{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 42 "Examples involving the numerical \+ functions" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C ., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2007" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "load numerical functions" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 8 "numfcn.m" }{TEXT -1 61 " contains the code for the alternative mathematical functions" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a \+ Maple session by a command similar to the one that follows, where the \+ file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \"K:\\\\Maple/procdrs/numfcn.m\";" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "load root-finding procedur es including: " }{TEXT 0 15 "secant,allroots" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 7 "roots.m" }{TEXT -1 38 " contai ns the code for the procedures " }{TEXT 0 6 "secant" }{TEXT -1 5 " and " }{TEXT 0 8 "allroots" }{TEXT -1 25 " used in this worksheet. " }} {PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives it s location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\ Maple/procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "load numerical integration procedures and data" } }{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 262 6 "intg.m " }{TEXT -1 5 " and " }{TEXT 262 8 "gkdata.m" }{TEXT -1 67 " contain t he code and data for the numerical integration procedure " }{TEXT 0 8 "quad/Int" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They can be read into a Maple session by commands simila r to those that follow, where the file paths give their location. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \"K:\\\\Maple/procdrs/i ntg.m\";\nread \"K:\\\\Maple/procdrs/gkdata.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 " f(x) = (cos(3*x)-sin(x))*(arcsin(x)+arccos(x^3))+exp(arctan(x))*ln(tan (x)+1);" "6#/-%\"fG6#%\"xG,&*&,&-%$cosG6#*&\"\"$\"\"\"F'F0F0-%$sinG6#F '!\"\"F0,&-%'arcsinG6#F'F0-%'arccosG6#*$F'F/F0F0F0*&-%$expG6#-%'arctan G6#F'F0-%#lnG6#,&-%$tanG6#F'F0F0F0F0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "We solve the equation " }{XPPEDIT 18 0 "f(x)=0" "6#/ -%\"fG6#%\"xG\"\"!" }{TEXT -1 41 " using the \"special\" numerical fun ctions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "f := x -> (cos_(2*x)-sin_(x))*(arcsin_(x)+arccos_(x^ 3))+\n exp_(arctan_(x))*ln_(tan_(x)+1);\ngraph(f(x),x=-0.75 ..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operato rG%&arrowGF(,&*&,&-%%cos_G6#,$*&\"\"#\"\"\"9$F5F5F5-%%sin_G6#F6!\"\"F5 ,&-%(arcsin_GF9F5-%(arccos_G6#*$)F6\"\"$F5F5F5F5*&-%%exp_G6#-%(arctan_ GF9F5-%$ln_G6#,&-%%tan_GF9F5F5F5F5F5F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7eo7$$!3++++++++v!#=$ !3]*****4HD*z`F*7$$!3++++D1kJuF*$!3%)*****RWH;Y%F*7$$!3++++]7GjtF*$!3! ******fedfn$F*7$$!3++++v=#\\H(F*$!3G+++]Kn()HF*7$$!3+++++DcEsF*$!33+++ `Z$QP#F*7$$!3++++]P%)*3(F*$!3(******p%H*3J\"F*7$$!3+++++]7`pF*$!3m**** **fR!)fS!#>7$$!3++++]iS;oF*$\"3;+++S-7sQFK7$$!3+++++vozmF*$\"3-+++<'zv 4\"F*7$$!3++++](oHa'F*$\"3,+++))=MWMF*7$$!3+++++]PfeF*$\"3-+++yNz'Q%F*7 $$!3+++++v$fe&F*$\"3a*****H()fJF&F*7$$!3++++++]7`F*$\"3-+++Rr+*4'F*7$$ !3+++++D1R]F*$\"3A+++K&3w(oF*7$$!3+++++]ilZF*$\"3S+++1!3zh(F*7$$!3)*** *****\\(=#\\%F*$\"3))*****zO>fK)F*7$$!3++++++v=UF*$\"3y******=Wd0!*F*7 $$!3+++++DJXRF*$\"3K+++2QEf'*F*7$$!3+++++](=n$F*$\"3\"******>()>)G5!#< 7$$!3+++++vV)R$F*$\"3!*******>jE*3\"F\\r7$$!3+++++++DJF*$\"3/+++i)>s9 \"F\\r7$$!3+++++]7yDF*$\"35+++'4v^D\"F\\r7$$!3++++++DJ?F*$\"3#*******Q ZI^8F\\r7$$!3+++++]P%[\"F*$\"3)******f2+N-]\"F\\r7$$!3+++++]iSmFK$\"3!******RfBo_\"F\\r7$$!3++++++D1RF K$\"3'*******=7f[:F\\r7$$!3+++++](=<\"FK$\"3/+++'fp_c\"F\\r7$$\"3+++++ +]i:FK$\"31+++7kgw:F\\r7$$\"3+++++](oH%FK$\"33+++]-P#e\"F\\r7$$\"3y*** ***Gt,dcFK$\"3-+++xo4$e\"F\\r7$$\"3++++++DJqFK$\"33+++h`N#e\"F\\r7$$\" 3-++++]il(*FK$\"3%******RA'Qw:F\\r7$$\"3+++++++]7F*$\"31+++2>Kk:F\\r7$ $\"3+++++vVB:F*$\"37+++#>gga\"F\\r7$$\"3+++++](oz\"F*$\"30+++N9a@:F\\r 7$$\"3+++++DJq?F*$\"31+++M@v!\\\"F\\r7$$\"3++++++vVBF*$\"3!*******o\"G PX\"F\\r7$$\"3)*********\\i!*GF*$\"3)******fc$Qh8F\\r7$$\"3++++++]PMF* $\"3!******4!=)eC\"F\\r7$$\"3+++++v$4r$F*$\"3!******H:I,=\"F\\r7$$\"3+ ++++]P%)RF*$\"3/+++-)G&46F\\r7$$\"3+++++D\"yD%F*$\"3'******Hj;X.\"F\\r 7$$\"3++++++DJXF*$\"3#)******)=Jfb*F*7$$\"3+++++vo/[F*$\"3(******>+zJt )F*7$$\"3+++++]7y]F*$\"3K+++c55$)yF*7$$\"3+++++Dc^`F*$\"3s*****4hvC,(F *7$$\"3+++++++DcF*$\"3]+++^,mGhF*7$$\"3+++++vV)*eF*$\"3$******4A'eR_F* 7$$\"3+++++](=<'F*$\"33+++$\\fPN%F*7$$\"3+++++DJXkF*$\"3u******f*o-[$F *7$$\"3++++++v=nF*$\"31+++F0\")GEF*7$$\"3)********\\(=#*pF*$\"37+++*oK (4=F*7$$\"3+++++]ilsF*$\"3&*******ff6M5F*7$$\"3+++++D1RvF*$\"3.++++]/R JFK7$$\"3++++++]7yF*$!31++++/zyLFK7$$\"3+++++v$f3)F*$!3[*******4#zn!*F K7$$\"3+++++]Pf$)F*$!3-+++]oIw8F*7$$\"3+++++D\"Gj)F*$!3)*******zV'os\" F*7$$\"3++++++D1*)F*$!34+++gEmL>F*7$$\"3++++](oH/*F*$!31+++q$pH(>F*7$$ \"3]+++#3P=3*F*$!37+++!Q)Gv>F*7$$\"3+++++voz\"*F*$!37+++q(e@'>F*7$$\"3 ++++]iS;$*F*$!3)********\\$[$*=F*7$$\"3+++++]7`%*F*$!35+++5h#fv\"F*7$$ \"3++++]P%)*e*F*$!3'*******>_=K:F*7$$\"3+++++DcE(*F*$!3)*******RLc!>\" F*7$$\"3++++v=#\\z*F*$!3c++++uBc&*FK7$$\"3++++]7Gj)*F*$!3p*******4\"oO lFK7$$\"3++++D1kJ**F*$!3<++++(3cK#FK7$$\"\"\"\"\"!$\"3O*******pl:S)FK- %'COLOURG6&%$RGBG$\"#5!\"\"$F_blF_blFibl-%+AXESLABELSG6$Q\"x6\"Q!F^cl- %%VIEWG6$;$!#v!\"#F]bl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "The three solutions can be found to 10 di gits using the procedure " }{TEXT 0 6 "secant" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "secant(f(x),x=-0.7..-0.6);\nsecant(f(x),x=0.7..0.8);\nsecant(f(x), x=0.98..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+(Q%>&)o!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]k#om(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+U*[%e**!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We can obtain more accurate values." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "x1 := eva lf[20](secant(f(x),x=-0.7..-0.6));\nx2 := evalf[20](secant(f(x),x=0.7. .0.8));\nx3 := evalf[20](secant(f(x),x=0.98..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!5cXU'GtQ%>&)o!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"5E!)))z]\\k#om(!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#x3G$\"5q7^)*RU*[%e**!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Now let's find the areas cut off above and below \+ the " }{TEXT 263 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(f(x),x=x1..x2) ;\nevalf[20](quad(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&* &,&-%%cos_G6#,$*&\"\"#\"\"\"%\"xGF/F/F/-%%sin_G6#F0!\"\"F/,&-%(arcsin_ GF3F/-%(arccos_G6#*$)F0\"\"$F/F/F/F/*&-%%exp_G6#-%(arctan_GF3F/-%$ln_G 6#,&-%%tan_GF3F/F/F/F/F//F0;$!5cXU'GtQ%>&)o!#?$\"5E!)))z]\\k#om(FN" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5A=7>1O]" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(-f(x),x=x2..x3);\nevalf[20](quad(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&,&-%%cos_G6#,$*&\"\"#\"\" \"%\"xGF/F/F/-%%sin_G6#F0!\"\"F/,&-%(arcsin_GF3F/-%(arccos_G6#*$)F0\" \"$F/F/F/F4*&-%%exp_G6#-%(arctan_GF3F/-%$ln_G6#,&-%%tan_GF3F/F/F/F/F4/ F0;$\"5E!)))z]\\k#om(!#?$\"5q7^)*RU*[%e**FN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5TZRFC![\\@2$!#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 73 "As a check we can repeat the calculation using standard Maple procedures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "g := x -> (cos(2*x)-sin(x)) *(arcsin(x)+arccos(x^3))+\n exp(arctan(x))*ln(tan(x)+1);\nx _1 := evalf[20](fsolve(g(x),x=-0.7..-0.6));\nx_2 := evalf[20](fsolve(g (x),x=0.7..0.8));\nx_3 := evalf[20](fsolve(g(x),x=0.98..1));\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&*&,&-%$cosG6#,$*&\"\"#\"\"\"9$F5F5F5-%$sinG6#F6!\"\"F5,&-%'arcsin GF9F5-%'arccosG6#*$)F6\"\"$F5F5F5F5*&-%$expG6#-%'arctanGF9F5-%#lnG6#,& -%$tanGF9F5F5F5F5F5F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_1G$! 5cXU'GtQ%>&)o!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_2G$\"5E!)))z] \\k#om(!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_3G$\"5q7^)*RU*[%e** !#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(g(x),x=x_1..x_2);\nevalf[20](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&,&-%$cosG6#,$*&\"\"#\"\"\"%\"xGF/F/F/ -%$sinG6#F0!\"\"F/,&-%'arcsinGF3F/-%'arccosG6#*$)F0\"\"$F/F/F/F/*&-%$e xpG6#-%'arctanGF3F/-%#lnG6#,&-%$tanGF3F/F/F/F/F//F0;$!5cXU'GtQ%>&)o!#? $\"5E!)))z]\\k#om(FN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5A=7>1O]" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "-Int(g(x),x=x_2..x_3);\nevalf[20](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$,&*&,&-%$cosG6#,$*&\"\"#\"\"\"%\"xGF0F0 F0-%$sinG6#F1!\"\"F0,&-%'arcsinGF4F0-%'arccosG6#*$)F1\"\"$F0F0F0F0*&-% $expG6#-%'arctanGF4F0-%#lnG6#,&-%$tanGF4F0F0F0F0F0/F1;$\"5E!)))z]\\k#o m(!#?$\"5q7^)*RU*[%e**FOF5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5TZRF C![\\@2$!#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+ 2" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=arccos(x/( 1+x^2))*arcsin(1/(cos(x)+3))+arctan(tan(x)^2+sin(x))+ln(x^2+1)*exp(-x^ 2/10)-2" "6#/-%\"fG6#%\"xG,**&-%'arccosG6#*&F'\"\"\",&F.F.*$F'\"\"#F.! \"\"F.-%'arcsinG6#*&F.F.,&-%$cosG6#F'F.\"\"$F.F2F.F.-%'arctanG6#,&*$-% $tanG6#F'F1F.-%$sinG6#F'F.F.*&-%#lnG6#,&*$F'F1F.F.F.F.-%$expG6#,$*&F'F 1\"#5F2F2F.F.F1F2" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "We find all the solutions of the equation " }{XPPEDIT 18 0 "f(x)=0" "6#/ -%\"fG6#%\"xG\"\"!" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[ -10,10]" "6#7$,$\"#5!\"\"F%" }{TEXT -1 59 " using the \"special\" nume rical functions and the procedure " }{TEXT 0 8 "allroots" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "f := x -> arccos_(x/(1+x^2))*arcsin_(1/(cos_(x)+3))+ \n arctan_(tan_(x)^2+sin_(x))+ln_(x^2+1)*exp_(-x^2/10)-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,** &-%(arccos_G6#*&9$\"\"\",&*$)F2\"\"#F3F3F3F3!\"\"F3-%(arcsin_G6#*&F3F3 ,&-%%cos_G6#F2F3\"\"$F3F8F3F3-%(arctan_G6#,&*$)-%%tan_GF@F7F3F3-%%sin_ GF@F3F3*&-%$ln_G6#F4F3-%%exp_G6#,$*&#F3\"#5F3F5F3F8F3F3F7F8F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(x),x=-10..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 520 271 271 {PLOTDATA 2 "6%-%'CURVESG6$7ax7$$!#5\"\"!$!*Wo2H%!\"*7$$!+e%G?y*F-$!*7 ]r)pF-7$$!+1+lF-7$$!+\\Bz*[)F-$!*)33HZF-7$$!+Ip6O %)F-$!*AGe7$F-7$$!+6:W#Q)F-$!*7&z#z\"F-7$$!+#4m(G$)F-$!)FhwuF-7$$!+ua# >A)F-$\")7GnhF-7$$!+c[3:\")F-$\"*45\">8F-7$$!+ZXmh!)F-$\"*#[U0:F-7$$!+ QUC3!)F-$\"*cq-h\"F-7$$!+HR#[&zF-$\"*7'=Z;F-7$$!+@OS,zF-$\"*Xj\\i\"F-7 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exp_(1+arctanh_(x/(1+x^2)))*ln_(tanh_(x)+2);\ngraph(f(x),x=-1..1 .7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operator G%&arrowGF(,&*&,&-%&cosh_G6#9$\"\"\"*&\"\"%F3-%&sinh_GF1F3F3F3,&-%)arc sinh_GF1F3-%)arccosh_G6#,&*$)F2\"\"#F3F3F3F3!\"\"F3F3*&-%%exp_G6#,&F3F 3-%)arctanh_G6#*&F2F3F>FBF3F3-%$ln_G6#,&-%&tanh_GF1F3FAF3F3F3F(F(F(" } }{PARA 13 "" 1 "" {GLPLOT2D 288 309 309 {PLOTDATA 2 "6%-%'CURVESG6$7en 7$$!\"\"\"\"!$\"3')*****>U,tF(!#<7$$!3c********\\7y&*!#=$\"3x*****f,5& 4mF-7$$!3A+++++Dc\"*F1$\"3g*****>z9M)fF-7$$!3y********\\PM()F1$\"3m*** **fj#*zR&F-7$$!3X+++++]7$)F1$\"35+++9ON_[F-7$$!3)*********\\i!*yF1$\"3 m*****zjSdM%F-7$$!3c*********\\(ouF1$\"3#******pnGv(QF-7$$!3A++++](o/( F1$\"3++++JLAZMF-7$$!3y***********\\i'F1$\"3#*******=%eW0$F-7$$!3W++++ ]7.iF1$\"3#*******fC**)p#F-7$$!3++++++D\"y&F1$\"39+++%p.2Q#F-7$$!3c*** *****\\Pf`F1$\"3))*****4a$e*4#F-7$$!3A+++++]P\\F1$\"35+++?$Gd&=F-7$$!3 y********\\i:XF1$\"3!******>;H$\\;F-7$$!3))*********\\P4%F1$\"3++++u+m 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"" 1 "" {XPPMATH 20 "6$%7approximati on~4~~->~~~G$\".%4f\\j&G\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7app roximation~5~~->~~~G$\".$)3'\\j&G\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$\".$)3'\\j&G\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+h\\j&G\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "We can obtain a more accurate value." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x1 := evalf[20](secant(f(x),x=1.2..1.4));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"5c%\\x$)3'\\j&G\"!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Now let's find the area i n the first quadrant bounded by the curve and the two axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "In t(f(x),x=0..x1);\nevalf[20](quad(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&,&-%&cosh_G6#%\"xG\"\"\"*&\"\"%F--%&sinh_GF+F-F-F-,&-% )arcsinh_GF+F--%)arccosh_G6#,&*$)F,\"\"#F-F-F-F-!\"\"F-F-*&-%%exp_G6#, &F-F--%)arctanh_G6#*&F,F-F8F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"57p F)*H]'Gx+$!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "As a check we can repeat the calculation using standard M aple procedures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 198 "g := x -> (cosh(x)+4*sinh(x))*(arcsinh(x)-arc cosh(1+x^2))+\n exp(1+arctanh(x/(1+x^2)))*ln(tan h(x)+2);\nx_1 := evalf(fsolve(g(x),x=1.2..1.4),20);\nInt(g(x),x=0..x_1 );\nevalf(%,20);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&*&,&-%%coshG6#9$\"\"\"*&\"\"%F3-%%sinhGF1F3 F3F3,&-%(arcsinhGF1F3-%(arccoshG6#,&*$)F2\"\"#F3F3F3F3!\"\"F3F3*&-%$ex pG6#,&F3F3-%(arctanhG6#*&F2F3F>FBF3F3-%#lnG6#,&-%%tanhGF1F3FAF3F3F3F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x_1G$\"5c%\\x$)3'\\j&G\"!#> " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&,&-%%coshG6#%\"xG\"\" \"*&\"\"%F--%%sinhGF+F-F-F-,&-%(arcsinhGF+F--%(arccoshG6#,&*$)F,\"\"#F -F-F-F-!\"\"F-F-*&-%$expG6#,&F-F--%(arctanhG6#*&F,F-F8F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"57pF)*H]'Gx+$!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }