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{PSTYLE "" 3 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 262 "" 0 "" {TEXT 258 26 "Tips for Maple Instruct ors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 48 " Posted with permission of Birkhauser Boston from" }}{PARA 269 "" 0 "" {TEXT -1 129 "Maple Tech, volume 3, number 3, pp 41-47, 1996,\nTips fo r Maple Instructors by Robert J. Lopez.\nCopyright 1996 Birkhauser Bos ton. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 441 "In the first column of this series w e addressed the question of how to present Maple as a pedagogical tool in the classroom. In addition, we described several features of Rele ase 4 that will very likely impact the use of Maple in the classroom. \+ In this second column of the series we again illustrate some Maple Re lease 4 functionality of significance in the classroom, and include a \+ discussion of Maple styles, a feature new in Release 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 611 "I continue to hope \+ this column becomes a forum where common problems can be identified an d resolved, where useful hints and strategems can be found, and where \+ pedagogical insights can be shared. Hence, I urge all readers who hav e their own experiences with Maple in instruction, successes and failu res alike, to communicate with me by e-mail (r.lopez@rose-hulman.edu), fax(812-877-3198), or letter (Math Dept., Rose-Hulman Institute of Te chnology, Terre Haute, IN 47803). As much as possible, I would like t his column to address real issues, from real classes. Only the column 's readers can make that happen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 339 "A key principle articulated in the previous colum n was the dictum of keeping things simple, and doing things only one w ay. We will illustrate the implications of this in the first section \+ where we demonstrate some strategies for \"matching coefficients\" in \+ several calculations. Then, we illustrate an improper integral that r equires the " }{TEXT 260 4 "surd" }{TEXT -1 338 " command. Third, we \+ discuss Maple as an n-digit computer in a section that started with an enigmatic computation and evolved into enlightenment under Keith Gedd es' insistance that \"when the Digits variable is set to n, Maple is a n n-digit computer.\" Finally, we pass along a few tips on using docu ment styles, new to Maple in Release 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Don't Build On Sand" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 5 "match" }{TEXT -1 93 " command works in some cases but not in others. It is confusin g to students to learn to use " }{TEXT 262 5 "match" }{TEXT -1 158 " i nitially, but then be forced to use a different approach later on for \+ a similar calculation. It's better to use initially the syntax that w ill work always." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 27 "Match and Partial Fractions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Consider, for example, the case of partial fractions where " }{TEXT 263 5 "match" }{TEXT -1 231 " is an elegant way to est ablish an identity between two different forms of a rational function. Match will set up and solve appropriate equations for the undetermin ed parameters, storing the computed paremeters in a set named 's'." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := (5*x^2+5*x+4)/(x+1)/(x^2+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(,(*$%\"xG\"\"#\"\"&F(F*\"\"%\"\"\"F,,&F(F,F,F,!\"\",&F'F ,F,F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f1 := a/(x+1);\n f2 := (b*x+c)/(x^2+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G*&%\"a G\"\"\",&%\"xGF'F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G*&, &*&%\"bG\"\"\"%\"xGF)F)%\"cGF)F),&*$F*\"\"#F)F)F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "match(f = normal(f1 + f2),x,'s');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "subs(s, f1 + f2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*$,&%\"xG\"\"\"F'F'!\"\"\"\"#*&,&F&\"\"$F)F'F',&*$F&F)F'F'F'F(F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Unfortunately, " }{TEXT 264 5 "match" }{TEXT -1 154 " is not robust e nough to support the coefficient matching needed in the method of Unde termined Coefficients from a first course in differential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Match and Differential Equations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Given the differential equation, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "qq := 3*diff(y(t),t, t)+4*diff(y(t),t)+5*y(t)=cos(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #qqG/,(-%%diffG6$-F(6$-%\"yG6#%\"tGF/F/\"\"$F*\"\"%F,\"\"&-%$cosGF." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " assume a particular solution of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "yp := A*cos(t)+B*sin (t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypG,&*&%\"AG\"\"\"-%$cosG6# %\"tGF(F(*&%\"BGF(-%$sinGF+F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "and force it to be a solution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "qq1 := simplify(subs(y(t)=yp,qq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$qq1G/,**&%\"AG\"\"\"-%$cosG6#%\"tGF)\"\"#*&%\"BGF)-%$sinGF,F) F.*&F(F)F1F)!\"%*&F0F)F*F)\"\"%F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "This requires recognizing equation qq1 as an identity in t. However," }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "match(qq1,t,'s');" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Solve/identity" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Thu s, the student needs to learn the new syntax" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "qq2:=solve(identi ty(qq1,t),\{A,B\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$qq2G<$/%\"AG #\"\"\"\"#5/%\"BG#F)\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "a complication that could have been avoid ed if " }{TEXT 265 5 "match" }{TEXT -1 24 " were more robust or if " } {TEXT 266 14 "solve/identity" }{TEXT -1 31 " had been introduced initi ally." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Moreover, " }{TEXT 267 5 "match" }{TEXT -1 42 " will not identify a solution in the form " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "y1 := exp(-3*t)*(5*cos(4*t) - 9*sin(4*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&-%$expG6#,$%\"tG!\"$\"\"\",&-%$cosG6#,$F*\"\"% \"\"&-%$sinGF0!\"*F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 " with the engineering preference " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "y2 : = exp(-3*t)*A*cos(4*t-phi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2 G*(-%$expG6#,$%\"tG!\"$\"\"\"%\"AGF,-%$cosG6#,&F*\"\"%%$phiG!\"\"F," } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 " Fortunately, the second approach of using " }{TEXT 268 14 "solve/ident ity" }{TEXT -1 129 " does work, provided y2 is given to Maple with the cosine term expanded. To prevent Maple from further expanding the tr ig terms " }{XPPEDIT 18 0 "sin(4*t)" "-%$sinG6#*&\"\"%\"\"\"%\"tGF'" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(4*t)" "-%$cosG6#*&\"\"%\"\"\"% \"tGF'" }{TEXT -1 41 " to expressions involving just powers of " } {XPPEDIT 18 0 "sin(t)" "-%$sinG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(t)" "-%$cosG6#%\"tG" }{TEXT -1 11 ", give the " }{TEXT 269 6 "expand" }{TEXT -1 83 " command the additional parameter 4*t to prev ent that argument from being expanded." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "q := solve(identity(exp and(y1=y2,4*t),t),\{A,phi\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"qG6$<$/%$phiG,$-%'arctanG6##\"\"*\"\"&!\"\"/%\"AG*$\"$1\"#\"\"\"\"\" #<$/F2,$F3F0/F(,&F*F0%#PiGF6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Transfer this information to the templ ate y2, selecting the positive value of the amplitude. This use of " }{TEXT 270 14 "solve/identity" }{TEXT -1 117 " performs the conversion of one form of a solution to another. It is, in fact, the simplest w ay to do this in Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Y2 := subs(q[1],y2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#Y2G*(-%$expG6#,$%\"tG!\"$\"\"\"\"$1\"#F,\"\"# -%$cosG6#,&F*\"\"%-%'arctanG6##\"\"*\"\"&F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Caution: In an early \+ version of this calculation, I called the phase angle " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 12 " instead of " }{XPPEDIT 18 0 "phi" "I$p hiG6\"" }{TEXT -1 25 ". Note the effect on y2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "exp(-3*t)*A *cos(4*t-phi);\nexp(-3*t)*A*cos(4*t-z);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$expG6#,$%\"tG!\"$\"\"\"%\"AGF*-%$cosG6#,&F(\"\"%%$ phiG!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$expG6#,$%\"tG!\"$ \"\"\"%\"AGF*-%$cosG6#,&F(!\"%%\"zGF*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "Apparently, the ordering of terms in Maple is lexicographically challenged. For the cosine fu nction the expressions are equivalent, but when commands like " } {TEXT 271 6 "expand" }{TEXT -1 219 ", which rely on pattern matching, \+ are applied, the differences in \"appearance\" lead to different behav iors in the ensuing calculations. This is a difficulty for authors of journal articles and classroom lessons, alike." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Two Equations i n Two Unknowns" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "To perform the a bove matching by setting up and solving two equations in the two unkno wns " }{XPPEDIT 18 0 "A" "I\"AG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 55 ", the student has to be aware of sev eral quirks of the " }{TEXT 272 6 "expand" }{TEXT -1 74 " command. Fi rst, note what happens when the equality y1 = y2 is expanded." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "expand(y1 = y2,4*t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&- %$expG6#%\"tG!\"$-%$cosG6#,$F)\"\"%\"\"\"\"\"&*&F&F*-%$sinGF-F0!\"*,&* *F&F*%\"AGF0F+F0-F,6#%$phiGF0F0**F&F*F8F0F3F0-F4F:F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The optio nal parameter 4*t keeps the trig functions on the left from being expa nded. If " }{XPPEDIT 18 0 "cos(4*t-z)" "-%$cosG6#,&*&\"\"%\"\"\"%\"tG F(F(%\"zG!\"\"" }{TEXT -1 34 " was used, then Maple's switch of " } {XPPEDIT 18 0 "cos(4*t-z)" "-%$cosG6#,&*&\"\"%\"\"\"%\"tGF(F(%\"zG!\" \"" }{TEXT -1 19 " to the equivalent " }{XPPEDIT 18 0 "cos(-4*t+z)" "- %$cosG6#,&*&\"\"%\"\"\"%\"tGF(!\"\"%\"zGF(" }{TEXT -1 95 " means the t rig terms on the right will not be frozen the same way they are frozen on the left." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "expand(y1 = exp(-3*t)*A*cos(4*t-z), 4*t);\n" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*&-%$expG6#%\"tG!\"$-%$cosG6#,$F)\" \"%\"\"\"\"\"&*&F&F*-%$sinGF-F0!\"*,,**F&F*%\"AGF0-F,6#%\"zGF0-F,F(F/ \"\")**F&F*F8F0F9F0F<\"\"#!\")*(F&F*F8F0F9F0F0*,F&F*F8F0-F4F:F0-F4F(F0 F<\"\"$F=*,F&F*F8F0FCF0FDF0F " 0 "" {MPLTEXT 1 0 48 "expa nd(y1 = exp(-3*t)*A*cos(4*t-z), 4*t, -4*t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$expG6#%\"tG!\"$-%$cosG6#,$F)\"\"%\"\"\"\"\"&*&F &F*-%$sinGF-F0!\"*,&**F&F*%\"AGF0-F,6#,$F)!\"%F0-F,6#%\"zGF0F0**F&F*F8 F0-F4F:F0-F4F>F0!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 150 "to keep all the trig terms from being ex panded in terms of t. In addition, the exponential terms could be tre ated more conventionally. Including an " }{TEXT 259 3 "exp" }{TEXT -1 8 " in the " }{TEXT 273 6 "expand" }{TEXT -1 77 " command will forc e Maple to treat the exponential terms more conventionally." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "q 1 := expand(y1 = y2, 4*t, -4*t, exp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G/,&*&-%$expG6#,$%\"tG!\"$\"\"\"-%$cosG6#,$F,\"\"%F.\"\"&*&F(F .-%$sinGF1F.!\"*,&**F(F.%\"AGF.F/F.-F06#%$phiGF.F.**F(F.F;F.F6F.-F7F=F .F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Match the coefficients of cos(4t) on both sides of the equation q1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "e1 := map(coeff,q1,cos(4*t));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G/,$-%$expG6#,$%\"tG!\"$\"\"&*(F'\"\"\"%\"AGF/-%$c osG6#%$phiGF/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Match the coefficients of sin(4t) on both sides of the equation q1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "e2 := map(coeff,q1,sin(4*t));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G/,$-%$expG6#,$%\"tG!\"$!\"**(F'\"\"\"%\"AGF/ -%$sinG6#%$phiGF/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 49 "Solve equations e1 and e2 for the unknowns A and \+ " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "q2 := solve (\{e1,e2\},\{A,phi\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2G<$/% $phiG-%'arctanG6$,$-%'RootOfG6#,&!$1\"\"\"\"*$%#_ZG\"\"#F1#!\"*\"$1\", $F,#\"\"&F7/%\"AGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "One cure for " }{TEXT 256 6 "RootOf" }{TEXT -1 4 " is " }{TEXT 257 9 "allvalues" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "q3 := allva lues(q2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q3G6$<$/%\"zG-%'arct anG6$,$*$\"$1\"#\"\"\"\"\"##!\"*F.,$F-#\"\"&F./%\"AGF-<$/F8,$F-!\"\"/F (-F*6$,$F-#\"\"*F.,$F-#!\"&F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "Another cure for " }{TEXT 274 6 "Root Of" }{TEXT -1 112 " is the environment variable _EnvExplicit. Setting this variable to \"true\" generally suppresses Maple's use of " } {TEXT 275 6 "RootOf" }{TEXT -1 8 ". Thus," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "_EnvExplicit := t rue;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-_EnvExplicitG%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(\{e1,e2\},\{A,phi\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%$phiG,$-%'arctanG6##\"\"*\"\"& !\"\"/%\"AG*$\"$1\"#\"\"\"\"\"#<$/F/,$F0F-/F%,&F'F-%#PiGF3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Unfortuna tely, setting _EnvExplicit to \"true\" does not suppress all occurance s of " }{TEXT 276 6 "RootOf" }{TEXT -1 137 ". In solving the differen tial equation for a body falling with quadratic drag, a problem posed \+ as early as Calculus II, we find Maple's " }{TEXT 277 6 "dsolve" } {TEXT -1 13 " solution to " }{XPPEDIT 18 0 "m*diff(v(t),t)=-m*g+r*v(t) ^2" "/*&%\"mG\"\"\"-%%diffG6$-%\"vG6#%\"tGF,F%,&*&F$F%%\"gGF%!\"\"*&% \"rGF%*$-F*6#F,\"\"#F%F%" }{TEXT -1 21 " given in terms of a " }{TEXT 278 6 "RootOf" }{TEXT -1 35 " containing an arctanh function, a " } {TEXT 279 6 "RootOf" }{TEXT -1 46 " that generates an error message wh en sent to " }{TEXT 280 9 "allvalues" }{TEXT -1 109 ". Separating var iables in the ODE, and integrating both sides, leads to a transcendent al equation for which " }{TEXT 281 5 "solve" }{TEXT -1 29 " again retu rns the offending " }{TEXT 282 6 "RootOf" }{TEXT -1 64 ". So, the use of the environment variable generally suppresses " }{TEXT 283 6 "Root Of" }{TEXT -1 24 ", but is not infallible." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "Returning to the main discussi on, we transfer the information in set q3 back into the template y2, s electing the solution with the positive value of the amplitude." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Y2 := subs(q3[1],y2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Y2 G*(-%$expG6#,$%\"tG!\"$\"\"\"\"$1\"#F,\"\"#-%$cosG6#,&F*!\"%-%'arctanG 6##\"\"*\"\"&!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We have recovered the same solution found by " }{TEXT 284 14 "solve/identity" }{TEXT -1 109 ", but Maple's casual app roach to minus signs in the argument of the cosine function causes stu dents problems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Finally, we show the solution with the single trig term i s the same as the original y1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "expand(Y2,4*t,exp);\nya1;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$expG6#,$%\"tG!\"$\"\"\"-%$cosG 6#,$F)\"\"%F+\"\"&*&F%F+-%$sinGF.F+!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$%\"tG!\"$\"\"\",&-%$cosG6#,$F(\"\"%\"\"&-%$sinGF.! \"*F*" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Overloading of Expand " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Notice the overloading of " } {TEXT 285 6 "expand" }{TEXT -1 12 ". In fact, " }{TEXT 286 6 "expand " }{TEXT -1 67 " in Maple is used for several related, but not identic al, purposes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand((x+y)*(x-y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "expand(sin(x+y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"xG\"\"\"-%$cosG6#%\"yGF)F)*&-F+F'F)-F&F ,F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expand(sin(3*x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"xG\"\"\"-%$cosGF' \"\"#\"\"%F%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 287 6 "Expand" }{TEXT -1 131 " is used to clear brackets \+ by multiplying out the product of two binomials, to apply the addition formulas in trig, and to express " }{XPPEDIT 18 0 "sin(n*x)" "-%$sinG 6#*&%\"nG\"\"\"%\"xGF'" }{TEXT -1 23 " in terms of powers of " } {XPPEDIT 18 0 "sin(x)" "-%$sinG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(x)" "-%$cosG6#%\"xG" }{TEXT -1 60 ". But the overloading is worse than that. A fourth use of " }{TEXT 288 6 "expand" }{TEXT -1 79 " is to force an integral (or a sum) to demonstrate its linearity, \+ provided the " }{TEXT 289 7 "student" }{TEXT -1 25 " package has been \+ loaded." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "F := Int(a*x+b*x^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%$IntG6$,&*&%\"aG\"\"\"%\"xGF+F+*&%\"bGF+F,\"\"#F+F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&%\"aG\"\"\"%\"xGF)F)*&%\"bGF)F*\"\"#F)F *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "With the " }{TEXT 290 7 "student" }{TEXT -1 17 " package loaded, " }{TEXT 291 6 "expand" }{TEXT -1 21 " behaves differently." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with (student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(F);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"aG\"\"\"-%$IntG6$%\"xGF*F&F&*& %\"bGF&-F(6$*$F*\"\"#F*F&F&" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 " Expand and ODEs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Finally, the be havior of " }{TEXT 292 6 "expand" }{TEXT -1 51 " is scrutinized one la st time since, in Release 4, " }{TEXT 293 6 "dsolve" }{TEXT -1 80 " no w yields, for simple differential equations, solutions in non-standard forms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "q := diff(y(t),t$3)-4*diff(y(t),t,t)+diff(y(t),t)+6*y (t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG,*-%%diffG6$-F'6$-F'6$-% \"yG6#%\"tGF0F0F0\"\"\"F)!\"%F+F1F-\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "q1 := rhs(dsolve(\{q,y(0)=1,D(y)(0)=-1,(D@@2)(y)(0)=0 \},y(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G*&,(#\"#6\"#7\"\" \"*&-%$expG6#,$%\"tG\"\"#F*-F-6#F0F*#F*\"\"$*&-F-6#,$F0F5F*F2F*#!\"\" \"\"%F*F2F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 166 "It is troubling to see, in a first course in different ial eqauations, a solution in such need of having its exponentials com bined. Standard form can be achieved via " }{TEXT 294 6 "expand" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "map(simplify,expand(q1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$%\"tG\"\"##\"\"\"\"\"$-F%6#,$F(F,#!\"\"\" \"%-F%6#,$F(F1#\"#6\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "but in more complex cases the overloading of " }{TEXT 295 6 "expand" }{TEXT -1 68 " requires extra syntax to re strain some of its latent functionality." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "q := diff(y(t),t$3)+ 5*diff(y(t),t,t)+17*diff(y(t),t)+13*y(t) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG/,*-%%diffG6$-F(6$-F(6$-%\"yG6#%\"tGF1F1F1\"\"\"F *\"\"&F,\"# " 0 "" {MPLTEXT 1 0 61 " q1 := rhs(dsolve(\{q,y(0)=1,D(y)(0)=-1,(D@@2)(y)(0)=0\},y(t)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G*&,(#\"\"*\"#5\"\"\"*(-%$expG6#, $%\"tG!\"#F*-%$cosG6#,$F0\"\"$F*-F-6#F0F*#F*F)*(F,F*-%$sinGF4F*F7F*#F* \"#IF*F7!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Here, a naive use of " }{TEXT 296 6 "expand" }{TEXT -1 61 " makes the expression less simple because of what happens to " }{XPPEDIT 18 0 "sin(n*x)" "-%$sinG6#*&%\"nG\"\"\"%\"xGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(n*x)" "-%$cosG6#*&%\"nG\"\"\"%\"xGF'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "expand(q1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, ,*$-%$expG6#%\"tG!\"\"#\"\"*\"#5*&F%!\"#-%$cosGF'\"\"$#\"\"#\"\"&*&F%F .F/\"\"\"#!\"$F,*(F%F.-%$sinGF'F6F/F3#F3\"#:*&F%F.F:F6#F)\"#I" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "I ncluding the parameter 3*t will keep the trig functions from being exp anded, and including the parameter " }{TEXT 297 3 "exp" }{TEXT -1 52 " will keep the exponentials from being converted to " }{XPPEDIT 18 0 " [exp(t)]^n" ")7#-%$expG6#%\"tG%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q2 := exp and(q1,3*t,exp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2G,(*$-%$expG6 #%\"tG!\"\"#\"\"*\"#5*&-F(6#,$F*!\"#\"\"\"-%$cosG6#,$F*\"\"$F4#F4F.*&F 0F4-%$sinGF7F4#F4\"#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "The solution \"in the back of the book\" \+ would have " }{XPPEDIT 18 0 "1/exp(t)" "*&\"\"\"\"\"\"-%$expG6#%\"tG! \"\"" }{TEXT -1 12 " written as " }{XPPEDIT 18 0 "exp(-t)" "-%$expG6#, $%\"tG!\"\"" }{TEXT -1 33 ". That takes the additional step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "m ap(simplify,q2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$%\"tG !\"\"#\"\"*\"#5*&-F%6#,$F(!\"#\"\"\"-%$cosG6#,$F(\"\"$F2#F2F,*&F.F2-%$ sinGF5F2#F2\"#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "All together, it's" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "map(simplify,expand(q1,3* t,exp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$%\"tG!\"\"#\" \"*\"#5*&-F%6#,$F(!\"#\"\"\"-%$cosG6#,$F(\"\"$F2#F2F,*&F.F2-%$sinGF5F2 #F2\"#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "To avoid the quirks of " }{TEXT 298 6 "expand" }{TEXT -1 27 ", the new functionality of " }{TEXT 299 7 "collect" }{TEXT -1 20 " can be exploited. " }{TEXT 300 7 "Collect" }{TEXT -1 82 " now takes \+ function names for the parameter against which to collect common terms ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "collect(q1, exp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&*&,&-%$cosG6#,$%\"tG\"\"$#\"\"\"\"#5-%$sinGF(#F-\"#IF--%$expG6#,$F*! \"#F-F-*$-F46#F*!\"\"#\"\"*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "Of course, \"textbook form\" still re quires" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "map(simplify,collect(q1,exp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$%\"tG!\"\"#\"\"*\"#5*&-F%6#,$F(!\"#\"\"\"- %$cosG6#,$F(\"\"$F2#F2F,*&F.F2-%$sinGF5F2#F2\"#I" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 301 8 "simplify" }{TEXT -1 98 " command \"uncollects\" the common exponentia l terms, so to achieve perfect \"textbook form\" requires" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "coll ect(map(simplify,collect(q1,exp)),exp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&-%$cosG6#,$%\"tG\"\"$#\"\"\"\"#5-%$sinGF(#F-\"#IF--%$expG6 #,$F*!\"#F-F--F46#,$F*!\"\"#\"\"*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The double " }{TEXT 302 7 "coll ect" }{TEXT -1 7 " and a " }{TEXT 303 3 "map" }{TEXT -1 97 " is a lot \+ of extra syntax to have to explain to students in a first differential equations class." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "S urd and an Improper Integral" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "We next give an example of an improper integral that yields to the use o f " }{TEXT 304 4 "surd" }{TEXT -1 25 ". New in Release 3, the " } {TEXT 305 4 "surd" }{TEXT -1 51 " command was introduced when the prin ciple root of " }{XPPEDIT 18 0 "(-8)^(1/3)" "),$\"\")!\"\"*&\"\"\"\"\" \"\"\"$F%" }{TEXT -1 110 " was redefined to be a complex number. For \+ calculus classes where students anticipate the root to be -2, the " } {TEXT 306 4 "surd" }{TEXT -1 51 " function provides the real branch of the nth-root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We illustrate its use in computing an improper integral w ith the following integrand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := 1/(x - 1)^(2/3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*$,&%\"xG\"\"\"!\"\"F(#!\"#\"\"$ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Notice how a naive usage results in a plot showing only part of th e expected graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 18 "plot(f, x = 0..3);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7N7$$\"+tiL-5!\"*$\"+>\")fzc!\")7$$\"1+++!4ET+ \"!#:$\"1__V[cB()Q!#97$$\"1+++1f\"f+\"F1$\"1X1acp;dIF47$$\"1+++Adq25F1 $\"1pUBU3DjDF47$$\"1+++Qb\\45F1$\"1OT9x\"z+B#F47$$\"1+++r^285F1$\"1o.8 W*zF17$$\"1+++UC%[V\"F1$\"1 ?O)HKcAu\"F17$$\"1+++J#>&)\\\"F1$\"1,U?1Ha!f\"F17$$\"1+++>:mk:F1$\"1(H %>=dxj9F17$$\"1+++w&QAi\"F1$\"1Kl&\\1D?P\"F17$$\"1+++uLU%o\"F1$\"1qt1V tg(G\"F17$$\"1+++bjm[&G@\"F17$$\"1+++zb^6=F1$\"1W:ieNR\\ 6F17$$\"1+++MaKs=F1$\"1zQ**4nL&4\"F17$$\"1+++6W%)R>F1$\"1InI\"*yAU5F17 $$\"1+++:K^+?F1$\"10_FH+e'***!#;7$$\"1+++8,Hl?F1$\"1Y_(*=z6(e*Fat7$$\" 1+++4w)R7#F1$\"1&o\"*pWn.D*Fat7$$\"1+++y%f\")=#F1$\"1)y\"p)HBU\"*)Fat7 $$\"1+++/-a[AF1$\"1]ez/ZXC')Fat7$$\"1+++ial6BF1$\"1qJJ&f@bM)Fat7$$\"1+ ++j@OtBF1$\"1[b:z(GO4)Fat7$$\"1+++fL'zV#F1$\"1#z(=S(p$\\yFat7$$\"1+++! *>=+DF1$\"13RD#363j(Fat7$$\"1+++E&4Qc#F1$\"1$\\V'*e&RAuFat7$$\"1+++%>5 pi#F1$\"1s/,g<@HsFat7$$\"1+++bJ*[o#F1$\"1on`n4RiqFat7$$\"1+++r\"[8v#F1 $\"1W%f\")Q'e#)oFat7$$\"1+++Ijy5GF1$\"1f\"ey_Q6t'Fat7$$\"1+++/)fT(GF1$ \"1[JqC4`ylFat7$$\"1+++1j\"[$HF1$\"1d`.:5JSkFat7$$\"\"$\"\"!$\"1mVZ\\_ g*H'Fat-%'COLOURG6&%$RGBG$\"#5!\"\"F`yF`y-%+AXESLABELSG6$%\"xG%!G-%%VI EWG6$;F`yF^y%(DEFAULTG" 2 305 305 305 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 5 0 0 0 0 0 0 }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Apparently, Maple does not see th e expression in f as the \"cube root of a square.\" Instead, Maple se es it as something raised to the power 2/3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Surd" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "For completeness, we remind the reader of the beh avior of the nth-root function in Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify((-8)^(1/3)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&%\"IGF$\"\"$#F$\"\"#F $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "convert((-8)^(1/3),sur d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 307 4 "surd " }{TEXT -1 44 " function is ordinarily available through a " }{TEXT 308 7 "readlib" }{TEXT -1 43 " call, but we will invoke it by use of t he " }{TEXT 309 7 "convert" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "fs := conve rt(f, surd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fsG*$-%%surdG6$*$,& %\"xG\"\"\"!\"\"F,\"\"#\"\"$F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "Conversion in the opposite direction \+ is achieved by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "f1 := convert(fs, power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G*$*$,&%\"xG\"\"\"!\"\"F)\"\"##F*\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "In this \+ form, Maple will indeed see the expression in f1 as the \"cube root of a square.\" In fact, the graph of f1 is the expected graph for Calcu lus II." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(f1, x = 0..3, y = 0..10);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7eo7$\"\"!$\"\"\"F(7$$\"1+++]i9Rl!#<$\"1KLq&y; h/\"!#:7$$\"1++vVA)GA\"!#;$\"1md[s3&34\"F17$$\"1++]Peui=F5$\"1CQ$Rn6t9 \"F17$$\"1++]i3&o]#F5$\"1.Zz**>:77F17$$\"1++voX*y9$F5$\"1/6b>5i'G\"F17 $$\"1++vVTAUPF5$\"11\"GaHZoO\"F17$$\"1++v$*zhdVF5$\"1/7>cx]k9F17$$\"1+ +v$>fS*\\F5$\"1\"HjL!\\9'e\"F17$$\"1++v=$f%GcF5$\"10$G9)*3ht\"F17$$\"1 +++Dy,\"G'F5$\"1QkMF17$$\"1++]7jw;Fgq7$$\"1kS\"p$pYv)*F5$\"1t#H*4OFh=F gq7$$\"1E1k1\\f'*)*F5$\"1?>MAZ(o5#Fgq7$$\"1)=nj(Gs<**F5$\"1`%=sXdOX#Fg q7$$\"1]P4Y3&)Q**F5$\"1E\">6rr/*HFgq7$$\"17.#e\")y*f**F5$\"1gB8!4-r'RF gq7$$\"1voa&y16)**F5$\"1')e7A*)GVlFgq7$$\"1Wt_vMA+5F1$\"1VAj&[![:F!#87 $$\"1++]siL-5F1$\"12'*yH*)fzcFgq7$$\"1v=n)3ET+\"F1$\"13kf#[Os)QFgq7$$ \"1]P%[!f\"f+\"F1$\"1&z%)[Nnr0$Fgq7$$\"1Dc,@dq25F1$\"1d4`g5DjDFgq7$$\" 1+v=Pb\\45F1$\"13vN/$z+B#Fgq7$$\"1]7`p^285F1$\"1]61z+y,=Fgq7$$\"1+](=! [l;5F1$\"10`qkmML:Fgq7$$\"1+DcmS\"Q-\"F1$\"1X\"H)eC737Fgq7$$\"1++DJL(4 .\"F1$\"1VYs\">IR,\"Fgq7$$\"1+]ig=HX5F1$\"1$GF!4V@qyF17$$\"1+++!R5'f5F 1$\"1YkS/*yJb'F17$$\"1+v=(4AH4\"F1$\"11lN$Q%Ru[F17$$\"1+]P/QBE6F1$\"1R BfeM*Q(RF17$$\"1+vo4.sb6F1$\"1ba*f03\\X$F17$$\"1+++:o?&=\"F1$\"1yK1bqr xIF17$$\"1+]Pa&4*\\7F1$\"1FZfR+X?DF17$$\"1+]7j=_68F1$\"13!>%os2w@F17$$ \"1++vVy!eP\"F1$\"1hm$yiU-#>F17$$\"1+](=WU[V\"F1$\"12vJBjDU&)\\\"F1$\"1XCn0Ha!f\"F17$$\"1+]P>:mk:F1$\"1?ia&G@\"F17$$\"1++vyb^6=F1$\"1/w&)eNR\\6F17$$\"1+]PMaKs=F1$\"1 m*z'4nL&4\"F17$$\"1++D6W%)R>F1$\"12>7\"*yAU5F17$$\"1+++:K^+?F1$\"10_FH +e'***F57$$\"1++]7,Hl?F1$\"1#4v>#z6(e*F57$$\"1+]P4w)R7#F1$\"1(>M\\Wn.D *F57$$\"1++]x%f\")=#F1$\"1QE>,LA9*)F57$$\"1+]P/-a[AF1$\"1W*oIqaWi)F57$ $\"1+](=Yb;J#F1$\"1'QVef@bM)F57$$\"1++]i@OtBF1$\"1!)*>6yGO4)F57$$\"1+] PfL'zV#F1$\"1BJ#)Q(p$\\yF57$$\"1+++!*>=+DF1$\"13RD#363j(F57$$\"1++DE&4 Qc#F1$\"1MC&))e&RAuF57$$\"1+]P%>5pi#F1$\"1#f**)e<@HsF57$$\"1+++bJ*[o#F 1$\"1on`n4RiqF57$$\"1++Dr\"[8v#F1$\"1lW](Q'e#)oF57$$\"1+++Ijy5GF1$\"1f \"ey_Q6t'F57$$\"1+]P/)fT(GF1$\"1;c#Q#4`ylF57$$\"1+]i0j\"[$HF1$\"1=v'e, 6.W'F57$$\"\"$F($\"1mVZ\\_g*H'F5-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXE SLABELSG6$%\"xG%\"yG-%%VIEWG6$;F(F^bl;F($FgblF(" 2 285 285 285 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 " The " }{TEXT 310 4 "surd" }{TEXT -1 124 " function becomes less strang e if we can make f1 look exactly like the original expression (by comb ining exponents). Hence," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(f1,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&%\"xG\"\"\"!\"\"F&#!\"#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Plotting \+ the " }{TEXT 311 4 "surd" }{TEXT -1 113 "-representation will also yie ld the expected graph. Additionally, the desired graph can be obtaine d by plotting " }{XPPEDIT 18 0 "[1/abs(x-1)^2]^(1/3)" ")7#*&\"\"\"\"\" \"*$-%$absG6#,&%\"xGF&\"\"\"!\"\"\"\"#F.*&\"\"\"F&\"\"$F." }{TEXT -1 3 ", " }{XPPEDIT 18 0 "1/abs(x-1)^(2/3)" "*&\"\"\"\"\"\")-%$absG6#,&% \"xGF$\"\"\"!\"\"*&\"\"#F$\"\"$F,F," }{TEXT -1 5 " or " }{XPPEDIT 18 0 "abs((x-1))^(-2/3)" ")-%$absG6#,&%\"xG\"\"\"\"\"\"!\"\",$*&\"\"#F(\" \"$F*F*" }{TEXT -1 15 ", or by typing " }{XPPEDIT 18 0 "1/((x-1)^2)^(1 /3)" "*&\"\"\"\"\"\")*$,&%\"xGF$\"\"\"!\"\"\"\"#*&\"\"\"F$\"\"$F*F*" } {TEXT -1 312 " into Maple and working with the equivalent of expressio n f1. Maple will integrate and simplify f1, will integrate but not s implify the second and third expressions containing absolute values, a nd will not integrate the first expression containing the absolute val ue. Hence, it would be possible to avoid the " }{TEXT 312 4 "surd" } {TEXT -1 61 " construct in this case if due care were exercised initia lly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Integrating Surd" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "M aple provides the indefinite integral of the " }{TEXT 313 4 "surd" } {TEXT -1 16 "-representation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "test := int(fs, x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG,$*&,&%\"xG\"\"\"!\"\"F)F)-%%s urdG6$*$F'\"\"#\"\"$F*F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "To express this antiderivative in the fo rm a calculus student would recognize, we convert back from the " } {TEXT 314 4 "surd" }{TEXT -1 9 " notation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "test1 := convert(tes t, power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&test1G,$*&,&%\"xG\"\" \"!\"\"F)F)*$F'\"\"##F*\"\"$F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "and simplify." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(te st1,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$,&%\"xG\"\"\"!\" \"F'#F'\"\"$F*" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The Improper \+ Integral" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The improper integral \+ of interest is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "q := Int(f, x = 0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG-%$IntG6$*$,&%\"xG\"\"\"!\"\"F+#!\"#\"\"$/F*;\"\" !F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The improper integral is handled by splitting it into two inte grals, each involving a limit, and each with an integrand expressed in " }{TEXT 315 4 "surd" }{TEXT -1 71 "-notation. We set up each of the integrals as if the small parameters " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 15 " were \+ positive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "q1 := Int(fs, x = 0..1 - a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G-%$IntG6$*$-%%surdG6$*$,&%\"xG\"\"\"!\"\"F/\"\"# \"\"$F0/F.;\"\"!,&F/F/%\"aGF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "q2 := Int(f, x = 1 + b .. 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#q2G-%$IntG6$*$,&%\"xG\"\"\"!\"\"F+#!\"#\"\"$/F*;,&F+F+%\"bGF+F/" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 " Evaluating each of these definite integrals,we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "q3 := value (q1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q3G,$*&,&%\"aG\"\"\"*$*$F( \"\"##F)\"\"$!\"\"F)F+#F/F.!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "q4 := value(q2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q4G,&*$ \"\"##\"\"\"\"\"$F**$%\"bGF(!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We are interested in the limits of these expressions as " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 35 " each approach zero fr om the right." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "q5 := limit(q3, a = 0, right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q5G\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "q6 := limit(q4, b = 0, right);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#q6G,$*$\"\"##\"\"\"\"\"$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The sum of these limits is the value of the improper integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q7 := q5 + q6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q7G,&\"\"$\"\"\"*$\"\"##F'F&F&" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Cauchy Principal Value" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "If we were to let the parameters " }{XPPEDIT 18 0 "a" "I \"aG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 86 " be equal, we would compute the Cauchy Principal Value of the give n improper integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "q8 := Int(fs, x = 0..1 - a) + Int(f, x = \+ 1 + a .. 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q8G,&-%$IntG6$*$-%% surdG6$*$,&%\"xG\"\"\"!\"\"F0\"\"#\"\"$F1/F/;\"\"!,&F0F0%\"aGF1F0-F'6$ *$F.#!\"#F3/F/;,&F0F0F8F0F3F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Evaluating the integrals we get" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "q9 := value(q8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q9G,(*&,& %\"aG\"\"\"*$*$F(\"\"##F)\"\"$!\"\"F)F+#F/F.!\"$*$F,F-F.*$F(F-F1" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Th e limit as " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 72 " approaches z ero from the right is therefore the Cauchy Principal Value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q 10 := limit(q9, a = 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q10G,&\" \"$\"\"\"*$\"\"##F'F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "Note that Maple has a built-in routine fo r computing the Cauchy Principal Value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "q11 := int(fs, x = 0.. 3, CauchyPrincipalValue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q11G,& \"\"$\"\"\"*$\"\"##F'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Maple is an n-digit Machine" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "In Maple, numeric differentiation of " }{XPPEDIT 18 0 "exp(x)" "-% $expG6#%\"xG" }{TEXT -1 341 " generates output I always found perplexi ng. An e-mail exchange with Keith Geddes found him clearly and emphat ically stating that \"Maple behaves like an n-digit computer.\" By th is he meant that when the user specifies \"Digits:=n\", Maple behaves \+ like a calculator with precisely n decimal digits. However, earlier e xperiences with Maple's " }{TEXT 316 6 "Digits" }{TEXT -1 507 " parame ter had left me with the impression that this might not be the case, a nd that vague sense prevented me from accepting Keith's explanation of the numeric differentiation results. It wasn't until I went back to \+ some numerical analysis texts and checked their examples of n-digit ar ithmetic - and found many of them wrong - that I uncovered the source \+ of my suspicion that Maple might not be an n-digit computer. I had fo rmed my impression by incorrectly assuming the examples in the texts w ere right!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 287 "Once I verified that Maple is indeed an n-digit machine, I was able to see through the original difficulty I had had with the numeri c differentiation results. The surprising outcome was the realization that many texts, my own included, have errors in the examples of n-di git arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Here are the details." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Numeric Differentiation" }} {PARA 0 "" 0 "" {TEXT -1 59 "Consider the numeric computation of f'(1) for the function " }{XPPEDIT 18 0 "f(x)=exp(x)" "/-%\"fG6#%\"xG-%$exp G6#F&" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "for k from 1 to 10 do\nevalf((exp(1+10^(-k))-exp(1))/10^(-k));\nod;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*'>%)eG!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")(=>t#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(U' >F!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'U=F!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&$=F!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" %>F!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$s#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" \"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 565 "Always at a loss \+ to explain the disappearance of digits in this calculation, I appealed to Keith Geddes for insight. \"Is Maple programmed to detect the los s of significance? Does Maple intelligently deliver only those digits it knows are correct?\" \"No,\" Keith replied. \"It's simply a matt er of subtractive cancellation on a finite precision machine.\" Of co urse, there was an extended explanation provided by Keith, but that ex planation could only be accepted if I believed that Maple was indeed b ehaving as a 10-digit computer. I just didn't believe that yet." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 209 "I didn't believe that, because my recollection of examples from numerical anal ysis texts was of discrepancies between Maple and the examples. The e xamples had to be revisited before this was going to be over." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Text Examples" }}{PARA 265 "" 0 "" {TEXT -1 247 "Every numerical anal ysis text I've examined contains examples of 3- or 4-digit arithmetic \+ intended to illustrate the effects of computation on a finite precisio n machine. My own text [1] contains, on page 183, the 3-digit example of row-reducing " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "A := matrix([[-.72e-1, .3, -.21], [.874, -.26 7, .133], [-.501, -.123, .125]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"AG-%'MATRIXG6#7%7%$!#s!\"$$\"\"$!\"\"$!#@!\"#7%$\"$u)F,$!$n#F,$\"$L \"F,7%$!$,&F,$!$B\"F,$\"$D\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "to" }}{PARA 266 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "MATRIX([[.874,-.267,.133],[0,.278,-.199],[0,0,.00339 ]])" "-%'MATRIXG6#7%7%$\"$u)!\"$,$$\"$n#!\"$!\"\"$\"$L\"!\"$7%\"\"!$\" $y#!\"$,$$\"$*>!\"$F.7%F3F3$\"$R$!\"&" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 96 "after interchanging the first and sec ond rows. Stepping through the calculation in Maple yields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 148 "a := swaprow(A,2,1):\nDigits := 3:\na1 := add row(a, 1,2,-a[2,1]/a[1,1]):\na2 := addrow(a1,1,3,-a1[3,1]/a1[1,1]);\na 3 := addrow(a2,2,3,-a2[3,2]/a2[2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a2G-%'MATRIXG6#7%7%$\"$u)!\"$$!$n#F,$\"$L\"F,7%\"\"!$\"$y#F,$!$* >F,7%F2$!$w#F,$\"$,#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a3G-%'MAT RIXG6#7%7%$\"$u)!\"$$!$n#F,$\"$L\"F,7%\"\"!$\"$y#F,$!$*>F,7%F2F2$\"\"$ F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 291 "The (3,3)-elements don't agree. It's tedious to check, but th ere is no other way to decide which version of the row reduction is co rrect. The arithmetic has to be checked by hand, a calculation showin g the error creeps in at the last moment. The final arithmetic for th e (3,3)-element is " }{XPPEDIT 18 0 ".201-[-.276/.278]*(-.199)" ",&$\" $,#!\"$\"\"\"*&7#,$*&$\"$w#!\"$F&$\"$y#!\"$!\"\"F1F&,$$\"$*>!\"$F1F&F1 " }{TEXT -1 17 " . The pivot is " }{XPPEDIT 18 0 ".276/.278=.99280575 5" "/*&$\"$w#!\"$\"\"\"$\"$y#!\"$!\"\"$\"*bd!G**!\"*" }{TEXT -1 278 " \+ which rounds to .993. The product of .993 and .199 is .197607 which s hould round to .198, leading to .201 - .198 = .003, Maple's correct re sult. However, if .197607 is not rounded to three places, we have the subtraction .201 - .197607 = .003393, the text's incorrect result." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "On page \+ 148 of [2] we find the matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "A := matrix([[-.002,4.,4.,7. 998],[-2.,2.906,-5.387,-4.481],[3.,-4.031,-3.112,-4.143]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7&$!\"#!\"$$\"\"%\"\"!F- $\"%)*zF,7&$F+F/$\"%1HF,$!%(Q&F,$!%\"[%F,7&$\"\"$F/$!%JSF,$!%7JF,$!%VT F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "row reduced, using 4-digit arithmetic without pivoting, to" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "MATRIX([[-.002,4.,4.,7.998],[0,-3.997,-4.005,-8.002],[0 ,0,-10.,0]])" "-%'MATRIXG6#7%7&,$$\"\"#!\"$!\"\"$\"\"%\"\"!$\"\"%F.$\" %)*z!\"$7&F.,$$\"%(*R!\"$F+,$$\"%0S!\"$F+,$$\"%-!)!\"$F+7&F.F.,$$\"#5F .F+F." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " Checking this result in Maple, find" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Digits := 4;\nA1 := pivot( A, 1,1,2..3);\nA2 := pivot(A1,2,2,3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'MATR IXG6#7%7&$!%+?!\"'$\"\"%\"\"!F-$\"%)*z!\"$7&F/$!%(*RF/$!%0SF/$!%-!)F/7 &F/$\"%'*fF/$\"%(*fF/$\"%+7\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#A2G-%'MATRIXG6#7%7&$!%+?!\"'$\"\"%\"\"!F-$\"%)*z!\"$7&F/$!%(*RF/$!%0 SF/$!%-!)F/7&F/F/$!#6F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 179 "The middle rows differ by a factor of 10 00 and the last row has -10 not -11. Hand calculation shows Maple is \+ correct. In fact, the first pivot is -2/(-.002) = 1000 so we compute " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "-2 - \+ [1000(-.002)] = -2 - [2] = 0" }}{PARA 0 "" 0 "" {TEXT -1 38 "2.906 - [ 1000(4)] = -3997.094 -> -3997" }}{PARA 0 "" 0 "" {TEXT -1 39 "-5.387 - [1000(4)] = -4005.387 -> -4005" }}{PARA 0 "" 0 "" {TEXT -1 61 "-4.481 - [1000(7.998)] = -4.481 - [7998] = -8002.481 -> -8002" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The second pivot is \+ 3/(-.002) = -1500 so we compute" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "3 - [-1500(-.002)] = 3 - 3 = 0" }}{PARA 0 "" 0 "" {TEXT -1 55 "-4.031 - [-1500(4)] = -4.031 + 6000 = 5.995.969 -> 5996" }}{PARA 0 "" 0 "" {TEXT -1 82 "-4.143 - [-1500(7.998)] = -4. 143 - [-11997] -> -4.143 + 12000 = 12004.143 -> 12000" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The third pivot is 599 6/(-3997)=1.50012 -> -1.5 so we compute" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "5996 - [-1.5(-3997)] = 5996 - [5995. 5] -> 5996 - 5996 = 0" }}{PARA 0 "" 0 "" {TEXT -1 59 "5997 - [-1.5(-40 05)] = 5997 - [6007.5] -> 5997 - 6008 = -11" }}{PARA 0 "" 0 "" {TEXT -1 57 "1200 - [-1.5(-8002)] = 12000 - 12003 -> 12000 - 12000 = 0" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 256 "It is on ly after grinding out one of these calculations that we appreciate why errors creep in, and why the arithmetic is so rarely checked by hand. And this one error in [2] is not isolated. In fact, on the very nex t page of that text we find the matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "A := matrix([[3.02, -1. 05, 2.53, -1.61], [4.33, .56, -1.78, 7.23], [-.83, -.54, 1.47, -3.38]] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7&$\"$-$!\"# $!$0\"F,$\"$`#F,$!$h\"F,7&$\"$L%F,$\"#cF,$!$y\"F,$\"$B(F,7&$!#$)F,$!#a F,$\"$Z\"F,$!$Q$F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "row-reduced, using pivoting and 3-digit arithmeti c, to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "MATRIX([[4.33, .56, -1.78, 7.23], [0, -1.44, 3.77, -6.65], [0, 0, -.362e-2, .962e-2]])" "-%'MATRIXG6#7%7&$\"$L%!\"#$\"#c !\"#,$$\"$y\"!\"#!\"\"$\"$B(!\"#7&\"\"!,$$\"$W\"!\"#F1$\"$x$!\"#,$$\"$ l'!\"#F17&F6F6,$$\"$i$!\"&F1$\"$i*!\"&" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "whereas Maple yields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Digits : = 3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a := swaprow(A,2,1) :\na1 := pivot(a,1,1,2..3);\na2:=pivot(a1,2,2,3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1G-%'MATRIXG6#7%7&$\"$L%!\"#$\"$g&!\"$$!$y\"F,$ \"$B(F,7&\"\"!$!$W\"F,$\"$x$F,$!$l'F,7&$\"$+\"!\"&$!$K%F/$\"$8\"F,$!$* >F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a2G-%'MATRIXG6#7%7&$\"$L%!\" #$\"$g&!\"$$!$y\"F,$\"$B(F,7&\"\"!$!$W\"F,$\"$x$F,$!$l'F,7&$\"$+\"!\"& F5F5$F>!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 103 "The third row of Maple's a1 is correct, since we get f or the pivot -.83/4.33 = -.19168 -> -.192. Then," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "-.83 - [-.192(4.33)] = -. 83 + [.83136] -> -.83 + .831 = .001" }}{PARA 0 "" 0 "" {TEXT -1 61 "-. 54 - [-.192(.56)] = -.54 + [.100752] -> -.54 + .108 = -.432" }}{PARA 0 "" 0 "" {TEXT -1 70 "1.47 - [-.192(-1.78)] = 1.47 - [.34176] -> 1.47 - .342 = 1.128 -> 1.13" }}{PARA 0 "" 0 "" {TEXT -1 65 "-3.38 - [-.192 (7.23)] = -3.38 + [1.38816] -> -3.38 + 1.39 = -1.99" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Moreover, the third ro w of Maple's a2 is correct since we get for the pivot -.432/(-1.44) = \+ .3. Then," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "-.432 - [.3(-1.44)] = -.432 + .432 = 0" }}{PARA 0 "" 0 "" {TEXT -1 52 "1.13 - [.3(3.77)] = 1.13 - [1.131] ->1.13 - 1.13 = 0" }} {PARA 0 "" 0 "" {TEXT -1 56 "-1.99 - [.3(-6.65)] = -1.99 + [1.995] -> \+ -1.99 + 2 = .01" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 331 "Wretchedly tedious as it is, I had to go through this ar ithmetic before I could accept Maple as an n-digit computer. My relia nce on text examples was not warranted, since errors appear in them so readily. Once Maple is accepted as an n-digit calculator, the anomal ous output from the numeric differentiation is easily understood." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Resolution" }}{PARA 0 "" 0 "" {TEXT -1 162 "It is now a simple matter of subtracting 10-digi t numbers to understand the numeric differentiation example. Consider , for example, the value obtained for k = 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Digits := 10:\nq1 \+ := evalf(exp(1+10^(-7)));\nq2 := evalf(exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G$\"++@G=F!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#q2G$\"+G=G=F!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "On a 10-digit machine, the subtraction \"elimi nates\" seven of the digits. There are only three digits left in the \+ mantissa of the difference." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "q1 - q2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$s#!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "Of \+ course I always understood fixed precision arithmetic. But only after I realized that Maple was truly an n-digit calculator could I see why the numeric differentiation produced the results seen above." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Styles" }}{PARA 0 "" 0 "" {TEXT -1 477 "Classroom users of Maple were among those who reque sted named styles that could be applied to worksheets. Before Release 4, most instructors using worksheets in class would have to keep mult iple copies of each worksheet, one copy for each style needed for a wo rksheet. There might be a \"working style,\" the style in which the w orksheet is authored. There might be a style for printing the workshe et for distribution to students, and a style for projecting to a large screen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 348 "In Release 4 there are now named styles that can be applied to a \+ single copy of a worksheet, converting the appearance of the worksheet on demand. However, it took me several months to begin to understand how to use \"styles,\" and it took a call to support to discover that the \"Merge Existing\" button was the way to \"apply\" a style to a w orksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 836 "Since the process of \"naming\" a style requires using the \"Save as Default\" button, the very first time you experiment with naming a style you will lose the pre-sets for the style parameters. This happ ens so soon in the learning cycle that it generally happens before the user is even aware of it. The good folks at support tell me, however , that the default settings are \"hard wired\" into Maple \"from the f actory.\" When these original defaults are changed, the change is rec orded, under Windows, in a file Windows/maplev4.ini. (I don't know th e situation for the Mac and UNIX platforms.) In the Windows directory , this file contains various sections, one of which is the [Files] sec tion. The original factory pre-sets can be re-established by changing , in this [Files] section, the line mentioning \"Template\" to read \" Template=.\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 364 "In other words, you can name styles and record those sty les as individual files, but the pre-set style does not have a compara ble name or file corresponding. If you want to give that pre-set styl e a name and a standing equal to that of any other style you create an d name, then before performing any experiments with styles, name and s ave the pre-sets as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 569 "From the FORMAT menu select STYLES. The \"Sty le Management\" dialog box appears. Click the \"Save As Default\" but ton and the \"Save Style Set\" dialog box appears. Enter the name of \+ a file in which the factory pre-set style is to be stored. I chose \" Original.mws\". There is also available in the \"Save Style Set\" dia log box a \"Folders\" section where you can choose where to save this \+ file. (I have created a special folder of just Maple styles where I s ave all my named styles.) Click \"OK.\" This returns you to the \"St yles Management\" window, and here, click \"Done\"." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 264 "If you now experiment \+ with styles and wish to revert to the factory pre-sets, you can either use the style file just created or you can edit your Windows/maplev4. ini file. I believe it is easier to use the style sheet \"Original.mw s\" and not edit the Windows file." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 395 "Now that we have the confidence our expe riments with styles can always be reversed, let's consider how to crea te and use an alternate style. Suppose the need is for a \"projection \" style with input, output, and text all black, text and input as bol d, and type sizes of 12 points. Since the Toolbar contains a zoom but ton for magnification, the projection style need not be inherently mag nified." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 479 "From the FORMAT menu, select STYLES. We'll change the characteri stics of text first. In the upper left portion of the \"Styles Manage ment\" window, in the panel \"Styles\" select \"Normal.\" This refers to text. Click the \"Modify\" button. The\010\"Paragraph Style\" wi ndow opens. Click the \"Font\" button in the lower left portion of th is window. You now have access to properties of the input font. Sele ct \"bold\" and \"12\" and click \"OK\". Click \"OK\" on the \"Paragr aph Style\" window." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 309 "Next, we change Maple input from red to black (and mak e it bold) by selecting \"Maple Input\" (very last item in the list) i n the Styles panel. Click the \"Modify\" button, and make the appropr iate changes in the \"Character Style\" window that opens. Hence, sel ect \"bold\", choose the color black, and click \"OK\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 228 "Finally, change the color of two-dimensional, true-math output. Select \"2D Output\" fro m the \"Styles\" panel and click the \"Modify\" button. A \"Character Style\" window opens from which you can select the color black. Clic k \"OK\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 509 "All the desired choices have been made for the new style. Next, \+ on the open \"Style Management\" window, click the \"Save As Default\" button. (I find it strange that the only way to save a style in a st yle file is to designate it as a default. That is why, after my initi al experiments with styles, I went back and investigated how to restor e the factory pre-sets. The instant you try to save your first test s tyle you lose the pre-sets because naming a style can only be done by \+ changing the default style.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 567 "Just as for the process by which we created th e named style \"Original.mws\", choose a name for the projection style , and select a folder in which to store it. Click \"OK\". You are ag ain looking at the \"Style Management\" window. You have already chan ged your default style to the projection style. Even if you click the button \"Revert to Default\" you have changed that setting in your Wi ndows/maplev4.int file. If you click the \"Done\" button, the \"Style Management\" window will disappear and you will be looking at a works heet that is in the new \"Projection\" style." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 658 "To change the style of t his worksheet back to your working style, say the factory pre-set styl e saved in the style file \"Original.mws\", choose STYLES from the FOR MAT menu, then click the \"Merge Existing\" button on the \"Style Mana gement\" window. The \"Merge Existing\" button opens the \"Merge Styl es Set\" window where you can select the folder in which your style fi les reside, and the file name of the style you wish to apply. Make th ese choices, thereby selecting \"Original.mws\". Click \"OK\" and you are back at the \"Styles Management\" window. If you click \"Done\" \+ your worksheet will appear in the factory pre-set style captured in th e file \"Original.mws\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 787 "But suppose you also want to restore the factory \+ pre-sets as the default style. When you have chosen \"Original.mws\" \+ from the \"Merge Style Set\" window, you can then click on the \"Save \+ As Default\" button in the \"Styles Management\" window. This opens t he \"Save Style Set\" window where you can navigate to the folder and \+ file name of the named style you wish to set as default. When you cho ose these options, and click \"OK\", you will be told that the name al ready exists and do you want to over-write this file. As far as I can tell, you have to say \"yes\" if you want to change your default styl e. Finally, if you click \"Done\" in the \"Styles Management\" window , you will have changed your default style back to the factory pre-set s recorded in the \"Original.mws\" file created initially." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 350 "Of course, the p rocess of applying any named style, and setting any named style as def ault, is the same as described above, mutatis mutandi. I did not find any of this intuitive, and had to spend a great deal of time experime nting. Perhaps those already familiar with styles from a word process or will not find managing styles as daunting as I did." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 110 "[1] Melvin J. Maron and Robert J. Lopez, Numerical Analysis: A Pr actical Approach, 3rd ed., Wadsworth, (1991)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "[2] Curtis F. Gerald and Patrick O. Wheatley, Applied Numerical Analysis, 5th ed., Addison-Wes ley, (1994)." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Summary" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 317 5 "match" }{TEXT -1 90 " comman d as a way of establishing mathematical identities is examined. It tu rns out that " }{TEXT 318 5 "match" }{TEXT -1 23 " is not as powerful \+ as " }{TEXT 319 14 "solve/identity" }{TEXT -1 169 ". To avoid having \+ students learn two different ways of handling these problems, the latt er syntax should be used from the very beginning. Incidental to the u se of the " }{TEXT 320 14 "solve/identity" }{TEXT -1 51 " syntax, the \+ overload of functionality for Maple's " }{TEXT 321 6 "expand" }{TEXT -1 21 " command is examined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "An improper integral requiring the use of the \+ " }{TEXT 322 4 "surd" }{TEXT -1 36 " command for a correct rendering o f " }{XPPEDIT 18 0 "x^(1/3)" ")%\"xG*&\"\"\"\"\"\"\"\"$!\"\"" }{TEXT -1 112 "is explored. The Cauchy Principal Value is also implemented a nd compared to Maple's built-in CPV functionality." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Numeric differentiation o f " }{XPPEDIT 18 0 "exp(x) " "-%$expG6#%\"xG" }{TEXT -1 255 " yields a n enigmatic result that only becomes transparent when Maple's behavior as an \"n-digit calculator\" is understood. A barrier to acceptance \+ of Maple as such a calculator is the prevalence of errors in n-digit a rithmetic in numerical analysis texts." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "Finally, a discussion of creating, na ming, and applying worksheet styles is given." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Biography" }} {PARA 0 "" 0 "" {TEXT -1 718 "With a Purdue University Ph.D. (1970) in Relativistic Cosmology, Robert J. Lopez is a classically trained appl ied mathematician. After a short stint at the University of Nebraska- Lincoln, he spent 12 years at Memorial University in St. John's, Newfo undland, Canada, an odyssey of cod fish, ice hockey, and long gray win ters. At Rose-Hulman Institute of Technology since 1985 where he pion eered the use of Maple in the classroom, he has authored books and pap ers, represented Maple \"on the road\" for 30 months, and received his Institute's awards for both teaching excellence and distinguished sch olarship. He continues to promote technology as an active partner in \+ undergraduate instruction and curriculum revision." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 267 "" 0 "" {HYPERLNK 17 "Part 1" 1 "tips1.mws " "" }{TEXT -1 3 " - " }{HYPERLNK 17 "Part 3" 1 "tips3.mws" "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }