{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {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 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 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "R-2 MCS 320 Wednesday 31 M arch 2004" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "1. Use rand() to generate a list of 100 numbers. Call the list l." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 " (a) Replace every element in \+ the list l by x mod 5." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 " (b) Remove all duplicates from the list l." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "l := ['rand()'$1 00];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"lG7`q\"-\"3p'>uU\"-qKp56K \"-(ptIjV$\"-jN9cUZ\"-w*=(e%e&\"-Q0$QvY(\",&3AA1K\"-o<7uHs\"-@Rh0Vg\"- 4u.!eX(\"-bE&>\")f#\"-jr[v+J\"-d/\\zrz\",gTfp\"R\",u;dI%))\"-&3M))\\g* \"-;zX?H\")\"-h%>qu`%\"-2`RJSk\"-\\t%\\i?*\"-'3IN0^*\"-)>2j[Y\"\"-mMw! fb\"\"-4Pn#RH%\"-t4^Ga_\"-\")*31gs#\"-u$*4g(>#\"-mP$H)fn\"-t%4NZX)\"-U $)yqkn\"-\">z(Q8G\"-$R+f\\#z\"-+$R&47v\"-6IWj$G'\"-Ql3YPJ\"+8\\mie\",? il8[(\"-WQWUQk\"-lXv0>8\"-\"Re`2s'\"-&z3r`N\"\"-ua:Q;**\"-4V(3h_%\"-g8 BnGu\"-#Q3Ba-&\"-#fvO0*>\"-8')yS_#)\"-xsc)*=**\"-\\Pz\")o6\"-N$Gx!3M\" -h7I%R&z\"-[0()z%H'\"-.S)4XO(\"-Bfq%f,%\"-&>1\\Jy\"\"-tp8D[\"*\"-*Gi[J \"G\"-P_u+TX\"-d+TM!p(\",*H6Jq!)\"-%f9r70'\"-G*47#Hb\"-:DC::*)\"-b!G[K ?'\"-2vAaG\"*\"-^8)zrD&\"-i-dtuW\"-(zF:\\\")*\"-)yo&\\MT\"-U/]()H**\"- (46vid\"\"-)zw$*\\#))\"-F-*G1R$\",FJg7f'\"-u%*e@&>\"\"-DQ^dTd\"-2]!4l] \"\"-#=[(3Mb\",H\\)f\"G)\"-_)pAjf$\"-/&4ee\"o\"-&[Q(R(4(\"-X![U5J#\"-) 45Z!Qp\"-Qbl-$)=\",wk')=3\"\",v\")y5V&\"-utSO@a\"-Bbg0i5\"-+o%RA(o\",> ?C7e#\"-,RdA57\"-\\$o*)Gl'\",Z\\T'R(*\"-8;tA/y\"-l-k&y()*\"-WGF)>u'\"- 6eO]S8\"-OEep[v\"-fo&3\"39" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "l := map(x -> x mod 5,l);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"l G7`q\"\"\"\"\"!\"\"#\"\"$F&F)F'F)F&\"\"%F'F)F(F'F*F'F&F&F(F*F&F)F&F*F) F&F*F&F)F(F&F)F'F&F)F)F'F*F'F&F'F*F*F'F(F(F)F(F*F'F&F)F)F)F'F)F*F(F(F* F*F)F'F'F(F&F(F(F)F(F(F)F(F(F*F'F(F(F*F(F*F'F'F)F)F&F'F*F)F'F*F&F*F(F) F'F*F&F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "s := \{op(l) \};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG<'\"\"!\"\"\"\"\"#\"\"$\" \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "l := [op(s)];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG7'\"\"!\"\"\"\"\"#\"\"$\"\"%" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "2. Explain why piecewise is often preferable to \+ a similar if-then-else instruction. Give a good example to illustrate your explanation." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "ANSWER: If a function is defined by piecewise, then Mapl e can differentiate and integrate it. This is not the case with a sim ilar if-then else. A good example follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f1 := piecewise(x<0,0,1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G-%*PIECEWISEG6$7$\"\"!2%\"xGF)7$\"\"\"%*otherwise G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f2 := x -> if x < 0 th en 0; else 1; end if;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2Gf*6#%\" xG6\"6$%)operatorG%&arrowGF(@%29$\"\"!F/\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(f1,x=-1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int (f2,x=-1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#f2G\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 77 "3. Write an indexed procedure with name rf, which \+ returns rf[n](x) defined by" }}{PARA 0 "" 0 "" {TEXT -1 92 " rf[0 ](x) = 1, rf[1](x) = x, and rf[n](x) = (x+1)(rf[n-1](x) - rf[n-2](x)), for n >= 2." }}{PARA 0 "" 0 "" {TEXT -1 87 "The index n is the degree of the polynomial. Make sure the recursion runs efficiently." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rf := proc(x)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 " option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " local n:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " n \+ := op(procname):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " if n = 0 then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " return 1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " elif n = 1 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " return x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " else" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " return expand((x+1)*rf[n-1](x) \+ - rf[n-2](x));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " end if;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#rfGf*6#%\"xG6#%\"nG6#%)rememberG6\"C$>8$-%#opG6#9!@' /F/\"\"!O\"\"\"/F/F8O9$O-%'expandG6#,&*&,&F;F8F8F8F8-&F$6#,&F/F8F8!\" \"6#F;F8F8-&F$6#,&F/F8\"\"#FGFHFGF,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rf[50](x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,bq!\" \"\"\"\"*&\"$D%F%)%\"xG\"\"#F%F%*&\"%7BF%)F)\"\"$F%F$*&\"&/3$F%)F)\"\" %F%F$*&\"&wa)F%)F)\"\"&F%F%*&\"'CR!*F%)F)\"\"'F%F%*&\"(!3z7F%)F)\"\"(F %F$*&\")MQ89F%)F)\"\")F%F$*&\"(#)*RuF%)F)\"\"*F%F%*&\"*y^oL\"F%)F)\"#5 F%F%*&\")/vXF%F%*&\"+!)zOA OF%)F)\"#?F%F%*&\"+8rvx#)F%)F)\"#;F%F$*&\",D(R$Rg\"F%)F)F^pF%F$*&\",tq IO5\"F%)F)\"#=F%F%*&\"(O.L'F%)F)\"#WF%F%*&\")yP\"f#F%)F)\"#VF%F%*&\")v G^xF%)F)\"#UF%F%*&\"*#QUG:F%)F)\"#TF%F%*&\"*\"yp36F%)F)\"#SF%F%*&\"*GD <5%F%)F)\"#RF%F$*&\"+ay " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 " 4. Divided differences for a function f(x) are defined as follows:" } }{PARA 0 "" 0 "" {TEXT -1 107 " f[x[1],x[2],..,x[n-1],x[n]] = (f[ x[2],..,x[n-1],x[n]] - f[x[1],x[2],..,x[n-1])/(x[1]-x[n]), for n > 1" }}{PARA 0 "" 0 "" {TEXT -1 33 "and f[x[k]] = f(x[k]), for all k." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Write a Mapl e procedure dvd which computes divided differences for any f, and is c alled like dvd[a,b,c,d](f) shows" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 76 " f(d) - f(c) f(c) - f(b) \+ f(c) - f(b) f(b) - f(a)" }}{PARA 0 "" 0 "" {TEXT -1 81 " -------------- - --------------- --------------- - ----- -----------" }}{PARA 0 "" 0 "" {TEXT -1 80 " d - c \+ b - c b - c a - b" }}{PARA 0 "" 0 "" {TEXT -1 81 " ---------------------------------- - --------- -----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 72 " \+ b - d a - c" }} {PARA 0 "" 0 "" {TEXT -1 80 " -------------------------------------- ---------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 49 " a - d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "For simplicity, assum e the user always make the correct call to dvd, i.e.: include no error handling features." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dvd \+ := proc(f)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " local x,n:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " x := op(procname);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 " n := nops([x]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " if n = 1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " t hen return f(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " else return \+ (dvd[x[2..n]](f) - dvd[x[1..n-1]](f))/(x[1]-x[n]); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end \+ proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$dvdGf*6#%\"fG6$%\"xG%\"nG6 \"F+C%>8$-%#opG6#9!>8%-%%nopsG6#7#F.@%/F4\"\"\"O-9$6#F.O*&,&-&F$6#&F.6 #;\"\"#F46#F>F;-&F$6#&F.6#;F;,&F4F;F;!\"\"FJFRF;,&&F.6#F;F;&F.6#F4FRFR F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dvd[a,b,c,d](f); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&,&*&,&-%\"fG6#%\"dG\"\"\"-F* 6#%\"cG!\"\"F-,&F0F-F,F1F1F-*&,&F.F--F*6#%\"bGF1F-,&F7F-F0F1F1F1F-,&F7 F-F,F1F1F-*&,&F3F-*&,&F5F--F*6#%\"aGF1F-,&F@F-F7F1F1F1F-,&F@F-F0F1F1F1 F-,&F@F-F,F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "5. What is a re member table in Maple? How is it used?" }}{PARA 0 "" 0 "" {TEXT -1 56 " Mention an example of a good use of a remember table." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 218 "ANSWER: a remember table stored t he arguments and results of all calls made to a procedure. A good use is to avoid repeated calls in a recursive definition of a function, l ike the computation of the Fibonacci numbers." }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 79 "6. Use the arrow operator to define the following \+ operations on a polynomial p:" }}{PARA 0 "" 0 "" {TEXT -1 59 " (a) r emove all terms with negative coefficients from p; " }}{PARA 0 "" 0 " " {TEXT -1 32 " (b) replace x by x^2 in p(x)." }}{PARA 0 "" 0 "" {TEXT -1 88 "Use these two functions to define a function which does b oth operations to a polynomial." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "ANSWER: we first generate a random polynomial for testing purposes :" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p : = randpoly(x,dense,degree=7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"p G,2*$)%\"xG\"\"(\"\"\"!#&)*&\"#bF*)F(\"\"'F*!\"\"*&\"#PF*)F(\"\"&F*F0* &\"#NF*)F(\"\"%F*F0*&\"#(*F*)F(\"\"$F*F**&\"#]F*)F(\"\"#F*F**&\"#zF*F( F*F*\"#cF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f1 := p -> re move(t->coeffs(t) < 0,p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1Gf*6 #%\"pG6\"6$%)operatorG%&arrowGF(-%'removeG6$f*6#%\"tGF(F)F(2-%'coeffsG 6#9$\"\"!F(F(F(F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f 1(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"#c\"\"\"*&\"#(*F%)%\"xG\" \"$F%F%*&\"#]F%)F)\"\"#F%F%*&\"#zF%F)F%F%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "f2 := p -> subs(x = x^2,p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2Gf*6#%\"pG6\"6$%)operatorG%&arrowGF(-%%subsG6$/%\" xG*$)F0\"\"#\"\"\"9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f2(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*$)%\"xG\"#9\"\"\"!#&)* &\"#bF()F&\"#7F(!\"\"*&\"#PF()F&\"#5F(F.*&\"#NF()F&\"\")F(F.*&\"#(*F() F&\"\"'F(F(*&\"#]F()F&\"\"%F(F(*&\"#zF()F&\"\"#F(F(\"#cF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f3 := f2@f1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3G-%\"@G6$%#f2G%#f1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f3(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"#c\"\"\" *&\"#(*F%)%\"xG\"\"'F%F%*&\"#]F%)F)\"\"%F%F%*&\"#zF%)F)\"\"#F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 59 "7. Create a function in the variables B and N whic h returns" }}{PARA 0 "" 0 "" {TEXT -1 11 " " }}{PARA 0 "" 0 "" {TEXT -1 23 " N" }}{PARA 0 "" 0 "" {TEXT -1 29 " ------- " }}{PARA 0 "" 0 "" {TEXT -1 36 " \+ B > r[k]^k" }}{PARA 0 "" 0 "" {TEXT -1 28 " \+ ------- " }}{PARA 0 "" 0 "" {TEXT -1 23 " \+ k=0" }}{PARA 0 "" 0 "" {TEXT -1 108 "where r[k] is random numbe r drawn from a normal distribution with a mean five and standard devia tion 0.1*k." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "r := k -> st ats[random,normald[5,0.1*k]](1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"rGf*6#%\"kG6\"6$%)operatorG%&arrowGF(-&%&statsG6$%'randomG&%(normald G6$\"\"&,$9$$\"\"\"!\"\"6#F8F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "r(10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+o&Re<'! \"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "s := (B,N) -> B*(5^0 + convert([seq(r(k)^k,k=1..N)],`+`));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6$%\"BG%\"NG6\"6$%)operatorG%&arrowGF)*&9$\"\"\",&F/F/-% (convertG6$7#-%$seqG6$)-%\"rG6#%\"kGF " 0 "" {MPLTEXT 1 0 8 "s(5,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?5I-B\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "8. Explain th e difference between symbolic and automatic differentiation. Illustra te with an example the difference between the two and give the Maple c ommands you need." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 180 "ANSWER: symbolic differentiation is the differentiation of a f ormula (done with diff). Automatic differentiation is the differentia tion of a procedure (done with D). For example:" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "diff(cos(x),x); # symboli c differentiation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$sinG6#%\"xG! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "D(cos); # automati c differentiation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%$sinG!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "9. What is the difference between int and Int?" } }{PARA 0 "" 0 "" {TEXT -1 60 " Give a good illustration why we need a command like Int." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 299 "ANSWER: Int is the \"inert\" version of int, i.e.: it on ly defines the integral and does not seek to compute the antiderivativ e. A good use it numerical integration on function which have no symb olic antiderivative. Then it does make no sense to let Maple search f or a symbolic answer. For example:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := exp(sin(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%$expG6#-%$sinG6#%\"xG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "int(f,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-%$expG6#-%$sinG6#%\"xG/F,;\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(Int(f,x=0..1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+3'p=j\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "11. The function g(x,t) = (1-t^2)/(1-2*x*t + t^2) is a g enerating function for the Chebyshev polynomials." }}{PARA 0 "" 0 "" {TEXT -1 199 " (a) Compute aTaylor series approximation for g(x,t) a round t=0 of order 10. Select the coefficient of t^8 and compare with the output of orthopoly[T](8,x). What is the difference between the t wo?" }}{PARA 0 "" 0 "" {TEXT -1 257 " (b) Make a function cp in n (n is the degree of the Chebyshev polynomial) which uses this generating function and returns the same expanded polynomial as the one returned by orthopoly[T](n,x). The function cp should work for any n, be care ful for n = 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "g := (1-t ^2)/(1-2*x*t+t^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&,&\"\"\" F'*$)%\"tG\"\"#F'!\"\"F',(F'F'*(F+F'%\"xGF'F*F'F,F(F'F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "ANSWER to (a):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ts10 := taylor(g,t=0,10);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ts10G+9%\"tG\"\"\"\"\"!,$%\"xG\"\" #F',&!\"#F'*&\"\"%F')F*F+F'F'F+,&F*F-*(F+F',&F+F'*&F/F'F0F'!\"\"F'F*F' F5\"\"$,(F+F'*&F/F'F0F'F5*(F+F',&F*\"\"'*&\"\")F')F*F6F'F5F'F*F'F5F/,( F*F;*&F=F'F>F'F5*(F+F',(F-F'*&\"#;F'F0F'F'*&FDF')F*F/F'F5F'F*F'F5\"\"& ,*F-F'*&FDF'F0F'F'*&FDF'FFF'F5*(F+F',(F*!#5*&\"#SF'F>F'F'*&\"#KF')F*FG F'F5F'F*F'F5F;,*F*FM*&FOF'F>F'F'*&FQF'FRF'F5*(F+F',*F+F'*&\"#OF'F0F'F5 *&\"#'*F'FFF'F'*&\"#kF')F*F;F'F5F'F*F'F5\"\"(,,F+F'*&FYF'F0F'F5*&FenF' FFF'F'*&FgnF'FhnF'F5*(F+F',*F*\"#9*&\"$7\"F'F>F'F5*&\"$C#F'FRF'F'*&\"$ G\"F')F*FinF'F5F'F*F'F5F=,,F*F`o*&FboF'F>F'F5*&FdoF'FRF'F'*&FfoF'FgoF' F5*(F+F',,F-F'*&FgnF'F0F'F'*&\"$?$F'FFF'F5*&\"$7&F'FhnF'F'*&\"$c#F')F* F=F'F5F'F*F'F5\"\"*-%\"OG6#F'\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p8 := coeff(ts10,t,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p8G,,\"\"#\"\"\"*&\"#OF')%\"xGF&F'!\"\"*&\"#'*F')F+\"\"%F'F'* &\"#kF')F+\"\"'F'F,*(F&F',*F+\"#9*&\"$7\"F')F+\"\"$F'F,*&\"$C#F')F+\" \"&F'F'*&\"$G\"F')F+\"\"(F'F,F'F+F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "orthopoly[T](8,x); expand(p8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\"\")\"\"\"\"$G\"*&\"$c#F()F&\"\"'F(!\"\"*&\" $g\"F()F&\"\"%F(F(*&\"#KF()F&\"\"#F(F.F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\"#\"\"\"*&\"#kF%)%\"xGF$F%!\"\"*&\"$?$F%)F)\"\"%F% F%*&\"$7&F%)F)\"\"'F%F**&\"$c#F%)F)\"\")F%F%" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 71 "From this we see that p8 is twice the Chebyshev polynom ial of degree 8:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "orthopoly[T](8,x) - expand(p8/2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "This i s so for all degrees, except for degree 0, when 1 (the coefficient wit h t^0) is the Chebyshev polynomial:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "orthopoly[T](0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "ANS WER to (b):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "cp := n -> piecewise(n=0,1,coeftayl(g,t=0,n)/2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#cpGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%*piec ewiseG6%/9$\"\"!\"\"\",$-%)coeftaylG6%%\"gG/%\"tGF1F0#F2\"\"#F(F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "cp(0);cp(8); cp(12);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\"\")\"\"\"\"$G\"*&\"$c#F()F&\"\"'F(!\"\"*&\"$g\"F()F& \"\"%F(F(*&\"#KF()F&\"\"#F(F.F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, 0\"\"\"F$*&\"%%e$F$)%\"xG\"\"'F$!\"\"*&\"%7pF$)F(\"\")F$F$*&\"$S)F$)F( \"\"%F$F$*&\"%[?F$)F(\"#7F$F$*&\"#sF$)F(\"\"#F$F**&\"%WhF$)F(\"#5F$F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "12. Consider p(x) = 5*x^2*a^2 + 61*x^2* a + 66*x^2 + 10*x*a^2 + 121*x*a + 121*x + a^2 + 15*a+ 44, as a polynom ial in x with parameter a." }}{PARA 0 "" 0 "" {TEXT -1 27 " (a) Find the roots of p." }}{PARA 0 "" 0 "" {TEXT -1 63 " (b) For which valu es of the parameter a is the answer valid?" }}{PARA 0 "" 0 "" {TEXT -1 62 " (c) Give the Maple command(s) to treat the special case(s). " }}{PARA 0 "" 0 "" {TEXT -1 172 " (d) As you can see the polynomial p is shown in expanded form. Give the Maple command to \"un-expand\" , i.e.: what is the command which reveals better the structure of p?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "p := 5 *x^2*a^2 + 61*x^2*a + 66*x^2 + 10*x*a^2 + 121*x*a + 121*x + a^2 + 15*a + 44;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,4*&)%\"xG\"\"#\"\"\") %\"aGF)F*\"\"&*(\"#hF*F'F*F,F*F**&\"#mF*F'F*F**(\"#5F*F(F*F+F*F**(\"$@ \"F*F(F*F,F*F**&F5F*F(F*F**$F+F*F**&\"#:F*F,F*F*\"#WF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "ANSWER to (a):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s := solve(p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG6$,$*&,&%\"aG\"\"&\"\"'\"\"\"!\"\",(F) !#5\"#6F-*$-%%sqrtG6#,(*$)F)\"\"#F,\"#!)*&\"$;\"F,F)F,F,\"#DF,F,F,F,#F ,F8,$*&F(F-,(F)F/F0F-F1F-F,F=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 " ANSWER to (b):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We can see that the answer is not valid when 5*a + 6 = 0, or a = - 6/5." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "ANSWER to (c):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(subs(a=-6/5,p),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#9 \"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "But there is another spe cial case:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(p,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!#6,$*&,(\"\"%\" \"\"*&\"#6F(%\"xGF(F(*&\"\"'F()F+\"\"#F(F(F(,(F+\"#5*&\"\"&F(F.F(F(F(F (!\"\"F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(a=-11,p); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "When a equals -11, then the entire polynomial vanishes an d every x is a solution." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 14 "ANSWER to (d):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 36 "Here we can find the -11 in general:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "collect(p,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,(*$)%\"aG\"\"#\"\"\"\"\"&*&\"#hF* F(F*F*\"#mF*F*)%\"xGF)F*F**&,(\"$@\"F**&F3F*F(F*F**&\"#5F*F'F*F*F*F0F* F*\"#WF*F&F**&\"#:F*F(F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(coeff(p,x,2),a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\"'\" \"&!#6" }}}}{MARK "84" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }