{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Purple Emphasis" -1 262 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 263 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Em phasis" -1 266 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -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 "Text \+ Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 60 "Computing the exponential functio n from its Maclaurin series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter S tone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: \+ 24.3.2007" }}{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 35 "Graphing partial sums of th e series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series for " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 6 " i s " }{XPPEDIT 18 0 "sum(x^n/n!,n = 0 .. infinity);" "6#-%$sumG6$*&)%\"xG%\"nG\"\"\"-%*factorialG6#F)!\" \"/F);\"\"!%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 54 "We can make a comparison of finite sums approximating " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 25 " graphically as follow s.\n" }}{PARA 0 "" 0 "" {TEXT -1 27 "First we define a function " } {TEXT 0 2 "MS" }{TEXT -1 132 " with two arguments or input parameters. The second argument n is one less than the number of terms in the fin ite sum approximating " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" } {TEXT -1 69 ", which is the same as the degree of the polynomial appro ximation to " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "unassign('x','\355', 'n'):\nMS := (n,x) -> sum(x^i/i!,i = 0..n);\nMS(1,x);\nMS(2,x);\nMS(3, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MSGf*6$%\"nG%\"xG6\"6$%)oper atorG%&arrowGF)-%$sumG6$*&)9%%\"iG\"\"\"-%*factorialG6#F3!\"\"/F3;\"\" !9$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$%\"xGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$%\"xGF$*&#F$\"\"#F$)F%F(F$F$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$%\"xGF$*&#F$\"\"#F$)F%F( F$F$*&#F$\"\"'F$)F%\"\"$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 62 "The degree 1, 2 and 3 approximations y = x + 1, y = 1 + x + " }{XPPEDIT 18 0 "x^2/2;" "6#*&%\"xG\"\"#F%!\"\" " }{TEXT -1 5 ".and " }{XPPEDIT 18 0 "1+x+x^2/2+x^3/6;" "6#,*\"\"\"F$% \"xGF$*&F%\"\"#F'!\"\"F$*&F%\"\"$\"\"'F(F$" }{TEXT -1 27 " can be plot ted as follows." }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph y = " } {XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 33 " is also plot ted for comparison. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "MS := (n,x) -> sum(x^i/i!,i=0..n):\nplot([exp(x), MS(6,x),MS(7,x),MS(8,x)],x=-5..3,-3..15,\n color=[black,magenta ,coral,cyan],linestyle=[2,1$3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7W7$$!\"&\"\"!$\"3+na3**p%zt'!#?7$$ !37nmmmFiD[!#<$\"3\\!*fPcnb@!)F-7$$!35LLLo!)*Qn%F1$\"3[#QTI*o!eL*F-7$$ !3AmmmwxE.XF1$\"3`Smp1aF26!#>7$$!3YmmmOk]JVF1$\"3:JFrMEx98F>7$$!3_LLL[ 9cgTF1$\"3S;\"3Uvz)f:F>7$$!3GmmmhN2-SF1$\"3.xB.j*px#=F>7$$!3!******\\` oz$QF1$\"3QlGWA4t`@F>7$$!3!omm;)3DoOF1$\"3up6;Iq5_DF>7$$!3?+++:v2*\\$F 1$\"3)*zz\"3HDD-$F>7$$!3BLLL8>1DLF1$\"3#Q)HQk\"Hqf$F>7$$!3kmmmw))yrJF1 $\"3'fPh9e_G>%F>7$$!3;+++S(R#**HF1$\"3WA`k(Q$\\#)\\F>7$$!30++++@)f#GF1 $\"3=U5/RQ/DfF>7$$!3-+++gi,fEF1$\"33D:aYmq,qF>7$$!3qmmm\"G&R2DF1$\"3[? ,w.'>![\")F>7$$!3XLLLtK5FBF1$\"3]%3mu[*zd(*F>7$$!3eLLL$HsV<#F1$\"3;,f_ ')[zO6!#=7$$!3?+++b)4n*>F1$\"3g8(>D*G\"yN\"F_q7$$!3rLLL$\\[%R=F1$\"3!) 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N);" "6#/&%\"SG6#%\"NG-%$ sumG6$*&)%\"xG%\"nG\"\"\"-%*factorialG6#F.!\"\"/F.;\"\"!F'" }{TEXT -1 44 " as an approximation for the infinite sum " }{XPPEDIT 18 0 "S[in finity] = sum(x^n/n!,n = 0 .. infinity);" "6#/&%\"SG6#%)infinityG-%$su mG6$*&)%\"xG%\"nG\"\"\"-%*factorialG6#F.!\"\"/F.;\"\"!F'" }{TEXT -1 100 ".\nIf we do this, the theoretical error is the sum of the remaini ng terms in the tail of the series " }{XPPEDIT 18 0 "E[N] = sum(x^n/n !,n = N+1 .. infinity);" "6#/&%\"EG6#%\"NG-%$sumG6$*&)%\"xG%\"nG\"\"\" -%*factorialG6#F.!\"\"/F.;,&F'F/F/F/%)infinityG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 18 "\nIf the series is " }{TEXT 262 18 "rapid ly convergent" }{TEXT -1 25 ", which is the case when " }{TEXT 264 1 " x" }{TEXT -1 34 " is close to 0, it turns out that " }{TEXT 262 107 "t he substantial part of this theoretical or trunction error is containe d in the first term which is omitted" }{TEXT -1 1 "." }{TEXT 261 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 202 "If t his is the case, then we can simply sum terms until the magnitude of t he term to be added is small relative to the final sum, or relative to the current sum, which amounts to almost the same thing ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Thus we keep addi ng terms until:\n" }}{PARA 0 "" 0 "" {TEXT -1 45 " \+ " }{XPPEDIT 18 0 "abs(term)/abs(sum) < epsilo n;" "6#2*&-%$absG6#%%termG\"\"\"-F&6#%$sumG!\"\"%(epsilonG" }{TEXT -1 9 " or " }{XPPEDIT 18 0 "abs(term) < epsilon*abs(sum);" "6#2-%$ab sG6#%%termG*&%(epsilonG\"\"\"-F%6#%$sumGF*" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where \"" } {XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 32 "\" is the relati ve error we want." }}{PARA 0 "" 0 "" {TEXT -1 110 "We check the validi ty of doing this in a numerical example. Suppose we want to use this m ethod of calculating " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" } {TEXT -1 68 " for x from 0 to 10. Then the worst situation will occur \+ for x = 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "evalf(exp(10),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0n![zlk-A!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 106 "If we are working with 10 digit precision, we would li ke to have a relative truncation error of less than " }{XPPEDIT 18 0 " 10^(-9);" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 64 ", which means that we ar e after an absolute truncation error of " }{XPPEDIT 18 0 "10^(-9)*`.`* exp(10);" "6#*()\"#5,$\"\"*!\"\"\"\"\"%\".GF)-%$expG6#F%F)" }{TEXT -1 21 ", or approximately " }{XPPEDIT 18 0 "2*`.`*10^(-5);" "6#*(\"\"# \"\"\"%\".GF%)\"#5,$\"\"&!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(exp(10 )*10^(-9),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0n![zlk-A!#>" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "By trial \+ and error we find that taking 35 terms leaves " }{XPPEDIT 18 0 "sum(10 ^n/n!,n = 35 .. infinity);" "6#-%$sumG6$*&)\"#5%\"nG\"\"\"-%*factorial G6#F)!\"\"/F);\"#N%)infinityG" }{TEXT -1 27 " , which is approximatel y " }{XPPEDIT 18 0 "1.3*`.`*10^(-5);" "6#*(-%&FloatG6$\"#8!\"\"\"\"\"% \".GF))\"#5,$\"\"&F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(sum(10^n/n!,n = 35 .. infinity),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0XB\\Sq[L\"!# >" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The first term in this series is " }{XPPEDIT 18 0 "10^35/35!;" "6#*&\"#5 \"#N-%*factorialG6#F%!\"\"" }{TEXT -1 27 " , which is approximately 0 " }{XPPEDIT 18 0 ".97*`.`*10^(-5);" "6#*(-%&FloatG6$\"#(*!\"#\"\"\"%\" .GF))\"#5,$\"\"&!\"\"F)" }{TEXT -1 69 ", which is indeed the substanti al part of the total truncation error " }{XPPEDIT 18 0 "1.3*`.`*10^(-5 );" "6#*(-%&FloatG6$\"#8!\"\"\"\"\"%\".GF))\"#5,$\"\"&F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf((10^35)/35!);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+fHfx'*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "If we sum until we have " }{XPPEDIT 18 0 "abs(term) < ep silon*abs(sum);" "6#2-%$absG6#%%termG*&%(epsilonG\"\"\"-F%6#%$sumGF*" }{TEXT -1 9 " , where " }{XPPEDIT 18 0 "epsilon = 10^(-10);" "6#/%(eps ilonG)\"#5,$F&!\"\"" }{TEXT -1 57 ", this will force the taking of at \+ least one more term. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 25 "relative truncation error" } {TEXT -1 84 " will then be smaller than the machine epsilon for a prec ision of 10 digits, namely " }{XPPEDIT 18 0 "5*`.`*10^(-10)" "6#*(\"\" &\"\"\"%\".GF%)\"#5,$F(!\"\"F%" }{TEXT -1 57 ". Any error in the compu ted result would then arise from " }{TEXT 262 15 "rounding errors" } {TEXT 258 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We check this numerically using different precisions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "evalf(sum(10^n/n!,n = 0 .. 35),15);\nevalf(exp(10),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0c8\"zlk-A!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0n![zlk-A!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "evalf(sum(10^n/n!,n = 0 .. 3 5),10);\nevalf(exp(10),10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zlk -A!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+zlk-A!\"&" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "Now we start to p ut this all together. First we give a loop to calculate a finite sum o f terms of the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "x := 'x': \nTERM := 1:\nSUM := TER M: \nfor k from 1 to 6 do\n TERM := TERM*x/k;\n SUM := SUM + TERM; \nend do:\nSUM;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$%\"xGF$*& #F$\"\"#F$)F%F(F$F$*&#F$\"\"'F$)F%\"\"$F$F$*&#F$\"#CF$)F%\"\"%F$F$*&#F $\"$?\"F$)F%\"\"&F$F$*&#F$\"$?(F$)F%F,F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Now we incorporate the converge nce criterion " }{XPPEDIT 18 0 "abs(term) < epsilon*abs(sum);" "6#2-% $absG6#%%termG*&%(epsilonG\"\"\"-F%6#%$sumGF*" }{TEXT -1 44 ", which w e check here in the computation of " }{XPPEDIT 18 0 "exp(10);" "6#-%$e xpG6#\"#5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "We print out some information as we approach the exit from the loop." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "eps \+ := Float(1,-10): \nTERM := evalf(1):\nSUM := TERM:\nfor k from 1 to 1 00 while abs(TERM) > eps*abs(SUM) do\n TERM := evalf(TERM*10/k);\n \+ SUM := evalf(SUM + TERM);\n if k > 34 then\n printf(\"When k = %d, |term| = %.14g and eps*|sum| = %14.8g\\n\\n\",k,abs(TERM),eps*abs (SUM));\n end if;\nend do:\nSUM;" }}{PARA 6 "" 1 "" {TEXT -1 73 "Whe n k = 35, |term| = 9.67759295700000e-06 and eps*|sum| = 2.20264658e-06 " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 73 "When \+ k = 36, |term| = 2.68822026600000e-06 and eps*|sum| = 2.20264658e-06" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 73 "When k \+ = 37, |term| = 7.26546017800000e-07 and eps*|sum| = 2.20264658e-06" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!e YE?#!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 76 "A procedur e for computing the exponential function from its Maclaurin series" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 93 "We are going to incorporate the ideas of the previous section i n a procedure for evaluating " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#% \"xG" }{TEXT -1 381 " by means of the Maclaurin series. To be on the s afe side, the variable \"eps\", used in the convergence check, is take n to be a bit smaller than the machine epsilon for the precision being used. If the condition is never satisfied, the summation terminates a fter \"5 times Digits\" terms are added, and an error message is given .\n\nNote the general scheme for constructing a procedure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 27 "name of procedure \+ := proc( " }{TEXT -1 27 ". . arguments or input . . " }{TEXT 0 1 ")" } }{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 0 5 "local" }{TEXT -1 63 " . \+ . list of variables.. used internally in the procedure . . " }{TEXT 0 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " . . . BODY OF PROCEDURE . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 260 73 "last statement giv ing the value or Maple object returned by the procedure" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 0 9 "end proc;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "exp_series" {MPLTEXT 1 0 450 "exp_series := proc(x::realcons)\n local xx,sum,term,eps,maxit,k;\n eps := Fl oat(1,-Digits);\n maxit := Digits*5;\n term := evalf(1);\n xx := evalf(x);\n sum := term; \n for k from 1 to maxit while abs(term) > eps*abs(sum) do\n term := term*xx/k;\n sum := sum + term; \n end do:\n if k=101 then\n print(`sum of`,100,`terms of ser ies is`,sum);\n error \"reached maximum iterations without conver gence\"\n end if;\n sum\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 8 "Examples" }{TEXT -1 25 " comparing t he procedure " }{TEXT 0 9 "expseries" }{TEXT -1 34 " with the built-in Maple function " }{TEXT 0 3 "exp" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "exp_series(2 );\nevalf(exp(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*4c!*Q(!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*4c!*Q(!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(exp_ series(2),30);\nevalf(exp(2),30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"?cguUIsA]1$*)4c!*Q(!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?eguUIs A]1$*)4c!*Q(!#H" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "expseries" }{TEXT -1 68 " doe s not perform so well for negative values of the input variable." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalf(exp_series(-5));\nevalf(exp(-5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^N%zt'!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+** p%zt'!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalf(exp_series(-5),20);\nevalf(exp(-5),20);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5-:na3**p%zt'!#A" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"5m4na3**p%zt'!#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(exp_series(-15),15 );\nevalf(exp(-15),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0e:0box0 $!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0E=]?B!fI!#@" }}}{PARA 0 " " 0 "" {TEXT -1 80 "\nWhen using the plot function it is necessary to \+ put use quotation marks around " }{TEXT 0 10 "exp_series" }{TEXT -1 5 " ... " }{TEXT 0 11 "'expseries'" }{TEXT -1 83 ".\nPutting quotes arou nd an expression delays evaluation. For more information type " } {TEXT 0 7 "?uneval" }{TEXT -1 20 " on a command line.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot('exp_series'(x),x=-3..3);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7V7$ $!\"$\"\"!$\"3Y.kyOoqy\\!#>7$$!3!******\\2<#pG!#<$\"3Wz*))o\\LVn&F-7$$ !3#)***\\7bBav#F1$\"3aWlPp$3#ejF-7$$!36++]K3XFEF1$\"3FB#R.nViA(F-7$$!3 %)****\\F)H')\\#F1$\"3tSY*=ja(>#)F-7$$!3#****\\i3@/P#F1$\"3A\"H7;6PTM* F-7$$!3;++Dr^b^AF1$\"3!Gb!4\"QaB0\"!#=7$$!3$****\\7Sw%G@F1$\"3?t^1A\\= !>\"FM7$$!3*****\\7;)=,?F1$\"3?tt(eyXFM7$$!3!******\\!)H%*\\\"F1$\"3h0HHTUdKAFM7$$!3/+++vl[p8F1$ \"3CK?;zXPUDFM7$$!3\"******\\>iUC\"F1$\"3'Q$)HwIM:)GFM7$$!3-++DhkaI6F1 $\"3G9X)[zn&GKFM7$$!3s******\\XF`**FM$\"3A*)[tlR-'p$FM7$$!3u*******>#z 2))FM$\"3:Mf7&HVk\"F17$$\"3A )****\\d6.B'FM$\"3mqX*RHrX'=F17$$\"3s****\\(o3lW(FM$\"3i$Q9)Qhq0@F17$$ \"35*****\\A))oz)FM$\"3G]rQ0'\\,T#F17$$\"3e******Hk-,5F1$\"3D03!pQt5s# F17$$\"36+++D-eI6F1$\"3[x`u4$F17$$\"3u***\\(=_(zC\"F1$\"3SynSEHG$[ $F17$$\"3M+++b*=jP\"F1$\"3m=?'Gt'HgRF17$$\"3g***\\(3/3(\\\"F1$\"3!QrbD XB'oWF17$$\"33++vB4JB;F1$\"37&=I/W[)p]F17$$\"3u*****\\KCnu\"F1$\"3MH(z xI$yNdF17$$\"3s***\\(=n#f(=F1$\"3g!3/m)['o_'F17$$\"3P+++!)RO+?F1$\"3]B `3dgu\"R(F17$$\"30++]_!>w7#F1$\"3C*e*)zGb[R)F17$$\"3O++v)Q?QD#F1$\"3Q; hP-?0C&*F17$$\"3G+++5jypBF1$\"3o?wDg1^p5!#;7$$\"3<++]Ujp-DF1$\"3v#=shk Q:A\"Fby7$$\"3++++gEd@EF1$\"3'*QVI?Utv8Fby7$$\"31+]PMh%\\o#F1$\"3I-\\$ G=TdY\"Fby7$$\"39++v3'>$[FF1$\"3+%GC:#ojh:Fby7$$\"39+++5h(*3GF1$\"3g)3 ;$>?Hf;Fby7$$\"37++D6EjpGF1$\"3#)=&*yuR0j " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 78 "When working with a precision of 10 digits, find the largest integer \+ value of " }{TEXT 265 1 "x" }{TEXT -1 24 " for which the function " } {TEXT 0 10 "exp_series" }{TEXT -1 3 " - " }{HYPERLNK 17 "exp_series" 1 "" "exp_series" }{TEXT -1 83 " converges before the maximum number o f terms allowed by the procedure is reached. " }}{PARA 0 "" 0 "" {TEXT -1 40 "________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "__ ______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 28 "Plot a graph of the function" }{TEXT 0 11 " exp_series" }{TEXT -1 3 " - " }{HYPERLNK 17 "exp_series" 1 "" "ex p_series" }{TEXT -1 28 " together with the graph of " }{XPPEDIT 18 0 " exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 109 " over the interval [-16.005, -15.995]. What do you notice, and why is this happening?\nHow could th e procedure " }{TEXT 0 10 "exp_series" }{TEXT -1 107 " still be used t o evaluate the exponential function for negative arguments, without th e problem occurring. " }}{PARA 0 "" 0 "" {TEXT 262 4 "Hint" }{TEXT -1 73 ": Think of the modified formula for the solution of a quadratic eq uation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "(As a technical detail, you will have to put " }{TEXT 266 12 "exps eries(x)" }{TEXT -1 23 " in quotation marks .. " }{TEXT 266 14 "'expse ries(x)'" }{TEXT -1 59 ", in order for the plot command to be able to \+ evaluate it.)" }}{PARA 0 "" 0 "" {TEXT -1 40 "________________________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }