observe th at the x - intercept is about 0.63. Note: This is only an approximati on and should be used as an exact answer. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 30 "An important example Follows. " }}{PARA 0 "" 0 "" {TEXT -1 47 "This shows two of Maples \"built in\" functions ; " }{XPPEDIT 18 0 "exp(g(x));" "6#-%$expG6#-%\"gG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin(g(x));" "6#-%$sinG6#-%\"gG6#%\"xG" } {TEXT -1 64 ". Other built in functions can be found in the help glos sary. " }{TEXT 270 20 "Remember to denote " }{XPPEDIT 18 0 "exp(g(x) );" "6#-%$expG6#-%\"gG6#%\"xG" }{TEXT -1 2 " " }{TEXT 271 14 "by exp( g(x)). " }{TEXT -1 134 "Plotting more than one function requires placi ng square backets around all of the functions and seperating them by c ommas. The number " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 464 " is \+ denoted by \"Pi\". You can choose the color of the individual graphs b y using something like \"color = [red,blue,green]\", but if you are us ing a black and white printer, the \"linestyle\" option is preferred. \+ Similarly, the default line thickness is sometimes not big enouph to s how up on some printers. You can alter this with the \"thickness\" com mand. Finally, you can label the axes with the \"labels\" command and give the plot a title with the \"title\" command. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "plot([f(x),sin(x^2),exp(x/2)], x = -Pi/2 .. Pi/2, y =-2..2, color=[red,green,blue],linestyle=[1,2,3],thickness =2, labels = [\"x\",\"y\"],title=\"Plotting Several Functions\");" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6%7S7$ $!1++lBjzq:!#:$!1`%G)*pQr\\#F*7$$!1NzQW&=B]\"F*$!1n%)efJ$zP#F*7$$!142! 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From this you can estimate the points of intersect ion. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# end of this sect ion" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "c) Numerical Features: The \"fsolve\" and \"int\" command s" }}{PARA 0 "" 0 "" {TEXT -1 126 "Here we will investigate two specif ic numerical features: Finding solutions to equations and evaluating a definite integral. " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 31 "(i) Solv ing equations with the " }{TEXT 268 9 "\"fsolve\" " }{TEXT 275 7 "comm and" }}{PARA 0 "" 0 "" {TEXT -1 453 "Consider the two functions f(x) = exp(x/10) and g(x) = x^2. If you plot these two functions on the sam e graph you will see that they intersect at two points. You can appro ximate these two points by clicking on the graph at the intersection a nd you see that the points of intersection are approximately (-0.95, 0 .90) and (1.06,1.10). However, you can have Maple approximate the x-v alue of these points to greater accuracy with the \"fsolve\" command: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fsolve(x^2 = exp(x/10) ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+r'>T0\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 254 "This command says: \"solve x^2 = exp(x/1 0) for x\". Notice, this found only one answer, but the graph suggests there are two. How do you tell Maple to find the other one? You rest rict the domain of your search by putting in a range of possible x-val ues. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve(x^2 = exp( x/10),x,-2..0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+? " 0 "" {MPLTEXT 1 0 33 "fsolv e(x^2 = exp(x/10),x,-4..-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'fso lveG6%/*$)%\"xG\"\"#\"\"\"-%$expG6#,$F)#\"\"\"\"#5F);!\"%!\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "It just returns your command sugg esting that no solution lies in this interval. If you don't use \"solv e\" instead of \"fsolve\" you are asking Maple to perform a much more \+ complicated task. This will be discussed in the section on symbolic f eatures. " }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 40 "(ii) Evaluating de finite integrals with " }{TEXT 274 6 "\"int\" " }{TEXT -1 7 "command" }}{PARA 0 "" 0 "" {TEXT -1 241 "Suppose we want to appoximate an indef inite integral that we just cannot do by hand. Generally this means we cannot find an antiderivative and therefore cannot apply the fundamen tal theorem of calculus. Here's an example of such a problem: " }} {PARA 0 "" 0 "" {TEXT -1 85 "The following definite integral gives the length of one arch of the curve y = sin(x)." }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "int(sqrt(1+cos(x)^2),x = 0 .. Pi);" "6#-%$ intG6$-%%sqrtG6#,&\"\"\"\"\"\"*$-%$cosG6#%\"xG\"\"#F+/F0;\"\"!%#PiG" } {TEXT -1 92 " and this has no \"closed form\" solution. But we can \+ have Maple approximate this length by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "int(sqrt(1+(cos(x))^2),x = 0..Pi);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&-%%sqrtG6#\"\"#\"\"\"-%*EllipticEG6#,$*$F%F)#\"\" \"F(F0F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This appears ugly but we can evaluate this number by the \"evalf\" command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)y(>?Q!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "This is more important than you may first guess because alot of defin ite integrals have no \"closed form\" solution. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 50 "2. Symbolic Features, Differentiation, Integration" }}{PARA 0 "" 0 "" {TEXT -1 147 "Here we will investigate what makes Ma ple so special. We will demonstrate Maple's ability to solve equation s, differentiate, anti-differentiate. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "a) Solving Equations Symbolically: The \"solve\" command " }}{PARA 0 "" 0 "" {TEXT -1 304 "In a previous example we found that \+ exp(x/10) = x^2 at two values of x. We had Maple numerically approxima te these solutions. Is it possible to have Maple find the exact soluti ons? To find exact solutions, if they can be found, you can use the \" solve\" command which works just like the \"fsolve\" command. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(exp(x/10) = x^2,x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%,$-%)LambertWG6##!\"\"\"#?!#?,$-F%6$F (F'F*,$-F%6##\"\"\"F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Is th is an exact solution? In a pure sense; yes it is, for all practical p urposes; no it isn't. But Maple can get the exact solution to other e quations. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(cos(a* x) = sin(a*x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#PiG\"\"\"% \"aG!\"\"#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Notice, Maple did a good job of finding one solution, however, there are infi nitely many. How about the quadratic equation " }{XPPEDIT 18 0 "a*x^2+ b*x+c = 0;" "6#/,(*&%\"aG\"\"\"*$%\"xG\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF' \"\"!" }{TEXT -1 3 " ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " solve(a*x^2 + b*x + c = 0, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*& ,&%\"bG!\"\"*$-%%sqrtG6#,&*$)F&\"\"#\"\"\"\"\"\"*&%\"aGF1%\"cGF1!\"%F0 F1F0F3!\"\"#F1F/,$*&,&F&F'F(F'F0F3F6F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "This second example results in the two solutions obtained from the quadratic formula. This is the " }{TEXT 277 3 "big" }{TEXT -1 12 " feature of " }{TEXT 276 24 "Computer Algebra Systems" }{TEXT -1 121 ": they can do symbolic manipulations. However, as is the case \+ in the first example, this is not always that practical. " }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 278 47 "b) Differentiation: The \"diff\" and \"D\" commands" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 283 "You can di fferentiate functions in Maple with the use of the \"diff\" or \"D\" \+ command. The \"D\" command is useful in defining new functions that a re derivatives of previous ones. Let's first take a brief look at the \+ \"diff\" command. Suppose we want the derivative of the polynomial \+ " }{XPPEDIT 18 0 "x^2+3-1" "6#,(*$%\"xG\"\"#\"\"\"\"\"$F'\"\"\"!\"\" " }{TEXT -1 108 ", with the \"diff\" command. You put in the function you want to differentiate and the independent variable. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "diff(x^2 + 3 * x - 1, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"#\"\"$\"\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 123 "Don't forget to specify the variable. For example we want to differentiate the following function with respect to t (no t a)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "diff(sin(a*t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#*&%\"aG\"\"\"%\"tGF)F)F( \"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Here Maple used the ch ain rule and got the correct answer. The syntax for taking the second derivative is a little strange. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "Suppose we want the " }{TEXT 281 17 "seco nd derivative" }{TEXT -1 116 " of sin(a t). Maple denotes the second d erivative with the \"$2\" sign or the n'th derivative with the \"$n\" \+ symbol. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(sin(a*t) ,t$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#*&%\"aG\"\"\"% \"tGF*F*)F)\"\"#\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "T he problem with the \"diff\" command is that it's difficult to assign \+ another function name to the derivative. You may try this by assignin g the derivative of \"g\" the name \"gprime\":" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "g := x -> exp(3*x); # g is an exponential fun ction" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operator G%&arrowGF(-%$expG6#,$9$\"\"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gprime := x -> diff(g(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gprimeGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%%diffG6$- %\"gG6#9$F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "This looks o k but problems occur if you try to evaluate gprime at a specific numbe r," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "gprime(0);" }}{PARA 8 "" 1 "" {TEXT -1 72 "Error, (in gprime) wrong number (or type) of pa rameters in function diff" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "whe re Maple returns the above error message. The problem is Maple replac es the \"x\" with a zero and then tries to differentiate with respect \+ to \"x\" which is now zero as well. To get around this problem the \" D\" command works well" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "g prime := 'gprime'; # clears gprime" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'gprimeGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "gprime := \+ x -> D(g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gprimeGR6#%\"xG6\" 6$%)operatorG%&arrowGF(,$-%$expG6#,$9$\"\"$F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "or " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gprime := D(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gprimeGR6#% \"xG6\"6$%)operatorG%&arrowGF(,$-%$expG6#,$9$\"\"$F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "Here we still must input the function na me as well as the variable but in a different format from the \"diff\" command. This results in a function that we can evaluate" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "gprime(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Change th e colon to a semicolon and see if it plots ok. " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "plot(gprime(x),x=-1..2):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 134 "This works out. Summary: Use \"diff(f(x),x)\" to differentiate. Use \"D(f)(x)\" when creating a new function for evalu ation or plotting. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# en d of this section" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 279 43 "c) Antidi fferentiation: The \"int\" command. " }}{PARA 0 "" 0 "" {TEXT -1 210 " The command for antidifferentiation is the same as the command used in evaluating definite integrals: \"int\", except, instead of putting i n the bounds of integration you just put in the independent variable. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "restart; #This clears all previous variables" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Suppos e we want to find the antiderivative (indefinite integral) of " } {XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 26 " using the \"int\" command. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "int(x^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"xG\"\"$\"\"\"#\"\"\"F'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Notice this gives only one antider ivative, " }{TEXT 280 61 "where the constant of integration is assigne d the value zero." }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The same command with integration bounds evaluates the definite in tegral" }{XPPEDIT 18 0 "int(x^2,x = 0 .. 2);" "6#-%$intG6$*$%\"xG\"\"# /F';\"\"!\"\"#" }{TEXT -1 4 " by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(x^2, x=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# #\"\")\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "This was an easy \+ example and hopefully one you could do in your head. Let's try a more \+ difficult example. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "h : = x -> exp(x)/(exp(x) + 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR 6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$expG6#9$\"\"\",&F-\"\"\"F3F3!\" \"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "recall that with the substitution u = 1 + e^x, the integrand has the form du/u and we fou nd the following antiderivative" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(h(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&-%$e xpG6#%\"xG\"\"\"F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "However, suppose we multiply the numerator and denominator of \"h\" by exp(-x ) to get the equivalent function" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g := x -> 1/(1 + exp(-x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&\" \"\"F/-%$expG6#,$9$!\"\"F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Here, no u-substitution is obvious. So we let Maple do it " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(g(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&\"\"\"F(-%$expG6#,$%\"xG!\"\"F(F (-F%6#F)F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 206 "Is this the same a nswer? It should be. Careful use of the properties of logarithms will result in the same answer. You may wish to avoid such trivial tasks b y requesting Maple to simplify the expression by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&\"\"\"F(-%$expG6#,$%\"xG!\"\"F(F(-F%6#F)F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "This didn't help at all. The \"simplify \" command can only do limited simpifications. For example:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify((cos(x))^2 + (sin(x ))^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 209 "The simplify command is useful for some trig expr essions, some exponential and logarithmic expressions, and some algebr aic expressions, however it by no means defines the simplest from of a ll expressions. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# e nd of this section" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "3. Solving Differential Equations" } {TEXT 260 20 ": The dsolve command" }}{PARA 0 "" 0 "" {TEXT -1 71 "Dif ferential equations and initial value problems are solved using the " }{TEXT 288 6 "dsolve" }{TEXT -1 10 " command. " }}{PARA 0 "" 0 "" {TEXT -1 7 "usage: " }}{PARA 0 "" 0 "" {TEXT 257 7 "dsolve(" }{TEXT -1 0 "" }{TEXT 258 1 " " }{TEXT -1 25 "the differential equation" } {TEXT 282 1 " " }{TEXT -1 24 ", the dependent variable" }{TEXT 283 4 " ) " }{TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 10 " dsolve( \{ " }{TEXT -1 0 "" }{TEXT 262 45 "a differential equation, in itial condition(s)" }{TEXT -1 0 "" }{TEXT 263 1 " " }{TEXT -1 0 "" } {TEXT 264 4 "\}, " }{TEXT -1 22 "the dependent variable" }{TEXT 259 2 " )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " I start most sections with the " }{TEXT 296 7 "restart" }{TEXT -1 55 " command to clear all previous variable definitions by " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 9 "Example 1" }{TEXT -1 71 ": Find the general solution to the first order differential equation: " } {TEXT 305 8 "y' = 3 y" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sol 1 := dsolve(diff(y(x),x) = 3 * y(x), y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G/-%\"yG6#%\"xG*&%$_C1G\"\"\"-%$expG6#,$F)\"\"$F ," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Notice: The " }{XPPEDIT 18 0 "_C1;" "6#%$_C1G" }{TEXT -1 34 " indicates an arbitrary constant. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 9 "Exampl e 2" }{TEXT -1 39 ": Consider the initial value problem " }{TEXT 297 8 "y' = 3 y" }{TEXT -1 5 " , " }{TEXT 298 12 "y(0) = 1/2. " }} {PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 299 1 "a" }{TEXT -1 21 ") Find t he solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sol2 := dso lve(\{diff(y(x),x) = 3*y(x), y(0) = 1/2\}, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G/-%\"yG6#%\"xG,$-%$expG6#,$F)\"\"$#\"\"\"\"\"# " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Notice: Maple automatically s olves for the arbitrary constant based on the initial condition. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 300 1 "b" }{TEXT -1 41 ") Plot the solution from x = 0 to x = 1. " }}{PARA 0 "" 0 "" {TEXT -1 96 "First we want to access the right hand side (rhs) of sol2 and make it a function of x called y2." }}{PARA 0 "" 0 "" {TEXT -1 22 "This is done with the " }{TEXT 301 7 "unapply" }{TEXT -1 5 " and " }{TEXT 302 3 "rhs" }{TEXT -1 24 " commands as follows. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y2 := unapply(rhs(sol2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2GR6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%$expG6 #,$9$\"\"$#\"\"\"\"\"#F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Y ou can plot the solution from x = 0 to x = 1 by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(y2(x),x=0..1,labels=[\"x\",\"y2(x)\"],ti tle=\"The solution to y' = 3y, y(0) = 1/2\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$\"\"!$\"1+++++++] !#;7$$\"1nmm;arz@!#<$\"1h&3XH%)yL&F+7$$\"1LL$e9ui2%F/$\"1gX!p&))R]cF+7 $$\"1nmm\"z_\"4iF/$\"1z`$G5lP-'F+7$$\"1mmmT&phN)F/$\"1RB>Jp_CkF+7$$\"1 LLe*=)H\\5F+$\"1;4C8U&)\\oF+7$$\"1nm\"z/3uC\"F+$\"1%QT)pBIpsF+7$$\"1++ DJ$RDX\"F+$\"1sDH'H-2t(F+7$$\"1nm\"zR'ok;F+$\"1TK6T/rQ#)F+7$$\"1++D1J: w=F+$\"1ulmc%4$y()F+7$$\"1LLL3En$4#F+$\"1R:)HB\\-P*F+7$$\"1nm;/RE&G#F+ $\"1*Hj'fbgC**F+7$$\"1+++D.&4]#F+$\"1%HMq#=!)e5!#:7$$\"1+++vB_BO\"Fco7$$\"1LLLLY.KNF+$\"1 \\SlCLiU9Fco7$$\"1++D\"o7Tv$F+$\"1bio*y4?a\"Fco7$$\"1LLL$Q*o]RF+$\"1&p c#f;oN;Fco7$$\"1++D\"=lj;%F+$\"1*oVFk8]u\"Fco7$$\"1++vV&R#*[l8z>Fco7$$\"1LLeR\"3Gy%F+$\"1c&fjo \"\\*4#Fco7$$\"1nm;/T1&*\\F+$\"1,(=\"=)GvB#Fco7$$\"1mm\"zRQb@&F+$\"1t \"=N=G0R#Fco7$$\"1***\\(=>Y2aF+$\"1FI&fN2A`#Fco7$$\"1mm;zXu9cF+$\"1bM& **>rYp#Fco7$$\"1+++]y))GeF+$\"1@)>rfmM(GFco7$$\"1****\\i_QQgF+$\"1i:#* ew&)fIFco7$$\"1***\\7y%3TiF+$\"1k&)yJ5q^KFco7$$\"1****\\P![hY'F+$\"1D! [d " 0 "" {MPLTEXT 1 0 16 "subs(x=1,y2(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $-%$expG6#\"\"$#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "an d get a numerical value to 6 significant digits by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(%,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'G/5!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The % symbol references Maple's last output. Not necessarily the line above it." }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 304 2 "d)" }{TEXT -1 74 " Find the value of x for which y2 is 100. Ie. Solve y2(x) = 100 for x. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(y2(x) = 100, x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#\"$+##\"\"\"\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(%,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'5m " 0 "" {MPLTEXT 1 0 43 "ode3 := diff( y(x),x$2)+diff(y(x),x)+y(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%o de3G/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2F*F2\"\" !" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "and then solve it using the \+ " }{TEXT 290 6 "dsolve" }{TEXT -1 8 " command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve(ode3, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(%$_C1G\"\"\"-%$expG6#,$F'#!\"\"\"\"#F +-%$cosG6#,$*&-%%sqrtG6#\"\"$\"\"\"F'F+#F+F2F+F+*(%$_C2GF+F,F<-%$sinGF 5F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Again, " }{XPPEDIT 18 0 "_C1;" "6#%$_C1G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "_C2;" "6#%$_C2 G" }{TEXT -1 31 " indicate arbitrary constants. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 9 "Example 4" }{TEXT -1 59 " : Find and plot the solution to the initial value problem: " }{TEXT 307 37 "y'' + y' + y = 0, y(0) = 1, y'(0) = 0" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dsolve(\{ode3,y(0)=1,D(y)(0) =0\}, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%$e xpG6#,$F'#!\"\"\"\"#\"\"\"-%$cosG6#,$*&-%%sqrtG6#\"\"$\"\"\"F'F1#F1F0F 1F1*(F7F;F*F;-%$sinGF4F1#F1F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 " Again, Maple solves for the arbitrary constants based on the initial \+ conditions. " }}{PARA 0 "" 0 "" {TEXT -1 55 "Define y4 as the right ha nd side of the previous output" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y4 := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #y4GR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&-%$expG6#,$9$#!\"\"\"\"#\"\" \"-%$cosG6#,$*&-%%sqrtG6#\"\"$\"\"\"F2F6#F6F5F6F6*(F " 0 "" {MPLTEXT 1 0 100 "plot(y4(x), x=0..10,labels=[\"x\",\"y4\"],title=\"The solution to y'' + y' + y = 0 , y(0) = 1, y'(0) = 0\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$\"\"!$\"\"\"F(7$$\"1LL$3FWYs#!#<$\"1#yg q'=K'***!#;7$$\"1mmmT&)G\\aF.$\"1mzdzAU&)**F17$$\"1++]7G$R<)F.$\"1$3o> K.v'**F17$$\"1LLL3x&)*3\"F1$\"1xE>(ymF%**F17$$\"1++]ilyM;F1$\"1?z:lekt )*F17$$\"1nmm;arz@F1$\"1#3@A\"Gmz(*F17$$\"1++D\"y%*z7$F1$\"1Mm-v[bh&*F 17$$\"1LL$e9ui2%F1$\"1x^lF3@\"G*F17$$\"1nmm\"z_\"4iF1$\"1w.tAgRk%)F17$ $\"1ommT&phN)F1$\"1Vlg^B!>X(F17$$\"1LLe*=)H\\5!#:$\"1:^i>\"*eKjF17$$\" 1nm\"z/3uC\"F\\o$\"1XO.[-Da_F17$$\"1++DJ$RDX\"F\\o$\"1E%oQ'[!f9%F17$$ \"1nm\"zR'ok;F\\o$\"1JQvaN%40$F17$$\"1++D1J:w=F\\o$\"1V2Psq4X?F17$$\"1 MLL3En$4#F\\o$\"1Qm8=`(\\7\"F17$$\"1nm;/RE&G#F\\o$\"1;[/#oBtB%F.7$$\"1 +++D.&4]#F\\o$!1+pX2R*>O#F.7$$\"1+++vB_c%*fc3Q\"F17$$\"1MLL347TL F\\o$!1\"y0(yQ2d:F17$$\"1MLLLY.KNF\\o$!1b6n*z`Ei\"F17$$\"1++D\"o7Tv$F \\o$!1X[K[x$yh\"F17$$\"1LLL$Q*o]RF\\o$!1y*e\\+hVb\"F17$$\"1,+D\"=lj;%F \\o$!1=Nz_&\\cV\"F17$$\"1++vV&RY2aF\\o$!1F suENugSF.7$$\"1nm;zXu9cF\\o$!1V?,(y%4VDF.7$$\"1+++]y))GeF\\o$!1$o&G1Hb q6F.7$$\"1++]i_QQgF\\o$!1@tbjT-=P!#>7$$\"1,+D\"y%3TiF\\o$\"1/1W&Q7*o&) !#=7$$\"1++]P![hY'F\\o$\"1$z[z/)e?;F.7$$\"1LLL$Qx$omF\\o$\"1j:5,Y^7@F. 7$$\"1+++v.I%)oF\\o$\"1mP?6Et_CF.7$$\"1mm\"zpe*zqF\\o$\"1y[SH:![h#F.7$ $\"1,++D\\'QH(F\\o$\"1deYx=.cEF.7$$\"1LLe9S8&\\(F\\o$\"1f::cee(e#F.7$$ \"1,+D1#=bq(F\\o$\"1Xl)e)GcGCF.7$$\"1LLL3s?6zF\\o$\"1:\"fikE(3AF.7$$\" 1++DJXaE\")F\\o$\"1wyY\\<]K>F.7$$\"1ommm*RRL)F\\o$\"1_xwmw\"4k\"F.7$$ \"1om;a<.Y&)F\\o$\"1FrgL]jL8F.7$$\"1NLe9tOc()F\\o$\"1?G'fquO.\"F.7$$\" 1,++]Qk\\*)F\\o$\"1&eVk)GlS'f@pas(Fgv7$ $\"1,+v=5s#y*F\\o$!1E " 0 "" {MPLTEXT 1 0 58 "ode3 :=5*diff(y(x),x,x)+10*sqrt(2)* diff(y(x),x)+10*y(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode3G/,( -%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"&*&-%%sqrtG6#F1\"\"\"-F(6$F* F-\"\"\"\"#5F*F;\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ds olve(\{ode3\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG *&%$_C1G\"\"\"-%$expG6#,$*&-%%sqrtG6#\"\"#\"\"\"F'F*!\"\"F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "HEY! That's not right! There shoul d be two linearly independent solutions. " }}{PARA 0 "" 0 "" {TEXT -1 50 "Check the roots of the characteristic equation by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solve(5*r^2 + 10*sqrt(2)*r + 10=0,r );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*$-%%sqrtG6#\"\"#\"\"\"!\"\"F# " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "There is a double root: r = " }{XPPEDIT 18 0 "-sqrt(2);" "6#,$-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 52 " so a second (linearly independent) solution is x " }{XPPEDIT 18 0 "exp(-sqrt(2)*x);" "6#-%$expG6#,$*&-%%sqrtG6#\"\"#\"\"\"%\"xGF,! \"\"" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 42 "Maple did not f ind this second solution. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# end of section" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "4. Mech anical Vibrations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart ; # clears previous variables" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Here we consider the differential equatio n describing the motion of a mass on a spring" }}{PARA 256 "" 0 "" {TEXT -1 8 "m u'' + " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT -1 16 " u' + k u = F(t)" }}{PARA 0 "" 0 "" {TEXT -1 74 "Here, u is the di placement from equilibrium, m is the mass of the object, " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT -1 136 " is the damping constant, k \+ is the spring constant, and F(t) is some external forcing term. All of the constants are positive: a larger " }{XPPEDIT 18 0 "gamma;" "6#%&g ammaG" }{TEXT -1 66 " implies greater damping and a larger k implies a stiffer spring. " }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 31 " > 0, the motion is considered " } {TEXT 309 6 "damped" }{TEXT -1 7 ", if " }{XPPEDIT 18 0 "gamma" "6#% &gammaG" }{TEXT -1 19 " = 0 the motion is " }{TEXT 308 9 "undamped." } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 90 "We can \+ define the left side (lhs) of this ordinary differential equation (ode ) as follows " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "lhsode := m *diff(u(t),t$2) + gam*diff(u(t),t) + k*u(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lhsodeG,(*&%\"mG\"\"\"-%%diffG6$-%\"uG6#%\"tG-%\"$G6 $F/\"\"#F(F(*&%$gamGF(-F*6$F,F/F(F(*&%\"kGF(F,F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "We can get a look at the general solution of the \+ homogeneous equation by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " dsolve(lhsode=0,u(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6#%\" tG,&*&%$_C1G\"\"\"-%$expG6#,$*&*&,&%$gamGF+*$-%%sqrtG6#,&*$)F3\"\"#\" \"\"F+*&%\"mGF+%\"kGF+!\"%F!\"\"#!\"\"F;F+F+*&%$_C2GF+-F- 6#,$*&*&,&F3FCF4F+F+F'FFA#F+F;F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 205 "This is a little messy but you can see how the parameters m, gam, and k effect our solution. If the term under the radical sign is neg ative the solution will contain sine and cosine terms of the form cos( " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 12 " t) and sin(" } {XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 10 " t) where " } {XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 " sqrt(abs(gam^2-4*mk))/(2*m)" "6#*&-%%sqrtG6#-%$absG6#,&*$%$gamG\"\"#\" \"\"*&\"\"%F.%#mkGF.!\"\"F.*&\"\"#F.%\"mGF.F2" }{TEXT -1 18 ". In th is case, " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 23 " (omega) is called the " }{TEXT 311 9 "frequency" }{TEXT -1 31 " of the sine a nd cosine terms. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 310 30 "Damped Oscillator, no Forcing:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Suppose m=gamma=k=1 and F(t) = 0. We assi gn these by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m := 1:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gam := 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k := 5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "F := t -> 0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 " Let's take a look at the general solution of this homogeneous equatio n. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dsolve(lhsode=F(t),u (t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6#%\"tG,&*(%$_C1G\"\" \"-%$expG6#,$F'!\"\"F+-%$sinG6#,$F'\"\"#F+F+*(%$_C2GF+F,\"\"\"-%$cosGF 3F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Notice, the " }{TEXT 256 9 "frequency" }{TEXT -1 24 " of the trig terms is " }{XPPEDIT 18 0 "sqrt(abs(2^2-4(1)(5)))/2(1);" "6#*&-%%sqrtG6#-%$absG6#,&*$\"\"# \"\"#\"\"\"--\"\"%6#\"\"\"6#\"\"&!\"\"F.-\"\"#6#\"\"\"F6" }{TEXT -1 80 " = 2. However, the solution is not periodic because of the expon ential term. " }}{PARA 0 "" 0 "" {TEXT -1 44 "In this case the soluti on is said to have a " }{TEXT 312 15 "quasi-frequency" }{TEXT -1 8 " o f 2. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "Suppose that we displ ace the mass 1 unit beyond equilibrium and let it go. The initial con ditions corresponding to this statement are u(0)=1 and u'(0) = 0. Thes e can be assigned values by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "uo := 1: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "uprimeo : = 0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Now we can find and plot \+ the solution to the initial value problem by " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "sol := dsolve(\{lhsode = 0,u(0) = uo, D(u)(0) \+ = uprimeo\},u(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"uG6# %\"tG,&*&-%$expG6#,$F)!\"\"\"\"\"-%$sinG6#,$F)\"\"#F1#F1F6*&F,\"\"\"-% $cosGF4F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u1 := unappl y(rhs(sol),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1GR6#%\"tG6\"6$% )operatorG%&arrowGF(,&*&-%$expG6#,$9$!\"\"\"\"\"-%$sinG6#,$F2\"\"#F4#F 4F9*&F.\"\"\"-%$cosGF7F4F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "plot(u1(t),t=0..10,labels=[\"t\",\"u(t)\"],title=\"t he solution to u'' + 2u' + 5u = 0, u(0)=1, u'(0)=0\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7co7$\"\"!$\"\" \"F(7$$\"1LLL3x&)*3\"!#;$\"1`]R,s%[s*F.7$$\"1nmm;arz@F.$\"1VsJ![,s)*)F .7$$\"1++D\"y%*z7$F.$\"1dHe6(e-2)F.7$$\"1LL$e9ui2%F.$\"1B>(z]RC)pF.7$$ \"1n;H2Q\\4YF.$\"1>)=&39xBjF.7$$\"1++voMrU^F.$\"1'[s?Kflk&F.7$$\"1L$3- 8Lfn&F.$\"1?!H0-+='\\F.7$$\"1nmm\"z_\"4iF.$\"1=Eg`&z%zUF.7$$\"1nm;zp!f u'F.$\"19obK(\\Tg$F.7$$\"1nmmm6m#G(F.$\"1;s6F9R[HF.7$$\"1om;a`T>yF.$\" 1$3))3re\">BF.7$$\"1ommT&phN)F.$\"1zgsFfLA*Fgo7$$\"1nm\"z/3uC\"F[p$!1K+wsa#oU\"F.7$$\"1LLe*ot*\\8F[p$!1Y$*oQdn* y\"F.7$$\"1++DJ$RDX\"F[p$!1VF)y)p\\+?F.7$$\"1LLe9x0z9F[p$!1ESlr1UK?F.7 $$\"1nm\"z4wb]\"F[p$!1qr+Sttb?F.7$$\"1++D\"[%4K:F[p$!1aENi<\"32#F.7$$ \"1LLekGhe:F[p$!1m$fL3=!y?F.7$$\"1nm\"zCJ^e\"F[p$!1j3zd!Qx2#F.7$$\"1++ DJ'\\;h\"F[p$!1KxY]$f.2#F.7$$\"1LLe9!o\"Q;F[p$!1D?\"Q!RFc?F.7$$\"1nm\" zR'ok;F[p$!1r:2vj(e.#F.7$$\"1LL3_(>/x\"F[p$!1`CPh`j**=F.7$$\"1++D1J:w= F[p$!1qka\\y4%p\"F.7$$\"1MLL3En$4#F[p$!11S3t>)3:\"F.7$$\"1nm;/RE&G#F[p $!1yt!H*p\"[Z'Fgo7$$\"1+++D.&4]#F[p$!1L#fCAP&)e\"Fgo7$$\"1+++vB_(R*G-\"*=Fgo7$$\"1+++Dg(=#GF[p$\"1[5?az)))*HFgo7$$\"1+++v'Hi#HF[p $\"12n&oMoYC%Fgo7$$\"1MLL347TLF[p $\"1J9^L7G\\RFgo7$$\"1MLLLY.KNF[p$\"19]S]Rd1JFgo7$$\"1++D\"o7Tv$F[p$\" 1Z$>kw@b*=Fgo7$$\"1LLL$Q*o]RF[p$\"1;'Q]ak#)p)!#=7$$\"1,+D\"=lj;%F[p$!1 1e:bHH=E!#>7$$\"1++vV&Rx))Fex7$$\"1nm;/T1&*\\F[p$!10?)H; 59`(Fex7$$\"1nm\"zRQb@&F[p$!1>qDO`B+_Fex7$$\"1++v=>Y2aF[p$!1lGG2^[5IFe x7$$\"1nm;zXu9cF[p$!15k06]!oF*F[y7$$\"1+++]y))GeF[p$\"1]-y3Pn'['F[y7$$ \"1++]i_QQgF[p$\"1)QR.\"fFW:Fex7$$\"1,+D\"y%3TiF[p$\"1G84Np$*e=Fex7$$ \"1++]P![hY'F[p$\"1*>Px7w/t\"Fex7$$\"1LLL$Qx$omF[p$\"1T+Enl6a8Fex7$$\" 1+++v.I%)oF[p$\"1=\"*e04?j%)F[y7$$\"1mm\"zpe*zqF[p$\"1X\")Hbk`J\"F[y7$$\"1NLe9tOc()F[p$!1vt[O!>x+%!#?7$$\"1,++]Qk\\*)F[p $\"16pr')[/nAF[`l7$$\"1NL$3dg6<*F[p$\"1#\\NF1;Vc'F[`l7$$\"1ommmxGp$*F[ p$\"1$GF)Q+b0!)F[`l7$$\"1++D\"oK0e*F[p$\"1+\"R;vu>j(F[`l7$$\"1,+v=5s#y *F[p$\"1>7<*\\J\"3hF[`l7$$\"#5F($\"1@sO2A2DRF[`l-%'COLOURG6&%$RGBG$Fga l!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q%u(t)Fdbl-%&TITLEG6#QTthe~solution~t o~u''~+~2u'~+~5u~=~0,~u(0)=1,~u'(0)=0Fdbl-%%VIEWG6$;F(Ffal%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Voila! A damped oscillating solution. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 293 33 "Undamped Oscillato r with Forcing:" }{TEXT -1 20 " m u'' + k u = F(t)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "restart: #clears all variable definitions . " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "lhsode := m*diff(u(t) ,t$2) + k*u(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lhsodeG,&*&%\"mG \"\"\"-%%diffG6$-%\"uG6#%\"tG-%\"$G6$F/\"\"#F(F(*&%\"kGF(F,F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "We can get a look at the general s olution of the homogeneous equation by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dsolve(lhsode=0,u(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6#%\"tG,&*&%$_C1G\"\"\"-%$sinG6#*&-%%sqrtG6#*&%\"kG\"\"\" %\"mG!\"\"F5F'F+F+F+*&%$_C2GF+-%$cosGF.F+F+" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 85 "In this case, the solution of the homogeneous equation \+ is periodic with frequency is " }{XPPEDIT 18 0 "sqrt(k/m);" "6#-%%sqrt G6#*&%\"kG\"\"\"%\"mG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 93 "The frequency of the solution to the homogeneous solution in th e undamped case is called the " }{TEXT 313 19 "natural frequency. " }} {PARA 0 "" 0 "" {TEXT -1 46 "Suppose k = 4 and m = 1. Let's see what w e get" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "m := 1: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k := 4:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 53 "The solution to the homogeneous equation is found \+ by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dsolve(lhsode=0,u(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6#%\"tG,&*&%$_C1G\"\"\"-% $sinG6#,$F'\"\"#F+F+*&%$_C2GF+-%$cosGF.F+F+" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Notice: The " }{TEXT 287 17 "natural frequency" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "sqrt(abs (0^2-4(1)(4)))/2(1);" "6#*&-%%sqrtG6#-%$absG6#,&*$\"\"!\"\"#\"\"\"--\" \"%6#\"\"\"6#\"\"%!\"\"F.-\"\"#6#\"\"\"F6" }{TEXT -1 6 " = 2. " }} {PARA 0 "" 0 "" {TEXT -1 42 "Now assign the function F(t) = sin(3t) by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "F := t -> sin(3*t): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "Notice: The frequency of the f orcing term is 3 and the natural frequency is 2. Suppose we start the \+ spring from the resting position by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "uo := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "uprimeo := 0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "and find the so lution " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sol := dsolve(\{ lhsode=F(t),u(0)=uo, D(u)(0)=uprimeo\},u(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"uG6#%\"tG,(*&,&-%$cosG6#,$F)\"\"&#!\"\"\"#? -F.F(#F3\"\"%\"\"\"-%$sinG6#,$F)\"\"#F8F8*&,&-F:F(F6-F:F/#F8F4F8-F.F;F 8F8F9#\"\"$\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "u2 := un apply( rhs(sol), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2GR6#%\"tG 6\"6$%)operatorG%&arrowGF(,(*&,&-%$cosG6#,$9$\"\"&#!\"\"\"#?-F06#F3#F6 \"\"%\"\"\"-%$sinG6#,$F3\"\"#FF9F:-F>F1#F " 0 "" {MPLTEXT 1 0 20 "plot(u2(t) ,t=0..60);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-% 'CURVESG6$7[dm7$\"\"!F(7$$\"1+++DJdpK!#;$\"1lHUP!*GH;!#<7$$\"1+++]i9Rl F,$\"1t,PZ?xZ5F,7$$\"1++]7G$R<)F,$\"1btIh:p@(3)*F,$ 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" 0 "" {MPLTEXT 1 0 17 " # end of section " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Assignment " }}{PARA 0 "" 0 "" {TEXT -1 82 "Consider the Undamped Oscillator with Forcing defined by the initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 257 43 "u'' + 4u = F(t), u(0) = 0 and u'(0) = 0" }}{PARA 0 "" 0 "" {TEXT -1 41 "with forcing terms F(t) described b elow. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 256 14 "(a) Resonance: " }{TEXT -1 75 " The forcing term is periodic with frequency equal to \+ the natural frequency" }}{PARA 0 "" 0 "" {TEXT 258 0 "" }{TEXT 259 12 " (b) Beats: " }{TEXT -1 75 "The forcing term is periodic with frequen cy close to the natural frequency." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "For each case: " }}{PARA 0 "" 0 "" {TEXT -1 27 "1. Define the forcing term." }}{PARA 0 "" 0 "" {TEXT -1 23 "2. \+ Find the solution. " }}{PARA 0 "" 0 "" {TEXT -1 79 "3. Plot the solut ion over a long enough time period that a pattern is apparent." }} {PARA 0 "" 0 "" {TEXT -1 58 "4. Describe what is going on in terms of \+ the oscillations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 37 "- You may work in pairs if you like. " }}{PARA 0 "" 0 " " {TEXT 315 53 "- Five of the points will be based on presentation. \+ " }}{PARA 0 "" 0 "" {TEXT 317 166 "- You may write up the lab in any t ext editor you want but make sure you can cut and paste Maple grap hics into the document and describe mathematical equations. " }} {PARA 0 "" 0 "" {TEXT 318 51 "- I suggest you hand in a printed Maple \+ document. " }}{PARA 0 "" 0 "" {TEXT 319 178 "- Only hand in the infor mation requested by 1 through 4 above. A lot of Maple garbage will co st you points. You should only need two pages; one for each type of \+ forcing term. " }}{PARA 0 "" 0 "" {TEXT 320 80 "- To enter the text ed itor in Maple, click on the \"T\" button from the tool bar. " }}{PARA 0 "" 0 "" {TEXT 321 72 "- To get a Maple prompt, click on the \"[>\" b utton from the tool bar. " }{TEXT -1 0 "" }}}}{MARK "5" 0 } {VIEWOPTS 1 1 0 1 1 1803 }