{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 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 256 "" 0 "" {TEXT -1 49 "Introduction to Maple, P re-Differential Equations" }}{PARA 0 "" 0 "" {TEXT -1 20 "Maple is a t ype of " }{TEXT 258 25 "Computer Algebra System. " }{TEXT -1 382 "Thi s lab introduces you to some of the features of Maple. Some of these f eatures are simple graphing and numerical capabilities similar to thos e of a graphing calculator, and some are symbolic manipulations exclu sive 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 and try topic or full text search." }}{PARA 257 "" 0 "" {TEXT -1 42 "Clic k 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%7[p7 $$!1++lBjzq:!#:$\"1+1e.'fEC'!#;7$$!1o*=SVdl`\"F*$\"1MzZcS%p.(F-7$$!1Nz QW&=B]\"F*$\"10%e8f;ot(F-7$$!1AVfpu_s9F*$\"1@!eFu;sE)F-7$$!142![ROFW\" F*$\"1kDl2\\1C()F-7$$!1VSW:KB49F*$\"1Z\")[q3f]\"*F-7$$!1xt3O+tv8F*$\"1 sIg&>gl[*F-7$$!1F-\"fx/?M\"F*$\"1X$*y5BDO(*F-7$$!1yIt:&z#38F*$\"1,[\"= !>/,**F-7$$!1.=yIq\\\"H\"F*$\"1p_X%yGG&**F-7$$!1H0$ea9ZF\"F*$\"1I79h!p `)**F-7$$!1#*[N.LKm7F*$\"1w0Sp$GY***F-7$$!1a#z31KzD\"F*$\"1&f&H&eF$*** *F-7$$!1\"F*$\"1u!4#H%[k*)*F-7$$!1:#)\\G:\"*y6F*$\"1iD0HmqO)*F-7$$!1b c#>e*oY6F*$\"16`iE0Ou'*F-7$$!1%4``jnW6\"F*$\"1YH&G0SWY*F-7$$!1oW()o'>y /\"F*$\"1xhhArk-*)F-7$$!1WfpKW&Q\")*F-$\"1o*RQpa(4#)F-7$$!1'e1Vw'\\I\" *F-$\"1wDKXF'RS(F-7$$!13*H!e\\fG&)F-$\"1H`GS,2\\mF-7$$!1`u@%3'*4&yF-$ \"1mj3$ei3y&F-7$$!1mnKC\\hqrF-$\"1fp\\:f==\\F-7$$!1I*3_gT\\^'F-$\"1%>2 ![w9=TF-7$$!1!RBoTF&>fF-$\"1))e;%)3\"GV$F-7$$!1-(RrNA:@&F-$\"1gHmNzs#o #F-7$$!1[5T-#\\F-$\"1ll5C\"eY*QF]v7$$!1SWU;s`+8F-$\"1`5(fS;8p\"F]v7$$!1IU\"3% yK9**F]v$\"1NzTt8BH)*!#=7$$!1gSQF -$\"1DvW-j'*GPF]v7$$\"11j*\\(z-/EF-$\"1wfG\\iwvnF]v7$$\"1#oMd]$=iKF-$ \"1')))fLmfF-$\"1=YWmu +LMF-7$$\"1\"oi?&HQMlF-$\"1O)H^!HDTTF-7$$\"1r$zA=*Q1sF-$\"1hMUqH!H'\\F -7$$\"1IJCuYpQyF-$\"1yy\\hI5ldF-7$$\"1'G\"*Q5O'*\\)F-$\"1T/^rr:7mF-7$$ \"1inYgr#e9*F-$\"1?*onvmFU(F-7$$\"1E5$3JHB#)*F-$\"1I#H!o\\C>#)F-7$$\"1 h>eG\")QZ5F*$\"1hFf;b_)*))F-7$$\"1K#)fG(=S6\"F*$\"1jYK-a?h%*F-7$$\"1p< 5h\"eq9\"F*$\"1W+!z(p\\w'*F-7$$\"10`g$f(4!=\"F*$\"1LF!))y.<%)*F-7$$\"1 @$pr\\x_>\"F*$\"1E$G)\\j>**)*F-7$$\"1OLt+uX57F*$\"1FKjI*)*F-7$$\"1wn3& [I:M\"F*$\"1[))4/$[\"R(*F-7$$\"1%yp@B_EP\"F*$\"1.(ek/xH^*F-7$$\"15:p0X $eS\"F*$\"1=ZF-7$F> $\"1GUs\"Qn3'[F-7$FH$\"1+:m)o![E]F-7$FR$\"1v:)y+\"*))>&F-7$Fjo$\"1<*HX A`jP&F-7$F^q$\"1iDp:0HYbF-7$Fhq$\"1iLf'>8zs&F-7$F]r$\"1r^,$41?#fF-7$Fb r$\"1@j\"eh@?7'F-7$Fgr$\"1)3Qlu7[L'F-7$F\\s$\"1bSWqsNGlF-7$Fas$\"1)\\A 56FLv'F-7$Ffs$\"1C4hI9-()pF-7$F[t$\"1^.+$px)>sF-7$F`t$\"1d4*H.]!QuF-7$ Fet$\"1?J_r\\21xF-7$Fjt$\"1j3rF)p1%zF-7$F_u$\"1\\Ea$zfDA)F-7$Fdu$\"1R5 \"4A=/[)F-7$Fiu$\"1i3^NAks()F-7$F_v$\"1xs))4KEg!*F-7$Fdv$\"1C%QJ$HUq$* F-7$F_w$\"1\\4l_)*ek'*F-7$Fiw$\"1Du'yr\\A***F-7$Fdx$\"1?jT`jVM5F*7$F^y $\"1i4XHm4m5F*7$Fcy$\"1EAv/*z85\"F*7$Fhy$\"1(4N]wd!R6F*7$F]z$\"14:-+_; x6F*7$Fbz$\"19(p(y\"\\_@\"F*7$Fgz$\"1.&*)y:!)*e7F*7$F\\[l$\"1ET-^]h*H \"F*7$Fa[l$\"1\\V\\s-XW8F*7$Ff[l$\"1^/NX?T'Q\"F*7$F[\\l$\"1?:&yP(yL9F* 7$F`\\l$\"1%f8X7T)z9F*7$Fe\\l$\"1#[;$*ei&H:F*7$Fj\\l$\"1'3K95*yz:F*7$F _]l$\"1TQ[i'RTj\"F*7$Fd]l$\"1i))oeED)o\"F*7$Fi]l$\"1a!RxpWau\"F*7$Fc^l $\"1\\BTYk2/=F*7$Fg_l$\"1?HNixof=F*7$Feal$\"1ws.(*f`D>F*7$F_bl$\"13HO# H/k)>F*7$Fibl$\"1dG9yMU`?F*7$Fccl$\"1i^*4?'o>@F*7$F]dl$\"1'>*GZ+G$>#F* -F`dl6&FbdlFcdlFcdlFddl-Fhdl6#\"\"$-%&TITLEG6#Q;Plotting~Several~Funct ions6\"-%+AXESLABELSG6$Q\"xFj^mQ\"yFj^m-%*THICKNESSGFidl-%%VIEWG6$;$!+ Fjzq:!\"*$\"+Fjzq:Fh_m;$!\"#Fcdl$FjdlFcdl" 1 2 0 1 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 210 "If you click on the graph a new content bar appears and shows the location of where you clicked as well as some alternative methods to \+ graph the function. From this you can estimate the points of intersect ion. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# end of this sect ion" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "c) Numerical Features: T he \"fsolve\" and \"int\" commands" }}{PARA 0 "" 0 "" {TEXT -1 126 "He re we will investigate two specific numerical features: Finding soluti ons to equations and evaluating a definite integral. " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 31 "(i) Solving equations with the " }{TEXT 268 9 "\"fsolve\" " }{TEXT 275 7 "command" }}{PARA 0 "" 0 "" {TEXT -1 453 "Consider the two functions f(x) = exp(x/10) and g(x) = x^2. If y ou plot these two functions on the same graph you will see that they i ntersect at two points. You can approximate these two points by click ing on the graph at the intersection and you see that the points of in tersection are approximately (-0.95, 0.90) and (1.06,1.10). However, \+ you can have Maple approximate the x-value of these points to greater accuracy with the \"fsolve\" command: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fsolve(x^2 = exp(x/10),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+r'>T0\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 254 "This command says: \"solve x^2 = exp(x/10) for x\". Notice, this \+ found only one answer, but the graph suggests there are two. How do y ou tell Maple to find the other one? You restrict the domain of your s earch by putting in a range of possible x-values. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve(x^2 = exp(x/10),x,-2..0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+? " 0 "" {MPLTEXT 1 0 33 "fsolve(x^2 = exp(x/10),x,- 4..-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'fsolveG6%/*$)%\"xG\"\"# \"\"\"-%$expG6#,$F)#\"\"\"\"#5F);!\"%!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 242 "It just returns your command suggesting that no solution lies in this interval. If you use \"solve\" instead of \"fsolve\" you are asking Maple to perform a much more complicated task. This will \+ be discussed in the section on symbolic features. " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 40 "(ii) Evaluating definite integrals with " } {TEXT 274 6 "\"int\" " }{TEXT -1 7 "command" }}{PARA 0 "" 0 "" {TEXT -1 241 "Suppose we want to appoximate an indefinite integral that we j ust cannot do by hand. Generally this means we cannot find an antideri vative and therefore cannot apply the fundamental 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 th e 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 thi s has no \"closed form\" solution. But we can have Maple approximate t his 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#,$*&-%%sqrt G6#\"\"#\"\"\"-%*EllipticEG6#,$*$F%F)#\"\"\"F(F0F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This appears ugly but we can evaluate this number by the \"evalf\" command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)y(>?Q!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "This is more important than you m ay first guess because alot of definite integrals have no \"closed for m\" solution. " }}}}}}{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 Maple so special. We will de monstrate Maple's ability to solve equations, differentiate, anti-diff erentiate. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "a) Solving Equati ons 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 approximate these solutions. Is it pos sible to have Maple find the exact solutions? To find exact solutions, if they can be found, you can use the \"solve\" command which works j ust 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 this an exact solut ion? In a pure sense; yes it is, for all practical purposes; no it is n't. But Maple can get the exact solution to other equations. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(cos(a*x) = sin(a*x),x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#PiG\"\"\"%\"aG!\"\"#\"\"\" \"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Notice, Maple did a goo d job of finding one solution, however, there are infinitely many. How about the quadratic equation " }{XPPEDIT 18 0 "a*x^2+b*x+c = 0;" "6#/ ,(*&%\"aG\"\"\"*$%\"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!\"%F0F1F0F3!\"\"#F1F/, $*&,&F&F'F(F'F0F3F6F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "This sec ond example results in the two solutions obtained from the quadratic f ormula. This is the " }{TEXT 277 3 "big" }{TEXT -1 12 " feature of " }{TEXT 276 24 "Computer Algebra Systems" }{TEXT -1 121 ": they can do \+ symbolic manipulations. However, as is the case in the first example, \+ this is not always that practical. " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 278 47 "b) Differentiation: The \"diff\" and \"D\" commands" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 283 "You can differentiate functions i n Maple with the use of the \"diff\" or \"D\" command. The \"D\" com mand is useful in defining new functions that are derivatives of previ ous ones. Let's first take a brief look at the \"diff\" command. Suppo se we want the derivative of the polynomial " }{XPPEDIT 18 0 "x^2+ 3-1" "6#,(*$%\"xG\"\"#\"\"\"\"\"$F'\"\"\"!\"\"" }{TEXT -1 108 ", with \+ the \"diff\" command. You put in the function you want to differentia te and the independent variable. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "diff(x^2 + 3 * x - 1, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"#\"\"$\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Don't forget to specify the variable. For example we wan t to differentiate the following function with respect to t (not 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 chai n rule and got the correct answer. The syntax for taking the second d erivative 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" }}}}}}{MARK "0 0 0" 22 }{VIEWOPTS 1 1 0 1 1 1803 }