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" }{TEXT -1 222 "This lab introduces \+ you to some of the features of Maple. Some of these features are simpl e graphing and numerical capabilities similar to those of a graphing \+ calculator, and some are symbolic manipulations exclusive to " } {TEXT 259 24 "Computer Algebra Systems" }{TEXT -1 137 ". If at anytim e you are at a loss as to how to do something, click the \"Help\" butt on on the menu bar and try topic or full text search. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 "1. Calculator, Graphing, and Numerical Fe atures" }}{PARA 0 "" 0 "" {TEXT -1 265 "Like your graphing calculator, Maple can plot functions but with greater resolution and more options . Maple can perform numerical routines such as finding the solutions \+ of an equation (using an alogorithm like Newton's Method) and approxim ating definite integrals." }}{SECT 1 {PARA 3 "" 0 "" {TEXT 277 17 "a) \+ Basic Commands" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 " The \">\" symb ol is the Maple prompt that lets you know Maple is ready for input. I n the following pages, everything after the \">\" represents something you would type. " }{TEXT 271 118 "In this, and all Maple lessons, be sure to evaluate all input (in red) by pressing the enter key somewhe re on the line" }{TEXT -1 460 ". If you don't do this, previous input will go unrecognized in current evaluations. Maple output appears in the center (in blue). All commands must end with a semicolon or a co lon. If you end the input with a colon, the results will not be printe d to the screen as they are in the examples below. You must use \"col on equals (:=)\" to assign a value or expression to a variable. The \+ \"=\" sign alone describes an equation which returns \"true\" or \"fal se\". " }}}{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##\"\"&\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Notice, Maple simplifies but does give a decimal expansion. In or der to do this, type " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ev alf(5/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nmmm;!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "The above command evaluates 5/3 as a flo ating point number. evalf(5/3,n) gives the result with n units of prec ision. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(5/3,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5nmmmmmmmm;!#>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 48 "When multiplying be sure to use the star symbo l:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a * b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$N\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "If \+ you had forgotten to use this and just typed" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a b;" }}{PARA 8 "" 1 "" {TEXT -1 31 "Error, missi ng operator or `;`\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "you get a n error message. " }}{PARA 0 "" 0 "" {TEXT -1 94 " To clear the value, or expression, assigned to a variable, you must use one of the follo wing" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a := 'a';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign('b');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "A # sign may be used for comments and everything after it is ignor ed by Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "3*a + b; \+ # Hi, How are you?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"aG\"\"$% \"bG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Notice, the values \+ assigned to a and b have been cleared. " }}{PARA 0 "" 0 "" {TEXT -1 62 "Finally you may reference previous output with the \"%\" symbol:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "2 * %;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&%\"aG\"\"'*&\"\"#\"\"\"%\"bGF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Here, the previous output \"3 a + b\" is multi plied 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 of this section" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 278 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#>%\"fGf*6#%\"xG6\"6$%)operat orG%&arrowGF(,**$)9$\"\"$\"\"\"F0*&\"\"(F1)F/\"\"#F1F1*&F5F1F/F1F1\"\" &!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "Here you may thi nk of the command as \"f\" takes a variable \"x\" and assigns it the v alue 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$7S7$$!\"$\"\"!$!#HF*7$$!3smm;HU ,\"*G!#<$!3c*z-&zd_wC!#;7$$!3=L$3FH'='z#F0$!3yMedER!\\9#F37$$!3gmmTgBa *o#F0$!3)QC^*HY\"4\"=F37$$!3wmm\"H_\">#e#F0$!3p4:*[!GB9:F37$$!3ML$3_!4 NvCF0$!3Sn2W2\"GhD\"F37$$!3'omTg(fHwBF0$!3KYbv0%R![5F37$$!3;+]PM.ttAF0 $!3IlpzT]/B')F07$$!3!omT5!oln@F0$!3?)G^'QF!***pF07$$!3%)**\\(oWB>1#F0$ !3uC&)e`k7icF07$$!3;LL$epjJ&>F0$!3;Dkrl(eab%F07$$!3amm\"z/ot&=F0$!3)y! H>J!e()y$F07$$!3))****\\P[_\\Pbfl'*o#F07$$!3'*****\\i^)o`\"F0$!3@T8[].5ICF07$$!3dm;/^?7U9F0$!3f w/f&)\\%QK#F07$$!3CLL$eaR%H8F0$!39vf;E$F07$$!3*z**\\7G-89)F[r$!3i\"H &y6/W2OF07$$!3%)GL$3Fp)pqF[r$!3-e!G%o6GvRF07$$!3YKL3-$ff3'F[r$!3)y4w>a ?2I%F07$$!38nm;z%zY-&F[r$!3QapShz>=YF07$$!35kmT5!3B#RF[r$!3mjXNP[d))[F 07$$!3C***\\iS!piHF[r$!3!*\\pZ@g7c]F07$$!3lim;/rFE>F[r$!3c\\6&>)*fp9&F 07$$!3Q&******\\2cb)!#>$!3aG(fq-_<7&F07$$\"3t9++DJE>>Fdt$!3osUhG],f\\F 07$$\"3t-+D1RU07F[r$!3-x0;Jt%>l%F07$$\"3+++](=S2L#F[r$!3'\\fgoG.c6%F07 $$\"3:jmm;p)=M$F[r$!3:.g'*y$zyV$F07$$\"3O-++v=]@WF[r$!3O'=S?G0z[#F07$$ \"3/JLe*[$z*R&F[r$!3YJU\\:'fmS\"F07$$\"3#e++]iC$pkF[r$\"37Hvdc^=yNFdt7 $$\"3ukm\"H2qcZ(F[r$\"3k9)R[\"pZg;F07$$\"3i.+DJ5fF&)F[r$\"3'zPV)*=zil$ F07$$\"3akmmTg.c&*F[r$\"3i,LOAUO@fF07$$\"3w**\\ilAFj5F0$\"31fR(Gi?mk)F 07$$\"3yLLL$)*pp;\"F0$\"31YB[/yUj6F37$$\"3)RL$3xe,t7F0$\"3aLkY`\\!z]\" F37$$\"3Cn;HdO=y8F0$\"3]\"f`OUA0*=F37$$\"3a+++D>#[Z\"F0$\"3Uwq$\\g**)z AF37$$\"3SnmT&G!e&e\"F0$\"3ae\"=3%p$Gx#F37$$\"3#RLLL)Qk%o\"F0$\"3K?BAI Z'yD$F37$$\"37+]iSjE!z\"F0$\"3'4**paigH#QF37$$\"3a+]P40O\"*=F0$\"3.T\" QE/,@T%F37$$\"\"#F*$\"#^F*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fb[l-%+AXES LABELSG6$Q\"x6\"Q!Fg[l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "and restrict the range by describing 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%-%'CURVE SG6$7S7$$!\"$\"\"!$!#HF*7$$!3smm;HU,\"*G!#<$!3c*z-&zd_wC!#;7$$!3=L$3FH '='z#F0$!3yMedER!\\9#F37$$!3gmmTgBa*o#F0$!3)QC^*HY\"4\"=F37$$!3wmm\"H_ \">#e#F0$!3p4:*[!GB9:F37$$!3ML$3_!4NvCF0$!3Sn2W2\"GhD\"F37$$!3'omTg(fH wBF0$!3KYbv0%R![5F37$$!3;+]PM.ttAF0$!3IlpzT]/B')F07$$!3!omT5!oln@F0$!3 ?)G^'QF!***pF07$$!3%)**\\(oWB>1#F0$!3uC&)e`k7icF07$$!3;LL$epjJ&>F0$!3; Dkrl(eab%F07$$!3amm\"z/ot&=F0$!3)y!H>J!e()y$F07$$!3))****\\P[_\\Pbfl'*o#F07$$!3'*****\\i^)o`\"F 0$!3@T8[].5ICF07$$!3dm;/^?7U9F0$!3fw/f&)\\%QK#F07$$!3CLL$eaR%H8F0$!39< UYA[+OBF07$$!3vf;E$F07$$!3*z**\\7G-89)F[r$!3i\"H&y6/W2OF07$$!3%)GL$3Fp)pqF[r$!3-e!G %o6GvRF07$$!3YKL3-$ff3'F[r$!3)y4w>a?2I%F07$$!38nm;z%zY-&F[r$!3QapShz>= YF07$$!35kmT5!3B#RF[r$!3mjXNP[d))[F07$$!3C***\\iS!piHF[r$!3!*\\pZ@g7c] F07$$!3lim;/rFE>F[r$!3c\\6&>)*fp9&F07$$!3Q&******\\2cb)!#>$!3aG(fq-_<7 &F07$$\"3t9++DJE>>Fdt$!3osUhG],f\\F07$$\"3t-+D1RU07F[r$!3-x0;Jt%>l%F07 $$\"3+++](=S2L#F[r$!3'\\fgoG.c6%F07$$\"3:jmm;p)=M$F[r$!3:.g'*y$zyV$F07 $$\"3O-++v=]@WF[r$!3O'=S?G0z[#F07$$\"3/JLe*[$z*R&F[r$!3YJU\\:'fmS\"F07 $$\"3#e++]iC$pkF[r$\"37Hvdc^=yNFdt7$$\"3ukm\"H2qcZ(F[r$\"3k9)R[\"pZg;F 07$$\"3i.+DJ5fF&)F[r$\"3'zPV)*=zil$F07$$\"3akmmTg.c&*F[r$\"3i,LOAUO@fF 07$$\"3w**\\ilAFj5F0$\"31fR(Gi?mk)F07$$\"3yLLL$)*pp;\"F0$\"31YB[/yUj6F 37$$\"3)RL$3xe,t7F0$\"3aLkY`\\!z]\"F37$$\"3Cn;HdO=y8F0$\"3]\"f`OUA0*=F 37$$\"3a+++D>#[Z\"F0$\"3Uwq$\\g**)zAF37$$\"3SnmT&G!e&e\"F0$\"3ae\"=3%p $Gx#F37$$\"3#RLLL)Qk%o\"F0$\"3K?BAIZ'yD$F37$$\"37+]iSjE!z\"F0$\"3'4**p aigH#QF37$$\"3a+]P40O\"*=F0$\"3.T\"QE/,@T%F37$$\"\"#F*$\"#^F*-%'COLOUR G6&%$RGBG$\"#5!\"\"$F*F*Fb[l-%+AXESLABELSG6$Q\"x6\"Q\"yFg[l-%%VIEWG6$; F(Fgz;$!#5F*$\"#?F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 448 "Suppo se you want an appoximation of the x-intercept in the above graph. Yo u can do this by clicking on the graph and then putting the arrow near the intercept and clicking there. A window will appear in the upper \+ left giving you the coordinates of the point to which the arrow is poi nting. Try this for the above curve and observe that the x - intercep t is about 0.63. Note: This is only an approximation and should be us ed as an exact answer. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 274 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 glossary. " } {TEXT 275 20 "Remember to denote " }{XPPEDIT 18 0 "exp(g(x));" "6#-%$ expG6#-%\"gG6#%\"xG" }{TEXT -1 2 " " }{TEXT 276 14 "by exp(g(x)). " } {TEXT -1 134 "Plotting more than one function requires placing square \+ backets around all of the functions and seperating them by commas. The number " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 464 " is denoted by \"Pi\". You can choose the color of the individual graphs by using so mething like \"color = [red,blue,green]\", but if you are using a blac k and white printer, the \"linestyle\" option is preferred. Similarly, the default line thickness is sometimes not big enouph to show up on \+ some printers. You can alter this with the \"thickness\" command. Fin ally, 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 0 181 "plot([f(x),sin(x^2),exp(x/2)], x = -Pi/2 .. 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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# end of this section" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "c) Numeri cal Features: The \"fsolve\" and \"int\" commands" }}{PARA 0 "" 0 "" {TEXT -1 126 "Here we will investigate two specific numerical features : Finding solutions to equations and evaluating a definite integral. \+ " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 31 "(i) Solving equations with th e " }{TEXT 273 9 "\"fsolve\" " }{TEXT 280 7 "command" }}{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 same graph you will see \+ that they intersect at two points. You can approximate these two poin ts by clicking on the graph at the intersection and you see that the p oints of intersection 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#$\"+r'>T0\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 254 "This command says: \"solve x^2 = exp(x/10) for x\". Notice, th is found only one answer, but the graph suggests there are two. How d o you tell Maple to find the other one? You restrict the domain of you r search by putting in a range of possible x-values. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve(x^2 = exp(x/10),x,-2..0);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+? " 0 "" {MPLTEXT 1 0 33 "fsolve(x^2 = exp(x/10),x, -4..-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'fsolveG6%/*$)%\"xG\"\"# \"\"\"-%$expG6#,$F)#F+\"#5F);!\"%!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "It just returns your command suggesting that no solution lies in this interval. If you don't use \"solve\" instead of \"fsolve \" you are asking Maple to perform a much more complicated task. This will be discussed in the section on symbolic features. " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 40 "(ii) Evaluating definite integrals with \+ " }{TEXT 279 6 "\"int\" " }{TEXT -1 7 "command" }}{PARA 0 "" 0 "" {TEXT -1 241 "Suppose we want to appoximate an indefinite integral tha t we just cannot do by hand. Generally this means we cannot find an an tiderivative and therefore cannot apply the fundamental theorem of cal culus. Here's an example of such a problem: " }}{PARA 0 "" 0 "" {TEXT -1 85 "The following definite integral gives the length of one a rch of the curve y = sin(x)." }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {XPPEDIT 18 0 "int(sqrt(1+cos(x)^2),x = 0 .. Pi);" "6#-%$intG6$-%%sqrt G6#,&\"\"\"F**$-%$cosG6#%\"xG\"\"#F*/F/;\"\"!%#PiG" }{TEXT -1 92 " \+ and this has no \"closed form\" solution. But we can have Maple approx imate this length by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "int( sqrt(1+(cos(x))^2),x = 0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& -%%sqrtG6#\"\"#\"\"\"-%*EllipticEG6#,$*$F%F)#F)F(F)F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This appears ugly but we can evaluate thi s number by the \"evalf\" command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)y(>? Q!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "This is more important \+ than you may first guess because alot of definite integrals have no \" closed form\" solution. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "2. Sym bolic Features" }}{PARA 0 "" 0 "" {TEXT -1 147 "Here we will investiga te what makes Maple so special. We will demonstrate Maple's ability t o solve equations, differentiate, anti-differentiate. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "a) Solving Equations Symbolically: The \" solve\" command" }}{PARA 0 "" 0 "" {TEXT -1 304 "In a previous example we found that exp(x/10) = x^2 at two values of x. We had Maple numeri cally approximate these solutions. Is it possible to have Maple find t he exact solutions? To find exact solutions, if they can be found, you can use the \"solve\" command which works just like the \"fsolve\" co mmand. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(exp(x/10) = x^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$-%)LambertWG6##!\"\"\"#? !#?,$-F%6$F(F'F*,$-F%6##\"\"\"F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Is this an exact solution? In a pure sense; yes it is, for al l practical purposes; no it isn't. But Maple can get the exact soluti on to other equations. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(cos(a*x) = sin(a*x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& %#PiG\"\"\"%\"aG!\"\"#F&\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Notice, Maple did a good 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\"\"\"*$%\"xG\"\"#F'F'*&%\"bGF'F)F'F '%\"cGF'\"\"!" }{TEXT -1 3 " ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(a*x^2 + b*x + c = 0, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&%\"aG!\"\",&%\"bGF&*$-%%sqrtG6#,&*$)F(\"\"#\"\"\"F1*(\"\"%F1 F%F1%\"cGF1F&F1F1F1#F1F0,$*&F%F&,&F(F&F)F&F1F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "This second example results in the two solutions obt ained from the quadratic formula. This is the " }{TEXT 282 3 "big" } {TEXT -1 12 " feature of " }{TEXT 281 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 283 47 "b) Differentiation: The \"diff \" and \"D\" commands" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 283 "You can \+ differentiate functions in Maple with the use of the \"diff\" or \"D \" command. The \"D\" command is useful in defining new functions tha t are derivatives of previous ones. Let's first take a brief look at t he \"diff\" command. Suppose we want the derivative of the polynomial \+ " }{XPPEDIT 18 0 "x^2+3-1" "6#,(*$%\"xG\"\"#\"\"\"\"\"$F'F'!\"\"" }{TEXT -1 108 ", with the \"diff\" command. You put in the function y ou 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(F )" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "Here Maple used the chain r ule and got the correct answer. The problem with the \"diff\" command is that it's difficult to assign another function name to the derivat ive. You may try this by assigning the derivative of \"g\" the name \+ \"gprime\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "g := x -> ex p(3*x); # g is an exponential function" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$9$\"\"$F(F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gprime := x -> diff(g( x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gprimeGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(-%%diffG6$-%\"gG6#9$F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "This looks ok but problems occur if you try to evaluat e gprime at a specific number," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "gprime(0);" }}{PARA 8 "" 1 "" {TEXT -1 73 "Error, (in gprime) \+ wrong number (or type) of parameters in function diff\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "where Maple returns the above error mess age. The problem is Maple replaces the \"x\" with a zero and then tri es 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 "gprime := 'gprime'; # clears gprim e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gprimeGF$" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "gprime := x -> D(g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gprimeGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%$expG6 #,$9$\"\"$F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "or " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gprime := D(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gprimeGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$ -%$expG6#,$9$\"\"$F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "He re we still must input the function name as well as the variable but i n a different format from the \"diff\" command. This results in a fun ction that we can evaluate" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "gprime(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "This works out. Summary: Use \"diff(f(x ),x)\" to differentiate. Use \"D(f)(x)\" when creating a new function \+ for evaluation or plotting. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# end of this section" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 284 43 "c) Antidifferentiation: The \"int\" command. " }}{PARA 0 "" 0 "" {TEXT -1 210 "The command for antidifferentiation is the same as the c ommand used in evaluating definite integrals: \"int\", except, instea d of putting in the bounds of integration you just put in the independ ent variable. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "restart; #This clears all previous variables" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Suppose we want to find the antiderivative (indefinite in tegral) of " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 26 " usi ng the \"int\" command. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "int(x^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"xG\"\"$\"\"\"# F(F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Notice this gives only on e antiderivative, " }{TEXT 285 61 "where the constant of integration i s assigned the value zero." }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The same command with integration bounds evaluates the de finite integral" }{XPPEDIT 18 0 "int(x^2,x = 0 .. 2);" "6#-%$intG6$*$% \"xG\"\"#/F';\"\"!F(" }{TEXT -1 4 " by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(x^2, x=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# #\"\")\"\"$" }}}{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#>%\"hGf *6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$expG6#9$\"\"\",&F-F1F1F1!\"\"F( F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "recall that with the sub stitution u = 1 + e^x, the integrand has the form du/u and we found t he following antiderivative" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(h(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&-%$expG6 #%\"xG\"\"\"F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "However, sup pose 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#>% \"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&F-F--%$expG6#,$9$! \"\"F-F4F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Here, no u-subs titution 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 answer? It should be. Carefu l use of the properties of logarithms will result in the same answer. \+ You may wish to avoid such trivial tasks by requesting Maple to simpli fy the expression by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sim plify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&\"\"\"F(-%$ex pG6#,$%\"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 com mand is useful for some trig expressions, some exponential and logarit hmic expressions, and some algebraic expressions, however it by no mea ns defines the simplest from of all expressions. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# end of this section" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Assignme nt" }}{PARA 0 "" 0 "" {TEXT -1 353 "You may use Maple's text editor to answer these questions. To use the text editor, click on the \"T\" b utton from the Tool bar. This removes the Maple prompt \">\" and leave s a \"[\". You may now use the keyboard as a typewriter with the abi lity to incorporate mathematical notation. Inserting mathematical exp ressions can best be done by clicking on the " }{XPPEDIT 18 0 "Sigma; " "6#%&SigmaG" }{TEXT -1 201 " button from the tool bar and typing in \+ Maple code for the expression you want to create. For example, if you type in the following: \"exp(x^2) * cos(Pi) = - exp(x^2)\" and hit e nter, the expression \" " }{XPPEDIT 18 0 "exp(x^2)*cos(Pi);" "6#*&-%$e xpG6#*$%\"xG\"\"#\"\"\"-%$cosG6#%#PiGF*" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "-exp(x^2);" "6#,$-%$expG6#*$%\"xG\"\"#!\"\"" }{TEXT -1 83 " \" i s printed at the cursor location. Another example: by typing \"Int(f( x),x)\" in " }{XPPEDIT 18 0 "Sigma;" "6#%&SigmaG" }{TEXT -1 47 " mode, the following is printed at the cursor: " }{XPPEDIT 18 0 "Int(f(x),x) ;" "6#-%$IntG6$-%\"fG6#%\"xGF)" }{TEXT -1 16 " . To exit the \"" } {XPPEDIT 18 0 "Sigma;" "6#%&SigmaG" }{TEXT -1 197 "\" mode, click on t he \"T\" from the menu bar. You can also do this sort of thing by cu tting and pasting Maple output, or using the palettes found in \"View/ Palettes\" from the menu bar. However the " }{XPPEDIT 18 0 "Sigma;" " 6#%&SigmaG" }{TEXT -1 21 " button works best. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "If you are familiar with Microsoft Word, it is possible to wri te your text and mathematical expressions in Word and import (or cut a nd paste) Maple graphs into this document." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 80 "Problem #1, Mathematical Defi nitions, The Fundamental Theorem(s) of Calculus: " }{TEXT 262 29 "Sup pose you have a function " }{TEXT 269 1 "f" }{TEXT 270 203 " that is differentiable and integrable on an interval I containing the numbers a and b. Answer the following questions using the Maple (or Word) te xt editor, with mathematical notation where necessary." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 2 "a)" }{TEXT -1 57 " \+ In words, what does the derivative of f(x) represent? " }}{PARA 0 "" 0 "" {TEXT 264 2 "b)" }{TEXT -1 29 " State the definition of an " } {TEXT 268 19 "indefinite integral" }{TEXT -1 27 " (antiderivative) of \+ f(x). " }{TEXT 265 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 301 2 " c)" }{TEXT -1 11 " Consider " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG " }{TEXT -1 39 " > 0. In terms of area, what does the " }{TEXT 302 18 "definite integral " }{XPPEDIT 18 0 "int(f(x),x = a .. b)" "6#-%$in tG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 11 " represent?" }}{PARA 0 "" 0 "" {TEXT 266 2 "d)" }{TEXT -1 148 " State the part of the Fundam ental Theorem of Calculus which relates the definite integral from par t (c) to the indefinite integral from part (b). " }}{PARA 0 "" 0 "" {TEXT 267 2 "e)" }{TEXT -1 63 " State the other part of the Fundament al Theorem of Calculus. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 261 23 "Problem #2, Sky-Diver:" }}{PARA 0 "" 0 "" {TEXT -1 74 "In the case of free fall motion retarded by air resistance, the velocity v" }{TEXT 286 3 "(t)" }{TEXT -1 21 " in feet /sec at time " }{TEXT 294 1 "t" }{TEXT -1 30 " (in seconds) is descri bed by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "v(t) = exp((-ct)/m)*(v[o]+mg/c)-mg/c;" "6#/-%\"vG6#%\"tG,&*&-%$expG 6#*&,$%#ctG!\"\"\"\"\"%\"mGF0F1,&&F%6#%\"oGF1*&%#mgGF1%\"cGF0F1F1F1*&F 8F1F9F0F0" }{TEXT -1 8 " " }{TEXT 290 3 ", " }{TEXT -1 7 "wher e " }{TEXT 291 1 "g" }{TEXT -1 18 " = 32 feet/sec^2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Here, " }{TEXT 287 2 "m " }{TEXT -1 27 "is the mass of the object, " }{XPPEDIT 18 0 "v[o]; " "6#&%\"vG6#%\"oG" }{TEXT -1 40 " is the initial velocity (in feet/se c), " }{TEXT 288 1 "c" }{TEXT -1 27 " is the drag constant, and " } {TEXT 289 2 "g " }{TEXT -1 166 "is the acceleration due to gravity. In the case of a 240 pound fully equipped sky diver with a terminal spee d of 120 feet/sec, the following values may be assigned: " }{TEXT 292 1 "m" }{TEXT -1 11 " = 240 and " }{TEXT 293 1 "c" }{TEXT -1 6 " = \+ 64." }}{PARA 0 "" 0 "" {TEXT -1 104 " Now suppose this sky-diver jumps out of an airplane at 10,000 feet and sadly, his chute does not open. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 295 3 "a) " }{TEXT -1 201 "Make a plot of the sky diver's veloci ty and acceleration (on the same graph) over the first 25 seconds of f ree fall. Comment on the relationship between the two and the notion o f \"terminal velocity\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 296 2 "b)" }{TEXT -1 102 " Use integration to derive a function descri bing the sky-divers height (in feet) above ground at time " }{TEXT 297 1 "t" }{TEXT -1 27 " (in seconds). Call this s(" }{TEXT 299 1 "t" }{TEXT -1 4 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 298 3 "c) " }{TEXT -1 49 " Make a plot of s(t) over the first 60 seconds. " }} {PARA 0 "" 0 "" {TEXT 300 4 "d) " }{TEXT -1 61 "How long (in seconds) does it take him to reach the ground. " }}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }