{VERSION 2 3 "SGI IRIS UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "helvetica" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "times" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 265 "courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "c ourier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "couri er" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE " Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 } 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 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 "" 0 256 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 "" 0 257 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 256 "" 0 "" {TEXT 256 7 "Mat 331" }}{PARA 257 "" 0 "" {TEXT 257 33 "Exercise 3: Some Elementary Maple" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "1. Evaluate " }{XPPEDIT 18 0 "Pi^(sqrt(2) )" ")%#PiG-%%sqrtG6#\"\"#" }{MPLTEXT 1 0 1 " " }{TEXT -1 21 "to 30 dec imal places." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(Pi^(sqrt(2)), 30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?R'fSO " 0 "" {MPLTEXT 1 0 81 "f:= x-> (sqrt(x)*(-116*x^3+8 *x^4+558*x^2-891*x))/\n(2*x^5-29*x^4+140*x^3-225*x^2);" }}{PARA 0 "" 0 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowGF(*(-%%sqrtG 6#9$\"\"\",**$F0\"\"$!$;\"*$F0\"\"%\"\")*$F0\"\"#\"$e&F0!$\"*)F1,**$F0 \"\"&F:F6!#HF3\"$S\"F9!$D#!\"\"F(F(" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "a. Write this in a simpler form (that is, factor and reduce it) ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"#!\"*\"\"\"F(,&F%F&!#6F(F(F%#! \"\"F&,&F%F(!\"&F(!\"#" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "b. Dr aw the graph of the function for " }{XPPEDIT 18 0 "x" "I\"xG6\"" } {TEXT -1 73 " between 0 and 10. Adjust the vertical range so some det ail can be seen." }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=0..10);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7co7$$\"+;arz@!#5$\"+8L&RZ)!\"*7$$\"1+++\"y%*z 7$!#;$\"1qn&y>212(!#:7$$\"1+++XTFwSF1$\"1)f^'=v$3>'F47$$\"1+++\"z_\"4i F1$\"1t(=')*435]F47$$\"1+++S&phN)F1$\"1QAS*[3FJ%F47$$\"1+++*=)H\\5F4$ \"1pqI`\"fB%QF47$$\"1+++[!3uC\"F4$\"10IT4?%y^$F47$$\"1+++J$RDX\"F4$\"1 -LE=;)HD$F47$$\"1+++)R'ok;F4$\"1fH7kmbIIF47$$\"1+++1J:w=F4$\"1(fqK`ua% GF47$$\"1+++3En$4#F4$\"1*z1eg5Eo#F47$$\"1+++/RE&G#F4$\"1d4B0LDcDF47$$ \"1+++D.&4]#F4$\"1=a)\\\"34GCF47$$\"1+++vB_F47$$\"1+++LY.KNF4$\"1h%p!)4_9)=F47$$\"1+++\"o7Tv$F4$\"1 lK8N:'>t\"F47$$\"1+++$Q*o]RF4$\"1J&phw1bb\"F47$$\"1+++\"=lj;%F4$\"17^ \\1Vpa7F47$$\"1+++V&R>RpuU'yF47 $$\"1+++J@(e$[F4$!1b[#)=L?1:!#97$$\"1+++Ah$*))[F4$!1%oGfvbb[$Fgs7$$\"1 +++=\"oa\"\\F4$!1)z#G:4sJhFgs7$$\"1+++8,+U\\F4$!1(eIa^+#>8!#87$$\"1+++ 7hEb\\F4$!1()3)pL6pA#Fgt7$$\"1+++4@`o\\F4$!1%e)[jAg7XFgt7$$\"1+++4^;v \\F4$!1O%\\8v<5D(Fgt7$$\"1+++2\")z\")\\F4$!1p+&F 4$!1EGj;f8r6F*7$$\"1+++[P%)3]F4$!1Sx(p&z86dF\\v7$$\"1+++qNt:]F4$!1>ihJ <(>!=F\\v7$$\"1+++\"RBE-&F4$!1o!)3m/?+()Fgt7$$\"1+++NISO]F4$!1)3q4*QpW LFgt7$$\"1+++yE=]]F4$!13@]l'4#\\ux]F4$!1Xt$GI`^;(Fgs7$$\"1+++4;_\"4&F4$!1#3%o(y@O6&Fgs7$$\"1++ +^7I0^F4$!1B(HOUTV\"QFgs7$$\"1+++D)>/;&F4$!1q#eu;*[M:Fgs7$$\"1+++(RQb@ &F4$!1?zZ]z#Rn(F47$$\"1+++e,]6`F4$!1)>U.k:ht#F47$$\"1+++=>Y2aF4$!1s.oV xP+()F17$$\"1+++yXu9cF4$\"1M050Kk8dF17$$\"1+++\\y))GeF4$\"1[z7p*HR0\"F 47$$\"1+++i_QQgF4$\"1$pWEcw.D\"F47$$\"1+++!y%3TiF4$\"1K;DSeET8F47$$\"1 +++O![hY'F4$\"17`Kz^3!R\"F47$$\"1+++#Qx$omF4$\"1GenU3()49F47$$\"1+++u. I%)oF4$\"1(R6=HnrT\"F47$$\"1+++(pe*zqF4$\"1BW\">ODkT\"F47$$\"1+++C\\'Q H(F4$\"1RqHf#>2T\"F47$$\"1+++8S8&\\(F4$\"1d#H-m'R-9F47$$\"1+++0#=bq(F4 $\"1>K1<'oE_7F4-%'COLOURG6&%$RGBG$Fial!\"\"FjalF jal-%+AXESLABELSG6$%\"xG%!G-%%VIEWG6$;FjalFhal%(DEFAULTG" 2 405 133 133 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 71 476 0 0 0 0 0 0 } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "That looks terrible (too much ra nge is wasted on the asymptote at " }{XPPEDIT 18 0 "x=5" "/%\"xG\"\"& " }{TEXT -1 166 "), so lets try looking at a restricted range, as sugg ested. After a bit of playing around, it seems that the following wor ks well, indicating both the asymptotes at " }{XPPEDIT 18 0 "x=0" "/% \"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=5" "/%\"xG\"\"&" } {TEXT -1 36 ", but also some of the detail when " }{XPPEDIT 18 0 "f(x )" "-%\"fG6#%\"xG" }{TEXT -1 12 " is between " }{XPPEDIT 18 0 "0" "\" \"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "4" "\"\"%" }{TEXT -1 7 " or s o." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(f(x),x=0..10,y=- 10..10);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7co7$$\"+;arz@!#5 $\"+8L&RZ)!\"*7$$\"1+++\"y%*z7$!#;$\"1qn&y>212(!#:7$$\"1+++XTFwSF1$\"1 )f^'=v$3>'F47$$\"1+++\"z_\"4iF1$\"1t(=')*435]F47$$\"1+++S&phN)F1$\"1QA S*[3FJ%F47$$\"1+++*=)H\\5F4$\"1pqI`\"fB%QF47$$\"1+++[!3uC\"F4$\"10IT4? %y^$F47$$\"1+++J$RDX\"F4$\"1-LE=;)HD$F47$$\"1+++)R'ok;F4$\"1fH7kmbIIF4 7$$\"1+++1J:w=F4$\"1(fqK`ua%GF47$$\"1+++3En$4#F4$\"1*z1eg5Eo#F47$$\"1+ ++/RE&G#F4$\"1d4B0LDcDF47$$\"1+++D.&4]#F4$\"1=a)\\\"34GCF47$$\"1+++vB_ F47$$\"1+++LY.KNF4$\"1h%p!)4_ 9)=F47$$\"1+++\"o7Tv$F4$\"1lK8N:'>t\"F47$$\"1+++$Q*o]RF4$\"1J&phw1bb\" F47$$\"1+++\"=lj;%F4$\"17^\\1Vpa7F47$$\"1+++V&R>RpuU'yF47$$\"1+++J@(e$[F4$!1b[#)=L?1:!#97$$\"1+++Ah$* ))[F4$!1%oGfvbb[$Fgs7$$\"1+++=\"oa\"\\F4$!1)z#G:4sJhFgs7$$\"1+++8,+U\\ F4$!1(eIa^+#>8!#87$$\"1+++7hEb\\F4$!1()3)pL6pA#Fgt7$$\"1+++4@`o\\F4$!1 %e)[jAg7XFgt7$$\"1+++4^;v\\F4$!1O%\\8v<5D(Fgt7$$\"1+++2\")z\")\\F4$!1p +&F4$!1EGj;f8r6F*7$$\"1+++[P%)3]F4$!1Sx(p&z86dF \\v7$$\"1+++qNt:]F4$!1>ihJ<(>!=F\\v7$$\"1+++\"RBE-&F4$!1o!)3m/?+()Fgt7 $$\"1+++NISO]F4$!1)3q4*QpWLFgt7$$\"1+++yE=]]F4$!13@]l'4#\\ux]F4$!1Xt$GI`^;(Fgs7$$\"1+++4;_\"4& F4$!1#3%o(y@O6&Fgs7$$\"1+++^7I0^F4$!1B(HOUTV\"QFgs7$$\"1+++D)>/;&F4$!1 q#eu;*[M:Fgs7$$\"1+++(RQb@&F4$!1?zZ]z#Rn(F47$$\"1+++e,]6`F4$!1)>U.k:ht #F47$$\"1+++=>Y2aF4$!1s.oVxP+()F17$$\"1+++yXu9cF4$\"1M050Kk8dF17$$\"1+ ++\\y))GeF4$\"1[z7p*HR0\"F47$$\"1+++i_QQgF4$\"1$pWEcw.D\"F47$$\"1+++!y %3TiF4$\"1K;DSeET8F47$$\"1+++O![hY'F4$\"17`Kz^3!R\"F47$$\"1+++#Qx$omF4 $\"1GenU3()49F47$$\"1+++u.I%)oF4$\"1(R6=HnrT\"F47$$\"1+++(pe*zqF4$\"1B W\">ODkT\"F47$$\"1+++C\\'QH(F4$\"1RqHf#>2T\"F47$$\"1+++8S8&\\(F4$\"1d# H-m'R-9F47$$\"1+++0#=bq(F4$\"1>K1<'oE_7F4-%'CO LOURG6&%$RGBG$Fial!\"\"FjalFjal-%+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;Fj alFhal;$F*FjalFhal" 2 401 276 276 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 60 103 0 0 0 0 0 0 }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 65 "c. Compute the area of the part of the curve that lies ab ove the " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 18 "-axis and betwee n " }{XPPEDIT 18 0 "x=0" "/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=5" "/%\"xG\"\"&" }{TEXT -1 61 " (i.e., integrate the function \+ over the appropriate range of " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 99 " values). Give your answer both in an exact form and as a decim al approximation to about 10 places." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "First, we need to notice where the function crosses the " } {XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 40 "-axis. This is easily seen to be where " }{XPPEDIT 18 0 "2*x-9=0" "/,&*&\"\"#\"\"\"%\"xGF&F&\"\" *!\"\"\"\"!" }{TEXT -1 25 " from the answer to part " }{TEXT 264 1 "a " }{TEXT -1 46 ", but we could also let maple do all the work:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"*\"\"##\"#6F%" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 51 "Now that we have that, the rest is straightforward." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int(f(x),x=0..9/2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"\"##\"\"\"F%#\"#d\"\"&*&F*F&-%(a rctanhG6#,$*&F%F&F*F&#\"\"$\"#5F'#!\"\"\"#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ \"yQff\"!\")" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 65 "d. What is the \+ value of the integral if you remove the factor of " }{XPPEDIT 18 0 "sq rt(x)" "-%%sqrtG6#%\"xG" }{TEXT -1 20 " from the numerator?" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This is most simply expressed as " } {XPPEDIT 18 0 "f(x)/sqrt(x)" "*&-%\"fG6#%\"xG\"\"\"-%%sqrtG6#F&!\"\"" }{TEXT -1 51 ", but of course there are many other ways to do it." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "int(f(x)/sqrt(x),x=0..9/2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The answer is infinite, because the new function loo ks like " }{XPPEDIT 18 0 "1/x" "*&\"\"\"\"\"\"%\"xG!\"\"" }{TEXT -1 6 " near " }{XPPEDIT 18 0 "x=0" "/%\"xG\"\"!" }{TEXT -1 48 ", which is d oes not bound a finite area (unlike " }{XPPEDIT 18 0 "1/sqrt(x)" "*&\" \"\"\"\"\"-%%sqrtG6#%\"xG!\"\"" }{TEXT -1 2 ")." }}}}{SECT 0 {PARA 4 " " 0 "" {TEXT -1 25 "e. Use the derivative of " }{XPPEDIT 18 0 "f(x)" " -%\"fG6#%\"xG" }{TEXT -1 28 " to determine for what real " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 93 " the function has a local maximum. \+ An approximation to about 8 decimal places is sufficient." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "We can do this all in one maple statemen t, supressing all the intermediate messiness. Note that we " }{TEXT 262 5 "don't" }{TEXT -1 91 " care what the derivative is, only where i t is zero. Also, we know from the graph in part " }{TEXT 263 1 "b" } {TEXT -1 154 " that the answer is about 6, and that it is clearly a lo cal maximum, so we needn't mess with second derivatives, although we c ould if we really wanted to." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf( solve ( diff(f(x),x)=0, x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+nXA_p!\"*,&$\"+ " 0 "" {MPLTEXT 1 0 30 "[6.952224567, f(6.952224567)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+nXA_p!\"*$\"+H5`<9F&" }}}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 20 "3. Use the commands " }{TEXT 265 3 "seq" }{TEXT -1 5 " \+ and " }{TEXT 266 8 "ithprime" }{TEXT -1 112 " generate a list of the f irst 20 primes. Compute the sum of the first 20 primes, and give its \+ integer factors. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq(ith prime(i),i=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "66\"\"#\"\"$\"\"& \"\"(\"#6\"#8\"#<\"#>\"#B\"#H\"#J\"#P\"#T\"#V\"#Z\"#`\"#f\"#h\"#n\"#r " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumprimes:= sum(ithprim e(i),i=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sumprimesG\"$R'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ifactor(sumprimes);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%!G6#\"\"$\"\"#-F%6#\"#r\"\"\"" }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 49 "4. Find the solutions to the sys tem of equations " }{XPPEDIT 18 0 "\{x^2-y^2=4, x-2*y=2" "<$/,&*$%\"xG \"\"#\"\"\"*$%\"yG\"\"#!\"\"\"\"%/,&F&F(*&\"\"#F(F*F(F,\"\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 3 "> " 0 "" {MPLTEXT 1 0 35 "solve(\{x^2-y^2=4 , x-2*y=2\},\{x,y\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"yG\"\" !/%\"xG\"\"#<$/F%#!\")\"\"$/F(#!#5F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "We should, of course, expect two solutions, because this is th e intersection of a hyperbola and a line. " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "plots[implicitplot](\{x^2-y^2=4, x-2*y=2\},x=-4..4, y=-4..4);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6_r7$7$$!\"%\"\"!$ !1!444444Y$!#:7$$!1$yM/8R<&RF-$!1<_cp3E3MF-7$7$$!1+++++]2RF-$!1++++++g LF-F.7$F47$$!1444444/QF-$!1!44444fB$F-7$7$$!1++++++!o$F-$!1*********** p3$F-F:7$7$FA$\"1++++++(3$F-7$$!1**********\\2RF-$\"1************fLF-7 $FI7$F($\"1\"444444Y$F-7$7$FA$!1++++++(3$F-7$$!1[!>w/>wl$F-$!1_4Q_4QiI F-7$7$$!1GFFFFFPOF-$!1++++++SIF-FV7$Ffn7$$!1+++++]7NF-$!1+++++]()GF-7$ 7$$!1XXXXXXvLF-$!1,+++++?FF-F\\o7$Fbo7$$!1`5Uot%*oLF-$!1[*y:j_5r#F-7$7 $F7$!1+++++v)p#F-Fho7$7$F7$\"1+++++v)p#F-7$Fco$\"1)***********>FF-7$7$ Fco$\"1************>FF-7$$!1&**********\\W$F-$\"1%**********\\!GF-7$7$ $!1EFFFFFPOF-$\"1************RIF-F\\q7$7$Fcq$\"1++++++SIF-FF7$7$F7$!1, ++++v)p#F-7$$!1AAAAAAFKF-$!1yxxxxxKDF-7$7$$!1++++++@JF-$!1,++++++CF-F_ r7$Fer7$$!1IN#)eqk(3$F-$!1skF-7$7$$!1-+++++?FF-$!1MLLLLLQ=F-F[u7$7$Feo$\"1M LLLLLQ=F-7$$!1666666\")GF-$\"1++++++!3#F-7$FjuFes7$7$FeoFdu7$$!1s&G9dG ko#F-$!1I9dG9d$z\"F-7$7$$!1,++++DhEF-$!1,+++++g=====GBF-$!1$======>\"F-7$7 $$!1dG9dG9(G#F-$!1,+++++?6F-F[z7$Faz7$$!1,+++++DAF-$!18+++++](*!#;7$7$ $!1++++++]@F-$!12++++++!)F\\[lFgz7$F^[l7$$!1XWWWWWM@F-$!1pbbbbbbuF\\[l 7$7$Fht$!1N+++++]bF\\[lFd[l7$7$Fht$\"1I+++++]bF\\[l7$F_[l$\"1%******** *****zF\\[l7$Fa\\l7$$!1**********\\-AF-$\"1r*********\\A*F\\[l7$7$Fbz$ \"1************>6F-Fe\\l7$F[]lFdx7$Fj[l7$$!1+++++Dh?F-$!19++++]()\\F\\ [l7$7$$!1++++++b?F-$!12++++++[F\\[lF`]l7$Ff]l7$$!1Vr&G9dG,#F-$!1'eG9dG 9F#F\\[l7$7$$!1nmmmmm,?F-$!12++++++;F\\[lF\\^l7$Fb^l7$Fc^l$\"1+lmmmmm \")!#<7$7$Fc^l$\"1$************f\"F\\[lFh^l7$F]_l7$$!1************\\?F -$\"1z***********\\%F\\[l7$7$Fg]l$\"1$************z%F\\[lFa_l7$Fg_lF^ \\l7$7$F]v$!1$***********\\bF\\[l7$$\"1,+++++b?F-Fi]l7$F_`l7$$\"1,++++ +]?F-$!17++++++XF\\[l7$7$$\"1nmmmmm,?F-Fe^lFc`l7$Fi`l7$Fj`l$!1pnmmmmm \")F[_l7$7$Fj`lF^_lF]al7$Faal7$$\"1Vr&G9dG,#F-$\"1i&G9dG9F#F\\[l7$7$$ \"1++++++b?F-Fh_lFcal7$Fial7$$\"1+++++Dh?F-$\"1\"*********\\()\\F\\[l7 $7$F]v$\"1))**********\\bF\\[lF]bl7$7$F[t$!1**********\\<8F-7$$\"1dG9d G9(G#F-Fdz7$Fjbl7$$\"1,++++]-AF-$!1?+++++D#*F\\[l7$7$$\"1++++++]@F-Fa[ lF^cl7$FdclF\\`l7$Fcbl7$$\"1WWWWWWM@F-$\"1\\bbbbbbuF\\[l7$7$FeclFb\\lF icl7$F_dl7$$\"1++++++DAF-$\"1!***********\\(*F\\[l7$7$F[clF\\]lFadl7$F gdl7$$\"1<=====GBF-$\"1\"======>\"F-7$7$F[t$\"1**********\\<8F-Fidl7$7 $Fjp$!1KLLLLLQ=F-7$$\"1+++++DhEF-F[w7$Ffel7$$\"1NLLLLLjDF-$!1PLLLLL.;F -7$7$$\"1+++++DhCF-FgwFjel7$F`flFgbl7$F_el7$$\"1LLLLL$3W#F-$\"1mmmmm;* R\"F-7$7$$\"1*********\\7Y#F-FjxFefl7$F[gl7$$\"1I#p2Bp2c#F-$\"1p2Bp2B* f\"F-7$7$$\"1*********\\7m#F-FfyF_gl7$Fegl7$$\"1q&G9dGko#F-$\"1G9dG9d$ z\"F-7$7$Fjp$\"1KLLLLLQ=F-Figl7$7$Feq$!18dG9dG%G#F-7$$\"1766666\")GF-F ht7$FfhlFcel7$F_hl7$$\"1mmmmmm;GF-$\"1LLLLLL$)>F-7$7$$\"1666666\")GF-F ]vF[il7$Fail7$$\"1********\\i]HF-$\"1++++]Pp@F-7$7$Feq$\"18dG9dG%G#F-F eil7$7$FL$!1*********\\()p#F-7$$\"1++++++@JF-Fhr7$7$Fcjl$!1+++++++CF-F chl7$F[jl7$$\"1HN#)eqk(3$F-$\"1pkw/>wl$F-$\"1^4Q_4QiIF-7$7$F]\\m$\"1***********p3$F-Fg^m7$7$$ \"1+++++++SF-F+7$$\"1,++++]2RF-F77$Fd_mF\\\\m7$F]_m7$$\"1344444/QF-$\" 1!44444fB$F-7$7$$\"1**********\\2RF-FLFi_m7$7$$\"1+++++]2RF-FL7$$\"1#y M/8R<&RF-$\"1<_cp3E3MF-7$7$$\"1**************RF-$\"1*344444Y$F-Ff`m-%' COLOURG6&%$RGBG\"\"\"F*F*-F$6go7$7$F($!1+++++++IF-7$$!1MLLLLL8QF-$!1nm mmmm1HF-7$7$$!1,+++++!o$F-$!1++++++SGF-F\\bm7$Fbbm7$$!1,++++++OF-$!1,+ +++++GF-7$7$$!1,+++++SMF-FeoFhbm7$F^cm7$$!1nmmmmm'Q$F-$!1MLLLLL$p#F-7$ 7$F7$!1,+++++!o#F-Fbcm7$Fhcm7$$!1MLLLLLtJF-$!1ommmmm'e#F-7$7$Fin$!1+++ +++?DF-F\\dm7$Fbdm7$$!1++++++gHF-$!1++++++![#F-7$7$$!1-++++++GF-FhrFfd m7$F\\em7$$!1ommmmmYFF-$!1MLLLLLtBF-7$7$Feo$!1++++++gBF-F`em7$Ffem7$$! 1NLLLLLLDF-$!1nmmmmmmAF-7$7$Fhr$!1,++++++AF-Fjem7$7$Fhr$!1+++++++AF-7$ $!1,+++++?BF-$!1,+++++g@F-7$7$$!1-+++++g@F-FhtFgfm7$F]gm7$$!1ommmmm1@F -$!1MLLLLL`?F-7$7$Fht$!1++++++S?F-Fagm7$Fggm7$$!1MLLLLL$*=F-$!1nmmmmmY >F-7$7$F[w$!1++++++!)=F-F[hm7$Fahm7$$!1,+++++!o\"F-$!1,+++++S=F-7$7$$! 1-+++++?:F-F[wFehm7$F[im7$$!1ommmmmm9F-$!1MLLLLLLF-$!1-nmmmmmEF[_l7$7$F]v$\"1))************RF[_lF\\d n7$Fbdn7$$\"1************f@F-$\"1j************zF[_l7$7$$\"1*********** *>BF-F^_lFfdn7$F\\en7$$\"1KLLLLLtBF-$\"1jmmmmmm=F\\[l7$7$F[t$\"1)***** ********>F\\[lF`en7$Ffen7$$\"1lmmmmm'e#F-$\"1HLLLLLLHF\\[l7$7$Fjp$\"1( ************f$F\\[lFjen7$F`fn7$$\"1*************z#F-$\"1&************* RF\\[l7$7$$\"1)***********fHF-Fh_lFdfn7$7$$\"1************fHF-Fh_l7$$ \"1KLLLLL8IF-$\"1immmmmm]F\\[l7$7$Feq$\"1'************>&F\\[lFagn7$Fgg n7$$\"1mmmmmmEKF-$\"1GLLLLLLhF\\[l7$7$FL$\"1&************z'F\\[lF[hn7$ Fahn7$$\"1************RMF-$\"1&************>(F\\[l7$7$$\"1************ *f$F-Fb\\lFehn7$F[in7$$\"1KLLLLL`OF-$\"1hmmmmmm#)F\\[l7$7$F]\\m$\"1%** **********R)F\\[lF_in7$Fein7$$\"1mmmmmmmQF-$\"1DLLLLLL$*F\\[l7$7$F]am$ \"0***************F-FiinFaam-%+AXESLABELSG6$%\"xG%\"yG" 2 310 290 290 2 0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 224 142 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 78 "5. Draw a graph showing both cos(x) and its fifth ta ylor polynomial (that is, " }{XPPEDIT 18 0 "1-x^2/2+x^4/24" ",(\"\"\" \"\"\"*&%\"xG\"\"#\"\"#!\"\"F)*&F&\"\"%\"#CF)F$" }{TEXT -1 6 ") for " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 9 " between " }{XPPEDIT 18 0 " -4" ",$\"\"%!\"\"" }{TEXT -1 121 " and 4. How many terms do you seem \+ to need to get ``good agreement'' in this range (you will need to use \+ something like " }{TEXT 271 36 "convert(taylor(cos(x),x,5), polynom)" }{TEXT -1 127 " to make this work. You should think about an appropri ate way to demonstrate that the approximation you've taken is ``good'' . " }}{EXCHG {PARA 0 "" 0 "" {TEXT 273 23 " Note that the command " } {TEXT 274 8 "taylor()" }{TEXT 275 135 " gives us an object which is a \+ series, rather than a polynomial, that is, it has an order of trunctat ion with it. Thus the result of " }{TEXT 276 9 "taylor() " }{TEXT 277 27 "cannot be plotted directly:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "taylor(cos(x),x,5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG\"\"\"\"\"!#!\"\"\"\"#\"\"##F% \"#C\"\"%-%\"OG6#F%\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " plot(\");" }}{PARA 8 "" 1 "" {TEXT -1 26 "Plotting error, empty plot" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "So, to get around that, we can \+ use " }{TEXT 278 9 "convert()" }{TEXT -1 16 " to discard the " } {XPPEDIT 18 0 "O(x^5" "-%\"OG6#*$%\"xG\"\"&" }{TEXT -1 31 " term and g ive us a polynomial." }}{PARA 0 "" 0 "" {TEXT -1 36 "Here are the two \+ functions together:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot (\{cos(x),convert(taylor(cos(x),x,5), polynom)\},x=-4..4); " }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7io7$$!\"%\"\"!$\"1mmmmmmmO!#:7$$!1 nmm\"p0k&RF-$\"1Gq,&[MEQ$F-7$$!1MLL$Q6G\"RF-$\"17oleid6JF-7$$!1+++vq@p QF-$\"1bTi2g;`GF-7$$!1nmmmFiDQF-$\"1]&>PO\"32EF-7$$!1++] Yc**)G2AF-7$$!1LLLo!)*Qn$F-$\"1jK8SY?U=F-7$$!1++]AHe)e$F-$\"1T`?]&y5Z \"F-7$$!1nmmwxE.NF-$\"1,CV>`aR6F-7$$!1mmm1rQkOe#F-$!1\\^;>e(*4[F en7$$!1+++:v2*\\#F-$!1n:$o@5\\(\\Fen7$$!1nm;9(p?T#F-$!1q&*Q50@')\\Fen7 $$!1LLL8>1DBF-$!1?:v#)=%H&[Fen7$$!1+++&RD%[AF-$!1AWntVBGYFen7$$!1nmmw) )yr@F-$!1\\8ACMx8VFen7$$!1+++S(R#**>F-$!18;#[/i#GLFen7$$!1++++@)f#=F-$ !18DVji)*Q?Fen7$$!1+++gi,f;F-$!1BvUr))z_g!#<7$$!1nmm\"G&R2:F-$\"1O+9\" G%z+zF\\t7$$!1LLLtK5F8F-$\"1ao?(>u0WaFen7$$!1PLLL\\[%R)Fen$\"19\"f6FLNo' Fen7$$!1)*****\\&y!pmFen$\"1)>Y9E$feyFen7$$!1******\\O3E]Fen$\"1h&R-]8 Nw)Fen7$$!1KLLL3z6LFen$\"1XK!>T9mX*Fen7$$!1MLL$)[`P)yFen7$$\"1#******4#32$)Fe n$\"162XE\"Q![nFen7$$\"1%*****\\#y'G**Fen$\"1dg%\\otfZ&Fen7$$\"1****** H%=H<\"F-$\"1@&[Ek<*4RFen7$$\"1mmm1>qM8F-$\"1G8Ep:9:CFen7$$\"1++++.W2: F-$\"1$\\%os\"yl*yF\\t7$$\"1LLLep'Rm\"F-$!1Vs%QR,p\\'F\\t7$$\"1+++S>4N =F-$!1pg&G^8E6#Fen7$$\"1mmm6s5'*>F-$!1;bkQeI2LFen7$$\"1+++lXTk@F-$!1(e hMG]\"zUFen7$$\"1LL$e;!pYAF-$!1#GW%)=p?i%Fen7$$\"1mmmmd'*GBF-$!1Zg`P\" *zh[Fen7$$\"1LL$ep+^T#F-$!1=%)3\"QR$))\\Fen7$$\"1+++DcB,DF-$!1oZ<%*\\l s\\Fen7$$\"1nm;*z$>%e#F-$!1G\\LfCV3[Fen7$$\"1MLLt>:nEF-$!1uUs@r>$[%Fen 7$$\"1LLL)o))>v#F-$!1HPd`/XoRFen7$$\"1LLL.a#o$GF-$!1p#RG%G4`KFen7$$\"1 nmm^Q40IF-$!1Djn&[PJ<\"Fen7$$\"1LL$eY/C3$F-$\"1NGMhrF-7$$\"1+++![,`u$F-$\"1#>9i Y')[=#F-7$$\"1++]Z\"Rdy$F-$\"1zK)zIoCR#F-7$$\"1+++:ohj3iVOlFen7$F>$!1ZuNSIK]xFen7$FH$!1YLZYJR;')Fen7$FR $!17'f<3aIN*Fen7$FW$!1lHJE;4A'*Fen7$Fgn$!1dG+V^??)*Fen7$F\\o$!1T_`W'4b %**Fen7$Fao$!12Q2y4?)***Fen7$Ffo$!1RLQJ+%=)**Fen7$F\\p$!1Q0;\"*)HG!**F en7$$!1LLL[5-?HF-$!1QrVMF`b(*Fen7$Fap$!1=2&R<#fU&*Fen7$F[q$!1P\\w+()\\ +*)Fen7$Feq$!1'z8<$>\"f+)Fen7$F_r$!1Z$3ZJOv%oFen7$Fir$!1<0*)=QhacFen7$ F^s$!1Pd')QTbaTFen7$Fcs$!1#QD>q^U_#Fen7$Fhs$!1=.([`a0\"))F\\t7$F^t$\"1 \\YJ@y&eL'F\\t7$Fct$\"1c_u.?)GT#Fen7$Fht$\"1#**e=w@7'QFen7$F]u$\"1%o#G !Qz1V&Fen7$Fbu$\"1\"Hy&**RtymFen7$Fgu$\"1_'zL%4QdyFen7$F\\v$\"1Nvo61Hj ()Fen7$Fav$\"1T:rAhfc%*Fen7$$!1LLLeGmCDFen$\"1\")49nI*Ho*Fen7$Ffv$\"1, #Qk,G%\\)*Fen7$F[w$\"10A=pLbg**FenF_w7$Few$\"1mmmZ#QX'**Fen7$Fjw$\"1)4 !o5iq^)*Fen7$$\"1hmmm7+#\\#Fen$\"1)G`c.+6p*Fen7$F_x$\"1pqC+qSt%*Fen7$F dx$\"1*4N\"4N'[\"))Fen7$Fix$\"1S\"GO[93)yFen7$F^y$\"15\"G\\!)HNu'Fen7$ Fcy$\"1S]R8.!HY&Fen7$Fhy$\"1hV]/)GY(QFen7$F]z$\"1ZjD^=2RBFen7$Fbz$\"1C ]_E]OJjF\\t7$Fgz$!1`+MG\"*e.$*F\\t7$F\\[l$!1pr<[SH7EFen7$Fa[l$!1]Dt**) Rg7%Fen7$Ff[l$!1MqEe\"ROf&Fen7$F`\\l$!1ON!oNMf(oFen7$Fj\\l$!1SA@4\\#)= !)Fen7$Fd]l$!1xGI4Q[&*))Fen7$F^^l$!1I]!*)>o\"R&*Fen7$$\"1++]F'f4#HF-$! 1'eIw3\"fd(*Fen7$Fc^l$!1`OzS\\)p!**Fen7$Fh^l$!1]!z^!*)[#)**Fen7$F]_l$! 1@nCLzN)***Fen7$Fb_l$!1ffKD&)4V**Fen7$Fg_l$!1&z)Qoa#)4)*Fen7$F\\`l$!1[ X:3NPD'*Fen7$Fa`l$!1#)=c#3*\\!Q*Fen7$F[al$!1m(HC#z2k')Fen7$F_bl$!1ZK!) *p?ou(Fen7$FcclF`dl-Ffcl6&FhclF*FiclF*-%+AXESLABELSG6$%\"xG%!G-%%VIEWG 6$;F(Fccl%(DEFAULTG" 2 399 190 190 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 0 289 529 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 307 "Now, we could just keep making that picture, increasing the nu mber of terms by two (every other term in the taylor series of cosine \+ is 0) until it looks good. But, we might as well be a bit fancier and do it all at once. Since I will need it later, I am also going to de fine a function which gives me the " }{XPPEDIT 18 0 "n^(th)" ")%\"nG%# thG" }{TEXT -1 8 " series." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "nthseries := n -> convert( taylor(cos(x),x,n), polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*nthseriesG:6#%\"nG6\"6$%)operatorG%&arrow GF(-%(convertG6$-%'taylorG6%-%$cosG6#%\"xGF59$%(polynomGF(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "Now I will plot the cosine and se veral of the approximations on the same graph, and see which looks goo d. I'll use " }{TEXT 281 6 "seq() " }{TEXT -1 82 "to generate them. \+ (Note that since I only want odd numbers, I will index them by " } {XPPEDIT 18 0 "2*i+1" ",&*&\"\"#\"\"\"%\"iGF%F%\"\"\"F%" }{TEXT -1 8 " so the " }{XPPEDIT 18 0 "5^(th)" ")\"\"&%#thG" }{TEXT -1 116 " polyno mial will correspond to i=2. That is, we are giving the number of non -constant terms rather than the order)." }}{PARA 0 "" 0 "" {TEXT -1 94 "We can see a bit better what is going on if we look at a slightly \+ different domain and range. 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