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}{PSTYLE "" 3 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 258 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 261 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 262 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 263 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 264 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 265 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 266 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 267 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 268 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 269 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 270 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 272 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 273 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 274 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 275 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 276 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 277 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 278 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 279 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 280 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 281 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 282 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 283 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 284 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 285 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 286 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 287 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 288 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 289 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 290 1 {CSTYLE "" -1 -1 "" 1 12 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 258 0 "" }{TEXT 256 36 "Introduct ion to Symbolic Computation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 12 "Problem book" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 " 1. Computing with elementary Maple procedures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 0 "" }{TEXT 259 2 "1." }{TEXT 263 21 " Define recursiv ely " }{XPPEDIT 18 0 "x[1]=2" "6#/&%\"xG6#\"\"\"\"\"#" }{TEXT 261 2 " , " }{XPPEDIT 18 0 "x[n]=2+1/(2+x[n-1])" "6#/&%\"xG6#%\"nG,&\"\"#\"\" \"*&\"\"\"F*,&\"\"#F*&F%6#,&F'F*\"\"\"!\"\"F*F3F*" }{TEXT 262 84 ". W hat is the limit of this sequence? \nDoes the limit depend on the init ial value " }{XPPEDIT 18 0 "x[1]=2" "6#/&%\"xG6#\"\"\"\"\"#" }{TEXT 260 1 "?" }{TEXT -1 0 "" }}}{SECT 0 {PARA 256 "" 0 "" {TEXT -1 2 "2." }{TEXT 419 124 " Write a Maple procedure that takes a numeric list as \+ an argument and \nthen computes the average of the elements of the lis t" }{TEXT -1 1 "." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 264 2 "3." }{TEXT 418 139 " Write a Maple procedure that takes a numer ic list L as an argument and \nthen computes the standard deviation o f the elements of the list." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The standard deviation " }{XPPEDIT 18 0 "sigma " "6#%&sigmaG" }{TEXT -1 18 " is determined by" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "sigma = sqrt((1/n)*(Sum((x[i]-mu)^2,i=1..n))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&sigmaG*$-%%sqrtG6#*&-%$SumG6$*$ ),&&%\"xG6#%\"iG\"\"\"%#muG!\"\"\"\"#\"\"\"/F3;F4%\"nGF8F;!\"\"F8" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Here " }{XPPEDIT 18 0 "mu" "6#%#muG " }{TEXT -1 32 " is the average of the elements " }{XPPEDIT 18 0 "x[1] ..x[n]" "6#;&%\"xG6#\"\"\"&F%6#%\"nG" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 265 1 " " }{TEXT 416 2 "4." }{TEXT 417 22 " Lege ndre polynomials." }}{PARA 4 "" 0 "" {TEXT 266 20 "Legendre polynomial s" }{TEXT -1 2 " " }{XPPEDIT 18 0 " L[n](x)" "6#-&%\"LG6#%\"nG6#%\"xG " }{TEXT -1 2 " " }{TEXT 267 35 "are defined recursively as follows: " }{TEXT 268 1 " " }{TEXT -1 1 " " }{XPPEDIT 18 0 "L[0](x)=1" "6#/-&% \"LG6#\"\"!6#%\"xG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "L[1](x)=x" "6#/-&%\"LG6#\"\"\"6#%\"xGF*" }{TEXT -1 2 ", " }{TEXT 269 3 "and" } {XPPEDIT 18 0 "L[n](x) = (n-1)/n*(x*L[n-1](x)-L[n-2](x))+x*L[n-1](x); " "6#/-&%\"LG6#%\"nG6#%\"xG,&*(,&F(\"\"\"\"\"\"!\"\"F.F(F0,&*&F*F.-&F& 6#,&F(F.\"\"\"F06#F*F.F.-&F&6#,&F(F.\"\"#F06#F*F0F.F.*&F*F.-&F&6#,&F(F .\"\"\"F06#F*F.F." }}{PARA 0 "" 0 "" {TEXT -1 85 "for n>1. Write a M aple procedure that computes Legendre polynomials for any given " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 15 ", and compute " }{XPPEDIT 18 0 "L[7](x)" "6#-&%\"LG6#\"\"(6#%\"xG" }{TEXT -1 2 ". " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 270 2 "5." }{TEXT 415 21 " Defin e the function " }{XPPEDIT 18 0 " f(x) = sin(x^2+1)+cos(x+1)" "6#/-% \"fG6#%\"xG,&-%$sinG6#,&*$F'\"\"#\"\"\"\"\"\"F/F/-%$cosG6#,&F'F/\"\"\" F/F/" }{TEXT 271 254 " first as a procedure and then using the arrow n otation. \nCompute the differential of the procude using the D nota tion (see help pages).\n Check that you get the same result using the \+ diff command applied to the function defined by the arrow notation. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "6. " }{TEXT 572 41 "Let m and n be positive integers. The " }{TEXT 573 23 "greatest common divisor" }{TEXT 574 406 " of m and n is the largest integer g that divides both m and n. Wri te a Maple procedure that finds g for any given positive integers m and n. Hint: Let m>n. Simple procedure tests first whether m \+ is divisible by n. If yes, then g=n. If not, then test whether m \+ and n are both divisible by n-1. If yes, then g = n-1. If not, \+ then continue the procedure with n-2, and so on. " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 4 "" 0 "" {TEXT -1 1 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 272 3 " 7." }{TEXT 414 43 " Let m and n be positive integ ers. The " }{TEXT 575 21 "least common multiple" }{TEXT 576 178 " of \+ m and n is mn/g,\n where g is the great common divisor of m an d n. \n Write a Maple procedure, LCM, that computes the least common multiple of two positive integers. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 " 8. " }{TEXT 273 112 "Absolute value. The following proc edure tries to compute the absolute value of a number. What is wrong \+ with it?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ABS:=proc(x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " if x<0 then x:= -x; fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 " 9 " }{TEXT 274 122 "Define a function f that has the value 1 on the interval [0,1] and 0 otherwise. Plot the function a nd integrate it. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 288 1 " " }{TEXT 412 4 " 10." }{TEXT 413 219 " Write a Maple procedure that first forms 10 lists of 10 random integers between -10 and 10 (these numbers inc luded). Then computes the sum of the numbers in each list, and finall y computes the average of these sums.\n" }}}{SECT 1 {PARA 264 "" 0 "" {TEXT -1 1 " " }{TEXT 411 4 "11. " }{TEXT -1 123 "Write a Maple proced ure that takes as its' input a numeric list and rearranges elements of the list into increasing order.\n" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 289 1 " " }{TEXT 409 3 "11." }{TEXT 410 116 " Define the numbers S[n] \+ as follows: S[1]=1, and S[n] = sqrt(1+S[n-1]). Compute \n20 first \+ terms of the sequence." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 " 2. \+ Elementary linear algebra" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " } {TEXT 318 2 "1." }{TEXT 420 132 " Use 3x3 matrices with unassigned ent ries to give a computational proof of the following law of matrix comp utations (A+B)C = AC + BC" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 270 "" 0 "" {TEXT -1 1 " " }{TEXT 421 1 "2" } {TEXT -1 86 ". Use the Maple commands addrow, mulrow, and swaprow to s olve the system \n " }{XPPEDIT 18 0 "3*x[1] + 2*x[2] -3*x[ 3] + 4*x[4] = 5" "6#/,**&\"\"$\"\"\"&%\"xG6#\"\"\"F'F'*&\"\"#F'&F)6#\" \"#F'F'*&\"\"$F'&F)6#\"\"$F'!\"\"*&\"\"%F'&F)6#\"\"%F'F'\"\"&" }{TEXT -1 16 " \n " }{XPPEDIT 18 0 " 2*x[1] + 7*x[2] + 11*x[3] - 3*x[4] = 41" "6#/,**&\"\"#\"\"\"&%\"xG6#\"\"\"F'F'*&\"\"(F'&F)6#\"\"# F'F'*&\"#6F'&F)6#\"\"$F'F'*&\"\"$F'&F)6#\"\"%F'!\"\"\"#T" }{TEXT -1 14 "\n " }{XPPEDIT 18 0 " 2*x[1] - 5*x[2] + 3*x[3] + 5*x[4 ] = 12" "6#/,**&\"\"#\"\"\"&%\"xG6#\"\"\"F'F'*&\"\"&F'&F)6#\"\"#F'!\" \"*&\"\"$F'&F)6#\"\"$F'F'*&\"\"&F'&F)6#\"\"%F'F'\"#7" }{TEXT -1 15 " \+ \n " }{XPPEDIT 18 0 " 6*x[1] + 3*x[2] - 49*x[3] + 2*x[4] = 34" "6#/,**&\"\"'\"\"\"&%\"xG6#\"\"\"F'F'*&\"\"$F'&F)6#\"\"#F'F'*&\"# \\F'&F)6#\"\"$F'!\"\"*&\"\"#F'&F)6#\"\"%F'F'\"#M" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 " 3. Functions " }}{SECT 1 {PARA 4 "" 0 "" {TEXT 290 1 " " }{TEXT 422 2 "1." }{TEXT 423 20 " Form the function " }{XPPEDIT 18 0 "f(x) = cos(x)*sin(x^2+1) " "6#/-%\"fG6#%\"xG*&-%$cosG6#F'\"\"\"-%$sinG6#,&*$F'\"\"#F,\"\"\"F,F, " }{TEXT 291 20 ". Plot it for x in " }{XPPEDIT 18 0 "[-2*Pi,2*Pi]" " 6#7$,$*&\"\"#\"\"\"%#PiGF'!\"\"*&\"\"#F'F(F'" }{TEXT 292 1 "." }}} {SECT 1 {PARA 265 "" 0 "" {TEXT -1 1 " " }{TEXT 424 2 "2." }{TEXT -1 145 " Form the function that takes the value 1 in the intervals [0,1] \+ and [2,3] and takes otherwise the value 0. Plot this function and int egrate it." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 12 " 4. Numbers" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 407 3 " 1." }{TEXT 425 68 " Let m and n be \+ positive integers. Show that (m,n) = (gcd(m,n))." }}}{SECT 0 {PARA 289 "" 0 "" {TEXT -1 1 " " }{TEXT 426 2 "2." }{TEXT -1 126 " Find the \+ smallest numbers m and n, m>n, such that the Euclidean algorithm fo r the computation of gcd(m,n) takes 15 loops.\n" }}}{SECT 0 {PARA 4 " " 0 "" {TEXT 408 1 " " }{TEXT 427 2 "3." }{TEXT 428 64 " For which val ues of k, k=0..10, 7 is in the ideal (5,k^2+1).\n" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 429 2 "4." }{TEXT -1 95 " Let a, b a nd c be three positive integers. Prove that\n gcd(gcd(a,b),c) = g cd(a,gcd(b,c))." }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 1 " " }{TEXT 430 2 "5." }{TEXT -1 148 " If an integer m is a factor of n, then we write m | n. Let a,b, c be three positive integers. \n Show that c \+ | ab ==> c | gcd(a,c)*gcd(b,c). " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 433 2 "6." }{TEXT -1 1 " " }{TEXT 275 134 "Prove that \+ there are infinitly many prime numbers. \nHint: If p is a prime, t hen 1 + 2*3*..*p is not divisible by any number <=p." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 276 1 " " }{TEXT 431 2 "7." }{TEXT 432 64 " Usin g the web, find out what is the largest known prime number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 " 5. \+ Modular arithmetics" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 1 " " }{TEXT 439 2 "1." }{TEXT -1 102 " Assume th at b<>0 mod p and that a/b is an integer. Show that \na/b mod p = (a \+ mod p)/(b mod p) mod p\n" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 277 1 " " } {TEXT 440 2 "2." }{TEXT 441 83 " Let a,b and p be integers. Show that \n (a + b) mod p = (a mod p) + (b mod p) mod p" }{TEXT -1 1 "\n" }}} {SECT 1 {PARA 259 "" 0 "" {TEXT -1 1 " " }{TEXT 442 2 "3." }{TEXT -1 49 " Show that ab mod p = (a mod p)(b mod p) mod p\n" }}}{SECT 1 {PARA 261 "" 0 "" {TEXT -1 1 " " }{TEXT 443 2 "4." }{TEXT -1 86 " Assu me that a and b are two positive integers, a>b. Show that a mod b < = (a-1)/2.\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 262 "" 0 "" {TEXT -1 1 " " }{TEXT 444 2 "5." }{TEXT -1 90 " Solve the followi ng system\n\n x = 2 mod 5\n x = 1 m od 3\n" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT 278 1 " " }{TEXT 437 1 "6" }{TEXT 438 78 ". Assume that g cd(m,n)>1. Find the kernel of the homomorphism\n " } {XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT 281 3 ": " }{XPPEDIT 18 0 "Z[m* n] -> Z[m]" "6#R6#&%\"ZG6#*&%\"mG\"\"\"%\"nGF*7\"6$%)operatorG%&arrowG 6\"&F&6#F)F0F0F0" }{TEXT 279 1 " " }{XPPEDIT 18 0 "Z[n]" "6#&%\"ZG6#% \"nG" }{TEXT 280 54 "; u -> (u mod m, u mod n)\nRecall that the kern el of " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT 282 32 " consists of al l those numbers " }{XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT 284 12 " such \+ that " }{XPPEDIT 18 0 "phi(u)=0" "6#/-%$phiG6#%\"uG\"\"!" }{TEXT 283 1 "." }{TEXT -1 1 "\n" }}}{SECT 1 {PARA 263 "" 0 "" {TEXT -1 1 " " } {TEXT 436 2 "7." }{TEXT -1 95 " Write Maple procedures msum and mprod \+ that compute sums and products in modular arithmetics. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "msum:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " input: integers " }{XPPEDIT 18 0 "m,n" "6$%\"mG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p" "6#%\"pG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " output: " }{XPPEDIT 18 0 "m+n mod p" "6#-%$modG6$,&%\" mG\"\"\"%\"nGF(%\"pG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "mprod" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " input: integers " } {XPPEDIT 18 0 "m,n" "6$%\"mG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p" "6#%\"pG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " output: " }{XPPEDIT 18 0 "m*n mod p" "6#-%$modG6$*&%\"mG\"\" \"%\"nGF(%\"pG" }}}{SECT 1 {PARA 266 "" 0 "" {TEXT 434 3 " 8." }{TEXT -1 167 " Find a numbers a and b such that \n \+ x = a mod 6\n x = b mod 8\nhas more than one solution in " }{XPPEDIT 18 0 "Z[48]" "6#&%\" ZG6#\"#[" }{TEXT -1 2 ".\n" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " \+ " }{TEXT 435 2 "9." }{TEXT -1 1 " " }{TEXT 293 139 "Let u, v be intege rs <145. \nWhat is the largest number of steps that the Euclidean alg orithm \nmay need in the computation of the gcd(u,v)?" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 " 6. Advanced procedures" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 285 1 " " }{TEXT 446 2 "1." }{TEXT 447 382 " The following Maple code implements the extended Euclidean algorithm. \nE xtEuclid:=proc(m::posint,n::posint)\nlocal N,M,L,i,u,v,U,V;\n(N,M):=(n ,m);\n (u,v):=(0,1);\n(U,V):=(1,0);\nwhile N<>0 do \n (M,N,u, v,U,V) := (N,irem(M,N),U-iquo(M,N)*u,V-iquo(M,N)*v,u,v);\n o d;\n(M,U,V);\nend; \n\nUsing mathematical induction, prove that at eac h step M= U*m + V*n provided that m>n.\n" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 2 " \n" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " " }{TEXT 445 3 "2. " }{TEXT 286 121 "Write a Maple proce dure that takes a list of sequences as its input and \ncomputes the pr oduct set of the input sequences." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 " 7. Debugging" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 287 3 " 1." }{TEXT 448 42 " What is wrong in the follo wing procedure?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x:=proc(N ::posint)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "local n, i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "for i to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " n:=n+1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 " 8. Polynomials" }}{SECT 1 {PARA 288 "" 0 "" {TEXT -1 1 " " }{TEXT 459 2 "1." }{TEXT -1 5 " Let \+ " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 72 " be a univariate polynomi al of degree >1. Explain why, for any number " }{XPPEDIT 18 0 "alpha " "6#%&alphaG" }{TEXT -1 13 ", we have " }{XPPEDIT 18 0 "P(x) = (x- alpha)*Q(x) + P(alpha)" "6#/-%\"PG6#%\"xG,&*&,&F'\"\"\"%&alphaG!\"\"F+ -%\"QG6#F'F+F+-F%6#F,F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 294 4 " 2. " }{TEXT 458 97 "Let\n P:=randpoly(x,degree=10,dense); and\nQ:=randpoly(x,degree=7,dense);\n Determine the constants " }{XPPEDIT 261 0 "alpha" "6#%&alphaG" } {TEXT 295 7 " and " }{XPPEDIT 262 0 "beta" "6#%%betaG" }{TEXT 296 40 " such that the degree of the polynomial " }{XPPEDIT 263 0 "R=P-alp ha*x^beta*Q" "6#/%\"RG,&%\"PG\"\"\"*(%&alphaGF')%\"xG%%betaGF'%\"QGF'! \"\"" }{TEXT 297 89 " is less than the degree of P. Compute the de gree of R. \nIn the above computation, if" }{TEXT 449 1 " " }{XPPEDIT 18 0 "beta>1" "6#2\"\"\"%%betaG" }{TEXT 298 2 ", " }{TEXT 450 75 "then the process can perhaps be repeated, and you may find another constan t" }{TEXT 451 1 " " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 299 1 " " }{TEXT 452 39 "such that the degree of the polynomial " }{TEXT 453 1 "\n" }{XPPEDIT 18 0 "S = P - alpha*x^beta*Q - delta*x^(beta-1)*Q " "6#/%\"SG,(%\"PG\"\"\"*(%&alphaGF')%\"xG%%betaGF'%\"QGF'!\"\"*(%&delta GF')F+,&F,F'\"\"\"F.F'F-F'F." }{TEXT 300 1 "\n" }{TEXT 454 55 "is less than the degree of R. Determine this constant" }{TEXT 455 1 " " } {XPPEDIT 18 0 "delta " "6#%&deltaG" }{TEXT 301 2 ", " }{TEXT 456 58 "i f possible, and compute the polynomial S and its degree.\n" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 404 2 " 3" }{TEXT 457 51 ". Construct an examp le of a non-zero polynomial in " }{XPPEDIT 18 0 "Z[5]" "6#&%\"ZG6#\"\" &" }{TEXT 405 1 "[" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 406 40 "] that takes the value 0 for every x in " }{XPPEDIT 18 0 "Z[5]" "6#&%\"ZG6# \"\"&" }{TEXT -1 3 ". \n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 " 9. Conversions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 303 4 "1. C" }{TEXT 302 22 "onsider the funct ion " }{TEXT 304 0 "" }{XPPEDIT 18 0 "sin(x^2+x)-cos(x)" "6#,&-%$sinG 6#,&*$%\"xG\"\"#\"\"\"F)F+F+-%$cosG6#F)!\"\"" }{TEXT 305 145 ". How l arge must the degree of the Taylor polynomial of this function be ino rder to get a \"good\" approximation of the function in the interval \+ " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT 306 2 "?\n" }} }{SECT 0 {PARA 267 "" 0 "" {TEXT 460 3 " 2." }{TEXT -1 56 " Is the pro duct of Taylor polynomials for the functions " }{XPPEDIT 18 0 " f(x)=s in(x)" "6#/-%\"fG6#%\"xG-%$sinG6#F'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "g(x) = cos(x)" "6#/-%\"gG6#%\"xG-%$cosG6#F'" }{TEXT -1 37 " a Ta ylor polynomial for the product " }{XPPEDIT 18 0 "f(x)*g(x)" "6#*&-%\" fG6#%\"xG\"\"\"-%\"gG6#F'F(" }{TEXT -1 1 "?" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 " 10. Elementary ideal membership problems" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 268 "" 0 "" {TEXT 461 1 "1" } {TEXT -1 37 ". The ideal generated by the numbers " }{XPPEDIT 18 0 "m " "6#%\"mG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 27 " is\n \+ (" }{XPPEDIT 18 0 "m,n,p" "6%%\"mG%\"nG%\"pG" }{TEXT -1 5 ") = \+ \{" }{XPPEDIT 18 0 " s*m+t*n+o*p " "6#,(*&%\"sG\"\"\"%\"mGF&F&*&%\"tGF &%\"nGF&F&*&%\"oGF&%\"pGF&F&" }{TEXT -1 2 "| " }{XPPEDIT 18 0 "s,t,o" "6%%\"sG%\"tG%\"oG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "Z" "6#%\"ZG" } {TEXT -1 102 "\}\n\nIs this ideal generated by one element only? If i t is, what is the element? Justify your answer.\n" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 " 2. " }{TEXT 307 78 "Determine whether 27 is \+ in the ideal (21713469,2349834699,115134983469)." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 308 1 " " }{TEXT 462 1 "3" }{TEXT 463 48 ". Determine \+ whether the polynomial \n " }{XPPEDIT 18 0 " P = 7225*x^4+9 350*x^3+9315*x^2+4070*x+1369 \n" "6#/%\"PG,,*&\"%Ds\"\"\"*$%\"xG\"\"% F(F(*&\"%]$*F(*$F*\"\"$F(F(*&\"%:$*F(*$F*\"\"#F(F(*&\"%qSF(F*F(F(\"%p8 F(" }{TEXT 309 61 "\nis in the ideal generated by the polynomials \n \+ " }{XPPEDIT 18 0 "A = 2975*x^4-6320*x^3-8290*x^2-6339*x-18 50 \n" "6#/%\"AG,,*&\"%vH\"\"\"*$%\"xG\"\"%F(F(*&\"%?jF(*$F*\"\"$F(! \"\"*&\"%!H)F(*$F*\"\"#F(F0*&\"%RjF(F*F(F0\"%]=F0" }{TEXT 310 20 "\nan d \n " }{XPPEDIT 18 0 "B = -6715*x^4-9105*x^3-10168*x^2-4 767*x-1813" "6#/%\"BG,,*&\"%:n\"\"\"*$%\"xG\"\"%F(!\"\"*&\"%0\"*F(*$F* \"\"$F(F,*&\"&o,\"F(*$F*\"\"#F(F,*&\"%nZF(F*F(F,\"%8=F," }{TEXT 311 1 "." }{TEXT -1 1 "\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 " 11. Plo tting" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 4 " 1. " }{TEXT 312 62 "Draw the Lemniscate of Bernoulli defi ned by\n(a) the equation " }{XPPEDIT 18 0 "(x^2+y^2)^2 = x^2-y^2" "6# /*$,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F)\"\"#,&*$F'\"\"#F)*$F+\"\"#!\"\" " }{TEXT 313 27 ",\n\n(b) the polar equation " }{XPPEDIT 18 0 "r^2 = \+ cos(2*phi)" "6#/*$%\"rG\"\"#-%$cosG6#*&\"\"#\"\"\"%$phiGF," }{TEXT 314 26 "\n\n(c) the parametrization " }{XPPEDIT 18 0 "x=cos(t)/(1+sin( t)^2)" "6#/%\"xG*&-%$cosG6#%\"tG\"\"\",&\"\"\"F**$-%$sinG6#F)\"\"#F*! \"\"" }{TEXT 315 1 "," }{XPPEDIT 18 0 " y=cos(t)*sin(t)/(1+sin(t)^2)" "6#/%\"yG*(-%$cosG6#%\"tG\"\"\"-%$sinG6#F)F*,&\"\"\"F**$-F,6#F)\"\"#F* !\"\"" }{TEXT 316 6 ", for " }{XPPEDIT 18 0 "t=-Pi..Pi" "6#/%\"tG;,$%# PiG!\"\"F'" }{TEXT 317 2 ".\n" }}}{SECT 1 {PARA 269 "" 0 "" {TEXT -1 1 " " }{TEXT 464 3 "2. " }{TEXT -1 41 "Draw the following parametric p lots\n\n(a) " }{XPPEDIT 18 0 "t -> [(t^2-1)/(t^2=1),2*t/(t^2+1)]" "6#R 6#%\"tG7\"6$%)operatorG%&arrowG6\"7$*&,&*$F%\"\"#\"\"\"\"\"\"!\"\"F0/* $F%\"\"#\"\"\"F2*(\"\"#F0F%F0,&*$F%\"\"#F0\"\"\"F0F2F*F*F*" }{TEXT -1 6 ", for " }{XPPEDIT 18 0 "t = -infinity..infinity" "6#/%\"tG;,$%)infi nityG!\"\"F'" }{TEXT -1 6 "\n\n(b) " }{XPPEDIT 18 0 "t -> [(4/9)*t^3-( 14/9)*t^2+(1/9)*t+1,-(4/9)*t^3-(1/9)*t^2+14/9*t]" "6#R6#%\"tG7\"6$%)op eratorG%&arrowG6\"7$,**(\"\"%\"\"\"\"\"*!\"\"F%\"\"$F/*(\"#9F/\"\"*F1F %\"\"#F1*(\"\"\"F/\"\"*F1F%F/F/\"\"\"F/,(*(\"\"%F/\"\"*F1F%\"\"$F1*(\" \"\"F/\"\"*F1F%\"\"#F1*(\"#9F/\"\"*F1F%F/F/F*F*F*" }{TEXT -1 5 " for \+ " }{XPPEDIT 18 0 "t = 0..1" "6#/%\"tG;\"\"!\"\"\"" }{TEXT -1 68 ".\n\n (c) Draw the previous parametric plot together with the plot of \n" } {XPPEDIT 18 0 "t->[cos(Pi/2*t),sin(Pi/2*t)]" "6#R6#%\"tG7\"6$%)operato rG%&arrowG6\"7$-%$cosG6#*(%#PiG\"\"\"\"\"#!\"\"F%F1-%$sinG6#*(F0F1\"\" #F3F%F1F*F*F*" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "t = 0..1" "6#/%\"t G;\"\"!\"\"\"" }{TEXT -1 2 ".\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 " 12. Varieties and ideals" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 358 3 " 1. " }{TEXT 465 16 "Affine varieties" }}{PARA 0 "" 0 "" {TEXT -1 42 "S ketch the following affine varieties in " }{XPPEDIT 18 0 "R^2" "6#*$% \"RG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "V(x^2+4* y^2+2*x-16*y+1)" "6#-%\"VG6#,,*$%\"xG\"\"#\"\"\"*&\"\"%F**$%\"yG\"\"#F *F**&\"\"#F*F(F*F**&\"#;F*F.F*!\"\"\"\"\"F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "V(x^2-y^2)" "6#-%\"VG6#,&*$%\"xG\"\"#\" \"\"*$%\"yG\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "V(2*x+y-1,3*x-y+2)" "6#-%\"VG6$,(*&\"\" #\"\"\"%\"xGF)F)%\"yGF)\"\"\"!\"\",(*&\"\"$F)F*F)F)F+F-\"\"#F)" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 357 2 "2." }{TEXT 466 22 " Equivalence o f ideals" }}{PARA 0 "" 0 "" {TEXT -1 46 "Are the following equivalen ces of ideals in " }{XPPEDIT 19 1 "Q" "6#%\"QG" }{TEXT -1 1 "[" } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "y" "6#%\"y G" }{TEXT -1 32 "] correct? Justify your answer." }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "x+y" "6#,&%\"xG\"\"\"%\"yGF%" } {TEXT -1 1 "," }{XPPEDIT 18 0 "x-y" "6#,&%\"xG\"\"\"%\"yG!\"\"" } {TEXT -1 5 ") = (" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "," } {XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " (" }{XPPEDIT 18 0 "x+x*y" "6#,&%\"xG\"\"\"*&F$F%%\"yGF%F%" }{TEXT -1 1 "," }{XPPEDIT 18 0 "y+x*y" "6#,&%\"yG\"\"\"*&%\"xGF%F$F%F%" }{TEXT -1 1 "," }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 1 "," } {XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 5 ") = (" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "2*x^2+3*y^2-11 " "6#,(*&\"\"#\"\"\"*$%\"xG\"\"#F&F&*&\"\"$F&*$%\"yG\"\"#F&F&\"#6!\"\" " }{TEXT -1 1 "," }{XPPEDIT 18 0 "x^2-y^2-3" "6#,(*$%\"xG\"\"#\"\"\"*$ %\"yG\"\"#!\"\"\"\"$F+" }{TEXT -1 5 ") = (" }{XPPEDIT 18 0 "x^2-4" "6# ,&*$%\"xG\"\"#\"\"\"\"\"%!\"\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "y^2-1 " "6#,&*$%\"yG\"\"#\"\"\"\"\"\"!\"\"" }{TEXT -1 1 ")" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 356 2 "3." } {TEXT 467 10 " Show that" }{TEXT -1 3 " V(" }{XPPEDIT 18 0 "x+x*y" "6# ,&%\"xG\"\"\"*&F$F%%\"yGF%F%" }{TEXT -1 1 "," }{XPPEDIT 18 0 "y+x*y" " 6#,&%\"yG\"\"\"*&%\"xGF%F$F%F%" }{TEXT -1 1 "," }{XPPEDIT 18 0 "x^2" " 6#*$%\"xG\"\"#" }{TEXT -1 1 "," }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 6 ") = V(" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "," } {XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 355 2 "4." }{TEXT 468 61 " De termine whether the given polynomial is in the given ideal" }}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "x^2-3*x+2" "6#,(*$%\"xG\"\"#\"\"\"*&\" \"$F'F%F'!\"\"\"\"#F'" }{TEXT -1 8 ", I = (" }{XPPEDIT 18 0 "x-2" "6# ,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 1 ")" }}}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "x^2-4*x+1" "6#,(*$%\"xG\"\"#\"\"\"*&\"\"%F'F%F'!\"\"\" \"\"F'" }{TEXT -1 7 ", I = (" }{XPPEDIT 18 0 "x^3-x^2+x" "6#,(*$%\"xG \"\"$\"\"\"*$F%\"\"#!\"\"F%F'" }{TEXT -1 1 ")" }}}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "x^2-4*x+4" "6#,(*$%\"xG\"\"#\"\"\"*&\"\"%F'F%F'!\" \"\"\"%F'" }{TEXT -1 7 ", I = (" }{XPPEDIT 18 0 "x^3-6*x^2+12*x-8" "6# ,**$%\"xG\"\"$\"\"\"*&\"\"'F'*$F%\"\"#F'!\"\"*&\"#7F'F%F'F'\"\")F," } {TEXT -1 1 "," }{XPPEDIT 18 0 "2*x^3-10*x^2+16*x-8" "6#,**&\"\"#\"\"\" *$%\"xG\"\"$F&F&*&\"#5F&*$F(\"\"#F&!\"\"*&\"#;F&F(F&F&\"\")F." }{TEXT -1 1 ")" }}}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "x^3-1" "6#,&*$%\"xG \"\"$\"\"\"\"\"\"!\"\"" }{TEXT -1 7 ", I = (" }{XPPEDIT 18 0 "x^9-1" " 6#,&*$%\"xG\"\"*\"\"\"\"\"\"!\"\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "x^5 +x^3-x^2-1" "6#,**$%\"xG\"\"&\"\"\"*$F%\"\"$F'*$F%\"\"#!\"\"\"\"\"F," }{TEXT -1 1 ")" }}}}{SECT 1 {PARA 275 "" 0 "" {TEXT 469 2 "5." }{TEXT -1 127 " Show that the polynomial \n \+ P= x + y \nis in the radical of the ideal generated by \n" }{XPPEDIT 18 0 "a =x^6*y-x^5*y+4*x^5*y^2-4*x^4*y^2+6*x^4*y^3-6*x^3*y^3 +4*x^3*y^4-4*y^4*x^2+y^5*x^2-y^5*x" "6#/%\"aG,6*&%\"xG\"\"'%\"yG\"\"\" F**&F'\"\"&F)F*!\"\"*(\"\"%F**$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**$F)\"\"%F*F'\"\"#F-*&F)\"\"&F'\"\"# F**&F)\"\"&F'F*F-" }{TEXT -1 6 " \nand\n" }{XPPEDIT 18 0 "b = x^6+6*x^ 5*y+15*x^4*y^2+20*x^3*y^3+15*y^4*x^2+6*y^5*x+y^6" "6#/%\"bG,0*$%\"xG\" \"'\"\"\"*(\"\"'F)*$F'\"\"&F)%\"yGF)F)*(\"#:F)*$F'\"\"%F)F.\"\"#F)*(\" #?F)*$F'\"\"$F)F.\"\"$F)*(\"#:F)*$F.\"\"%F)F'\"\"#F)*(\"\"'F)*$F.\"\"& F)F'F)F)*$F.\"\"'F)" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 287 "" 0 "" {TEXT -1 1 " " }{TEXT 470 1 "6" }{TEXT -1 74 ". Does the polynomial x+1 belong to the radical of the ideal ge nerated by " }{XPPEDIT 18 0 "P := -85*x^9-650*x^8-2207*x^7-4389*x^6-56 77*x^5-5005*x^4-3045*x^3-1247*x^2-314*x-37;" "6#>%\"PG,6*&\"#&)\"\"\"* $%\"xG\"\"*F(!\"\"*&\"$]'F(*$F*\"\")F(F,*&\"%2AF(*$F*\"\"(F(F,*&\"%*Q% F(*$F*\"\"'F(F,*&\"%xcF(*$F*\"\"&F(F,*&\"%0]F(*$F*\"\"%F(F,*&\"%XIF(*$ F*\"\"$F(F,*&\"%Z7F(*$F*\"\"#F(F,*&\"$9$F(F*F(F,\"#PF," }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "Q := -35*x^9-148*x^8-6*x^7+1162*x^6+3220*x^5+441 0*x^4+3542*x^3+1694*x^2+447*x+50;" "6#>%\"QG,6*&\"#N\"\"\"*$%\"xG\"\"* F(!\"\"*&\"$[\"F(*$F*\"\")F(F,*&\"\"'F(*$F*\"\"(F(F,*&\"%i6F(*$F*\"\"' F(F(*&\"%?KF(*$F*\"\"&F(F(*&\"%5WF(*$F*\"\"%F(F(*&\"%UNF(*$F*\"\"$F(F( *&\"%%p\"F(*$F*\"\"#F(F(*&\"$Z%F(F*F(F(\"#]F(" }{TEXT -1 2 "?\n" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 " 13. Equivalence of ideals" }} {PARA 0 "" 0 "" {TEXT -1 78 "In the following three problems show the \+ following equivalences of ideals in " }{XPPEDIT 19 1 "Q" "6#%\"QG" } {TEXT -1 1 "[" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 30 "] using Groebner basis method." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 " 1. (" }{XPPEDIT 18 0 "x+y,x-y" "6 $,&%\"xG\"\"\"%\"yGF%,&F$F%F&!\"\"" }{TEXT -1 5 ") = (" }{XPPEDIT 18 0 "x,y" "6$%\"xG%\"yG" }{TEXT -1 1 ")" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 " 2. (" }{XPPEDIT 18 0 "x +xy,y+xy,x^2,y^2" "6&,&%\"xG\"\"\"%#xyGF%,&%\"yGF%F&F%*$F$\"\"#*$F(\" \"#" }{TEXT -1 5 ") = (" }{XPPEDIT 18 0 "x,y" "6$%\"xG%\"yG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 471 4 "3. (" }{XPPEDIT 472 0 "2x^2+3y^2-11,x^2-y ^2-3" "6$,(*&\"\"#\"\"\"*$%\"xG\"\"#F&F&*&\"\"$F&*$%\"yG\"\"#F&F&\"#6! \"\",(*$F(\"\"#F&*$F-\"\"#F0\"\"$F0" }{TEXT 473 5 ") = (" }{XPPEDIT 474 0 "x^2-4,y^2-1" "6$,&*$%\"xG\"\"#\"\"\"\"\"%!\"\",&*$%\"yG\"\"#F' \"\"\"F)" }{TEXT 475 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT 476 8 " 4. Let " }{XPPEDIT 477 0 "J[1]" "6#&%\"JG6#\"\"\"" }{TEXT 478 4 " = (" }{XPPEDIT 479 0 "x^2+z, x*y+y^2 +z,x*z-y^3-2*y*z, y^4+3*y^2*z+z^2" "6&,&*$%\"xG\"\"#\"\"\"%\"zGF',(*&F %F'%\"yGF'F'*$F+\"\"#F'F(F',(*&F%F'F(F'F'*$F+\"\"$!\"\"*(\"\"#F'F+F'F( F'F2,(*$F+\"\"%F'*(\"\"$F'*$F+\"\"#F'F(F'F'*$F(\"\"#F'" }{TEXT 480 6 " ) and " }{XPPEDIT 481 0 "J[2]" "6#&%\"JG6#\"\"#" }{TEXT 482 4 " = (" } {XPPEDIT 483 0 "x^2+z,x*y+y^2+z,x^3-y*z" "6%,&*$%\"xG\"\"#\"\"\"%\"zGF ',(*&F%F'%\"yGF'F'*$F+\"\"#F'F(F',&*$F%\"\"$F'*&F+F'F(F'!\"\"" }{TEXT 484 5 "). \n" }{TEXT 319 52 "Determine which of the following (if any ) is true: " }{XPPEDIT 18 0 "J[1]" "6#&%\"JG6#\"\"\"" }{TEXT 320 9 " \+ contains" }{TEXT -1 1 " " }{XPPEDIT 18 0 "J[2]" "6#&%\"JG6#\"\"#" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "J[2] " "6#&%\"JG6#\"\"#" }{TEXT -1 2 " " }{TEXT 321 8 "contains" }{TEXT -1 1 " " }{XPPEDIT 18 0 "J[1]" "6#& %\"JG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "J[1]=J[2]" "6#/&%\"JG6 #\"\"\"&F%6#\"\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 271 " " 0 "" {TEXT 322 1 " " }{TEXT -1 2 "5." }{TEXT 486 6 " Let " }{TEXT -1 2 " \n" }{TEXT 327 0 "" }{TEXT -1 7 " " }{XPPEDIT 18 0 "P=x^4 *y^2+z^2-4*x*y^3*z-2*y^5*z" "6#/%\"PG,**&%\"xG\"\"%%\"yG\"\"#\"\"\"*$% \"zG\"\"#F+**\"\"%F+F'F+F)\"\"$F-F+!\"\"*(\"\"#F+*$F)\"\"&F+F-F+F2" } {TEXT -1 4 " \n " }{TEXT 323 6 "and \n" }{TEXT -1 6 " " } {XPPEDIT 18 0 "Q=x^2+2*x*y^2+y^4" "6#/%\"QG,(*$%\"xG\"\"#\"\"\"*(\"\"# F)F'F)%\"yG\"\"#F)*$F,\"\"%F)" }{TEXT -1 6 " \n S" }{TEXT 324 24 "ho w that the polynomial " }{TEXT -1 8 "\n " }{XPPEDIT 18 0 "f=y*z- x^3" "6#/%\"fG,&*&%\"yG\"\"\"%\"zGF(F(*$%\"xG\"\"$!\"\"" }{TEXT -1 2 " \n" }{TEXT 325 35 "belongs to the radical of the ideal" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "P,Q" "6$%\"PG%\"QG" }{TEXT -1 4 "). " }{TEXT 326 44 "Find the smallest power of f that belongs to" }{TEXT -1 2 " (" } {XPPEDIT 18 0 "P,Q" "6$%\"PG%\"QG" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 " 6. " }{TEXT 354 3 "Let" }{TEXT -1 2 " " }{XPPEDIT 18 0 "I[1]" "6#&%\"IG6#\"\"\"" }{TEXT -1 1 " " } {TEXT 353 41 "be the ideal generated by the polynomials" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 " P=49*x^2+63*x*y+57*x-59*y^2+ 45*y-8" "6#/%\"PG,.*&\"#\\\"\"\"*$%\"xG\"\"#F(F(*(\"#jF(F*F(%\"yGF(F(* &\"#dF(F*F(F(*&\"#fF(*$F.\"\"#F(!\"\"*&\"#XF(F.F(F(\"\")F5" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 7 " " } {XPPEDIT 18 0 "Q=-93*x^2+92*x*y+43*x-62*y^2+77*y+66" "6#/%\"QG,.*&\"#$ *\"\"\"*$%\"xG\"\"#F(!\"\"*(\"##*F(F*F(%\"yGF(F(*&\"#VF(F*F(F(*&\"#iF( *$F/\"\"#F(F,*&\"#xF(F/F(F(\"#mF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "I[2] " "6#&%\"IG6#\"\"#" }{TEXT -1 26 " be the ideal generated by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "A=-882*x^3+38 5*x^2*y-2300*x^2+3015*x*y^2-681*x*y-1338*x-1829*y^3+2929*y^2-1418*y+20 8" "6#/%\"AG,6*&\"$#))\"\"\"*$%\"xG\"\"$F(!\"\"*(\"$&QF(*$F*\"\"#F(%\" yGF(F(*&\"%+BF(*$F*\"\"#F(F,*(\"%:IF(F*F(F1\"\"#F(*(\"$\"oF(F*F(F1F(F, *&\"%Q8F(F*F(F,*&\"%H=F(*$F1\"\"$F(F,*&\"%HHF(*$F1\"\"#F(F(*&\"%=9F(F1 F(F,\"$3#F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "B=1674*x^3-4539*x^2*y+1644*x^2+3968*x*y^2-2445*x*y-2306*x-1922*y ^3+3999*y^2+44*y-1716" "6#/%\"BG,6*&\"%u;\"\"\"*$%\"xG\"\"$F(F(*(\"%RX F(*$F*\"\"#F(%\"yGF(!\"\"*&\"%W;F(*$F*\"\"#F(F(*(\"%oRF(F*F(F0\"\"#F(* (\"%XCF(F*F(F0F(F1*&\"%1BF(F*F(F1*&\"%A>F(*$F0\"\"$F(F1*&\"%**RF(*$F0 \"\"#F(F(*&\"#WF(F0F(F(\"%; " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 272 "" 0 "" {TEXT -1 1 " " } {TEXT 490 1 "2" }{TEXT -1 25 ". Solve the linear system" }}{PARA 0 "" 0 "" {TEXT -1 34 " " }{XPPEDIT 18 0 " x+y+z+u+v=0" "6#/,,%\"xG\"\"\"%\"yGF&%\"zGF&%\"uGF&%\"vGF&\"\"!" }} {PARA 0 "" 0 "" {TEXT -1 32 " " } {XPPEDIT 18 0 " 2*x+3*y+z+u+v=0" "6#/,,*&\"\"#\"\"\"%\"xGF'F'*&\"\"$F' %\"yGF'F'%\"zGF'%\"uGF'%\"vGF'\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 33 " \+ " }{XPPEDIT 18 0 " x+2*y+3*z+u+v=0" "6 #/,,%\"xG\"\"\"*&\"\"#F&%\"yGF&F&*&\"\"$F&%\"zGF&F&%\"uGF&%\"vGF&\"\"! " }}{PARA 0 "" 0 "" {TEXT -1 33 " " } {XPPEDIT 18 0 " x+y+2*z+3*u+v=0" "6#/,,%\"xG\"\"\"%\"yGF&*&\"\"#F&%\" zGF&F&*&\"\"$F&%\"uGF&F&%\"vGF&\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ " }{XPPEDIT 18 0 " x+y+z+2*u+3*v=0" "6#/,,%\"xG\"\"\"%\"yGF&%\"zGF&*&\"\"#F&%\"uGF&F&*&\"\"$F&%\"vGF&F&\" \"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "f irst by expressing it in the form AX=0 for some matrix A and then app lying row reduction and back substitution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Then solve the same system by u sing the method of Grobner basis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 488 2 "3." }{TEXT -1 1 " " }{TEXT 329 42 "Find the critical points of the fu nction " }{XPPEDIT 485 0 "f(x,y,z)=(x^2+y^2)*(x^2+y^2-1)*z+x^3+x+y" " 6#/-%\"fG6%%\"xG%\"yG%\"zG,**(,&*$F'\"\"#\"\"\"*$F(\"\"#F/F/,(*$F'\"\" #F/*$F(\"\"#F/\"\"\"!\"\"F/F)F/F/*$F'\"\"$F/F'F/F(F/" }{TEXT 487 1 ". " }{TEXT -1 1 " " }{TEXT 330 76 "Critical points are the points where \+ the partial derivatives with respect to" }{TEXT 496 1 " " }{XPPEDIT 494 0 "x" "6#%\"xG" }{TEXT -1 0 "" }{TEXT 497 0 "" }{TEXT 333 1 "," } {TEXT -1 2 " " }{TEXT 492 1 " " }{XPPEDIT 493 0 "y" "6#%\"yG" }{TEXT 331 5 ", and" }{TEXT -1 1 " " }{XPPEDIT 495 0 "z" "6#%\"zG" }{TEXT -1 2 " " }{TEXT 332 24 "vanish at the same time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 276 "" 0 "" {TEXT -1 1 " " }{TEXT 491 2 "4." }{TEXT -1 56 " Find singular points of the curve \n \+ \{ (x,y) | " }{XPPEDIT 18 0 "y^3-3*y^2+3*y-5-x^2+4*x=0 " "6#/,.*$%\" yG\"\"$\"\"\"*&\"\"$F(*$F&\"\"#F(!\"\"*&\"\"$F(F&F(F(\"\"&F-*$%\"xG\" \"#F-*&\"\"%F(F2F(F(\"\"!" }{TEXT -1 3 "\}.\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 " 15. Elimination" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 334 1 " " }{TEXT 498 2 "1." }{TEXT 499 4 " Let" }{TEXT -1 2 " " } {TEXT 335 5 "V in " }{XPPEDIT 502 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT 503 0 "" }{TEXT 500 46 " be the subset parametrized by \n " } {XPPEDIT 501 0 "t -> (t,t^3,t^4)" "6#R6#%\"tG7\"6$%)operatorG%&arrowG6 \"6%F%*$F%\"\"$*$F%\"\"%F*F*F*" }{TEXT 336 49 ". \nFind an affine vari ety containing this curve.\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 " 16. Partial fraction decompositi on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let \+ " }{XPPEDIT 18 0 "Q = x^4+3*x^2+2" "6#/%\"QG,(*$%\"xG\"\"%\"\"\"*&\"\" $F)*$F'\"\"#F)F)\"\"#F)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 504 3 "1. " }{TEXT -1 31 "Convert the rat ional function " }{XPPEDIT 18 0 "1/Q" "6#*&\"\"\"\"\"\"%\"QG!\"\"" } {TEXT -1 122 " to partial fractions using the convert command with th e option 'parfrac'. Integrate the partial fraction decomposition." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 505 2 "2." } {TEXT -1 10 " Convert " }{XPPEDIT 18 0 "1/Q" "6#*&\"\"\"\"\"\"%\"QG! \"\"" }{TEXT -1 145 " to partial fractions using the fullparfrac optio n. Integrate the partial fraction decomposition. Compare your answer to the answer in part 1. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 " 1 7. Resultants" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 399 2 "1." }{TEXT 513 14 " Let\n " }{TEXT 506 3 " \+ " }{XPPEDIT 507 0 "P := 79*y+56*y^2*z^2+49*x^3*y^2+63*x^2*z^2+57*x*y^2 *z-59*x^2*y^3;" "6#>%\"PG,.*&\"#z\"\"\"%\"yGF(F(*(\"#cF(*$F)\"\"#F(%\" zG\"\"#F(*(\"#\\F(*$%\"xG\"\"$F(F)\"\"#F(*(\"#jF(*$F3\"\"#F(F.\"\"#F(* *\"#dF(F3F(F)\"\"#F.F(F(*(\"#fF(*$F3\"\"#F(F)\"\"$!\"\"" }{TEXT 400 16 " \nand \n " }{TEXT 509 1 " " }{XPPEDIT 508 0 " Q := 77*x^3 *y+66*y^2*z^2+54*y*z^2-5*x*z^3+99*x*y*z^2-61*x*z^4" "6#>%\"QG,.*(\"#x \"\"\"*$%\"xG\"\"$F(%\"yGF(F(*(\"#mF(*$F,\"\"#F(%\"zG\"\"#F(*(\"#aF(F, F(F1\"\"#F(*(\"\"&F(F*F(F1\"\"$!\"\"**\"#**F(F*F(F,F(F1\"\"#F(*(\"#hF( F*F(F1\"\"%F9" }{TEXT 401 30 ". \nFind the leading terms of " } {XPPEDIT 510 0 "P" "6#%\"PG" }{TEXT 511 0 "" }{TEXT 402 5 " and " } {XPPEDIT 512 0 "Q" "6#%\"QG" }{TEXT 403 196 " with respect to the pure lexicographic ordering z>x>y. \nHow are these leading terms ordered \+ (i.e., which of the two leding terms is larger with respect to the pur e lexicographic ordering z>x>y)?" }{TEXT -1 1 "\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 273 "" 0 "" {TEXT -1 1 " " }{TEXT 520 2 "2." }{TEXT -1 66 " Let P, Q, and R be polynomials. Express \n \+ " }{XPPEDIT 18 0 "Res(P*Q,R,x) \n" "6#-%$ResG6%*&%\"PG \"\"\"%\"QGF(%\"RG%\"xG" }{TEXT -1 13 "\nin terms of " }{XPPEDIT 18 0 "Res(P,R,x)" "6#-%$ResG6%%\"PG%\"RG%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Res(Q,R,x)" "6#-%$ResG6%%\"QG%\"RG%\"xG" }{TEXT -1 1 ". " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 337 3 " 3." }{TEXT 521 10 " Let P( x)" }{TEXT 339 1 "=" }{XPPEDIT 514 0 "product(x-a[j],j = 1 .. m);" "6# -%(productG6$,&%\"xG\"\"\"&%\"aG6#%\"jG!\"\"/F,;\"\"\"%\"mG" }{TEXT 515 2 " " }{TEXT 338 26 "and Q(x)=x-t. Show that " }{XPPEDIT 516 0 "Res(P,Q,x)=P(t)" "6#/-%$ResG6%%\"PG%\"QG%\"xG-F'6#%\"tG" }{TEXT 340 1 "." }}}{SECT 1 {PARA 274 "" 0 "" {TEXT -1 1 " " }{TEXT 522 2 "4." } {TEXT -1 6 " Let " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "Q" "6#%\"QG" }{TEXT -1 27 " be polynomials of degre e " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 37 " respectively. \nDefine polynomials " } {XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "S" " 6#%\"SG" }{TEXT -1 6 " by " }{XPPEDIT 18 0 "R(x)= P(2*x+1)" "6#/-%\" RG6#%\"xG-%\"PG6#,&*&\"\"#\"\"\"F'F.F.\"\"\"F." }{TEXT -1 7 " and " }{XPPEDIT 18 0 "S(x) = Q(2*x+1)" "6#/-%\"SG6#%\"xG-%\"QG6#,&*&\"\"#\" \"\"F'F.F.\"\"\"F." }{TEXT -1 11 ". Express " }{XPPEDIT 18 0 "Res(R,S ,x)" "6#-%$ResG6%%\"RG%\"SG%\"xG" }{TEXT -1 15 " in terms of " } {XPPEDIT 18 0 "Res(P,Q,x)" "6#-%$ResG6%%\"PG%\"QG%\"xG" }{TEXT -1 1 ". " }}}{SECT 1 {PARA 290 "" 0 "" {TEXT -1 0 "" }{TEXT 341 4 " 5. " } {TEXT 342 62 "Find a polynomial with integer coefficients whose root i s\n(a) " }{XPPEDIT 517 0 "sqrt(3)+sqrt(5)+1\n" "6#,(-%%sqrtG6#\"\"$\" \"\"-F%6#\"\"&F(\"\"\"F(" }{TEXT 518 0 "" }{TEXT 343 5 "\n(b) " } {XPPEDIT 18 0 "sqrt(3)*sqrt(5)+sqrt(3)" "6#,&*&-%%sqrtG6#\"\"$\"\"\"-F &6#\"\"&F)F)-F&6#\"\"$F)" }{TEXT 344 5 "\n(c) " }{XPPEDIT 18 0 "1/(sqr t(3)*sqrt(5)+sqrt(3))" "6#*&\"\"\"\"\"\",&*&-%%sqrtG6#\"\"$F%-F)6#\"\" &F%F%-F)6#\"\"$F%!\"\"" }{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 362 1 " " }{TEXT 523 2 "6." }{TEXT 524 36 " Find a polynomial in Z[x] that has " }{XPPEDIT 519 0 "sqrt(7)+1/(15)^(1/3)" "6#,&-%%sqrtG6 #\"\"(\"\"\"*&\"\"\"F()\"#:*&\"\"\"F(\"\"$!\"\"F0F(" }{TEXT 363 11 " a s a root." }{TEXT -1 1 "\n" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" } {TEXT 345 3 "7. " }{TEXT 525 68 "Solve the polynomial system \n \+ " }{XPPEDIT 18 0 "x^2+y^2+y = 1;" "6#/ ,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F(F*F(\"\"\"" }{TEXT -1 2 " " }{TEXT 346 39 "\n " }{XPPEDIT 18 0 "x^2+ 2*y^2-x = 4;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"#F(*$%\"yG\"\"#F(F(F&!\"\" \"\"%" }{TEXT -1 1 "\n" }{TEXT 347 17 "using resultants." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 359 1 " " }{TEXT 526 2 "8." }{TEXT 527 108 " Con struct examples of different polynomials P and Q (in Q[x]) such th at \n " }{XPPEDIT 18 0 "Res(P,x,x) = Res( Q,x,x)" "6#/-%$ResG6%%\"PG%\"xGF(-F%6%%\"QGF(F(" }{TEXT 360 62 "\ni.e. even if resultants agree, the polynomials need not agree" }{TEXT -1 2 ".\n" }}}{SECT 1 {PARA 285 "" 0 "" {TEXT -1 1 " " }{TEXT 528 2 "9." }{TEXT -1 97 " Construct an example of a polynomial P in Q[x] of degr ee at least 2 and with the property that " }{XPPEDIT 18 0 "Res(P(x),x, x)=Res(P(x),x^2,x)" "6#/-%$ResG6%-%\"PG6#%\"xGF*F*-F%6%-F(6#F**$F*\"\" #F*" }{TEXT -1 2 ".\n" }}}{SECT 1 {PARA 286 "" 0 "" {TEXT -1 1 " " } {TEXT 529 3 "10." }{TEXT -1 55 " For which values of the parameter k \+ the polynomial \n" }{XPPEDIT 18 0 "P :=x^5+3*x^4+x^4*k+3*x^3*k-2*x^3* k^2-6*x^2*k^2-x^3-3*x^2-x^2*k-3*x*k+2*k^2*x+6*k^2; " "6#>%\"PG,:*$%\"x G\"\"&\"\"\"*&\"\"$F)*$F'\"\"%F)F)*&F'\"\"%%\"kGF)F)*(\"\"$F)*$F'\"\"$ F)F0F)F)*(\"\"#F)*$F'\"\"$F)F0\"\"#!\"\"*(\"\"'F)*$F'\"\"#F)F0\"\"#F:* $F'\"\"$F:*&\"\"$F)*$F'\"\"#F)F:*&F'\"\"#F0F)F:*(\"\"$F)F'F)F0F)F:*(\" \"#F)*$F0\"\"#F)F'F)F)*&\"\"'F)*$F0\"\"#F)F)" }{TEXT -1 21 "\nhas mult iple roots.\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 " 18. Systems o f polynomial equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 348 1 " " }{TEXT 532 1 "1" }{TEXT 533 167 ". Sol ve the following system of polynomial equations Find three univariate \+ polynomials of degree 8 which define candidates for solutions for x, \+ y and z, respectively." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "-76*x^2-65*x*y+25*x*z+28*x-61*y^2-60*y*z+9*y+29*z^ 2-66*z-32=0" "6#/,6*&\"#w\"\"\"*$%\"xG\"\"#F'!\"\"*(\"#lF'F)F'%\"yGF'F +*(\"#DF'F)F'%\"zGF'F'*&\"#GF'F)F'F'*&\"#hF'*$F.\"\"#F'F+*(\"#gF'F.F'F 1F'F+*&\"\"*F'F.F'F'*&\"#HF'*$F1\"\"#F'F'*&\"#mF'F1F'F+\"#KF+\"\"!" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 " 78*x^2+39*x*y+94*x*z+68*x-17*y^2-98*y* z-36*y+40*z^2+22*z+5=0" "6#/,6*&\"#y\"\"\"*$%\"xG\"\"#F'F'*(\"#RF'F)F' %\"yGF'F'*(\"#%*F'F)F'%\"zGF'F'*&\"#oF'F)F'F'*&\"# " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 349 3 " 2." }{TEXT 534 39 " Find the \+ singular points of the curve\n" }{XPPEDIT 530 0 "y^3-9*y^2+32*y+83-x^3 +15*x^2-72*x-x*y" "6#,2*$%\"yG\"\"$\"\"\"*&\"\"*F'*$F%\"\"#F'!\"\"*&\" #KF'F%F'F'\"#$)F'*$%\"xG\"\"$F,*&\"#:F'*$F1\"\"#F'F'*&\"#sF'F1F'F,*&F1 F'F%F'F," }{TEXT 531 0 "" }{TEXT 350 5 " =0.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 " 19. Newton iteration" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 3 "" 0 "" {TEXT 351 1 " " }{TEXT 535 2 "1." }{TEXT 536 29 " Write a Maple procedure that" }}{PARA 0 "" 0 "" {TEXT -1 40 "appr oximates a solution of the equation " }{XPPEDIT 18 0 "sin(x)=5/7" "6#/ -%$sinG6#%\"xG*&\"\"&\"\"\"\"\"(!\"\"" }{TEXT -1 66 " by performing 10 loops of the Newton iteration for the function " }{XPPEDIT 18 0 "sin (x)-5/7" "6#,&-%$sinG6#%\"xG\"\"\"*&\"\"&F(\"\"(!\"\"F," }{TEXT -1 2 " . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 352 1 " " }{TEXT 537 3 "2. " } {TEXT 538 39 " Show that the Newton iteration for the" }}{PARA 0 "" 0 "" {TEXT -1 12 "polynomial " }{XPPEDIT 18 0 "x^3/2-x+1" "6#,(*&%\"xG \"\"$\"\"#!\"\"\"\"\"F%F(\"\"\"F)" }{TEXT -1 52 " does not converge if the initial values are either " }{XPPEDIT 18 0 "x[0]=0" "6#/&%\"xG6# \"\"!F'" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x[0]=1" "6#/&%\"xG6#\"\"! \"\"\"" }{TEXT -1 1 "." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 361 1 " " } {TEXT 539 3 "3. " }{TEXT 540 93 "Use 10 loops of the Newton iteration \+ to approximate the positive solution \nfor the equation " }{XPPEDIT 541 0 "exp(x)=2*cos(x)" "6#/-%$expG6#%\"xG*&\"\"#\"\"\"-%$cosG6#F'F*" }{TEXT 542 1 "." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 " 20. Squaref ree polynomials" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 398 1 " " }{TEXT 543 2 "1." }{TEXT 544 76 " Compute the first and the second deflation of t he polynomial P given below " }{XPPEDIT 549 0 "P := -32+208*x-592*x^2+ 968*x^3-1002*x^4+681*x^5+x^9-14*x^8+86*x^7-304*x^6;\n" "6#>%\"PG,6\"#K !\"\"*&\"$3#\"\"\"%\"xGF*F**&\"$#fF**$F+\"\"#F*F'*&\"$o*F**$F+\"\"$F*F **&\"%-5F**$F+\"\"%F*F'*&\"$\"oF**$F+\"\"&F*F**$F+\"\"*F**&\"#9F**$F+ \"\")F*F'*&\"#')F**$F+\"\"(F*F**&\"$/$F**$F+\"\"'F*F'" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 364 1 " " }{TEXT 545 2 "2." }{TEXT 546 57 " Show that a polynomial P is squarefree if and only if " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " " } {XPPEDIT 18 0 "resultant(P(x),diff(P(x),x),x)<>0" "6#0-%*resultantG6%- %\"PG6#%\"xG-%%diffG6$-F(6#F*F*F*\"\"!" }{TEXT -1 1 "." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 365 1 " " }{TEXT 547 2 "3." }{TEXT 548 19 " Muss er's algorithm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Implement Musser's algorithm for squarefree factorization ." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 " 21. Automatic differentia tion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 5 " 1. " }{TEXT 366 71 "Let f, g, and h be the following multivari ate real functions\n \n" }{TEXT 554 17 " " } {XPPEDIT 550 0 "f(x,y):=ln(1+x^4+y^4)/sqrt(x^2+y^2)" "6#>-%\"fG6$%\"xG %\"yG*&-%#lnG6#,(\"\"\"\"\"\"*$F'\"\"%F/*$F(\"\"%F/F/-%%sqrtG6#,&*$F' \"\"#F/*$F(\"\"#F/!\"\"" }}{PARA 4 "" 0 "" {TEXT 367 12 " \+ " }{XPPEDIT 551 0 "g(x,y,z):=1/sqrt((x-a)^2+(y-b)^2+(z-c)^2)\n" "6#>-% \"gG6%%\"xG%\"yG%\"zG*&\"\"\"\"\"\"-%%sqrtG6#,(*$,&F'F,%\"aG!\"\"\"\"# F,*$,&F(F,%\"bGF4\"\"#F,*$,&F)F,%\"cGF4\"\"#F,F4" }{TEXT 368 14 "\n \+ " }{XPPEDIT 552 0 " h(x,y,z):=z/(x^2+y^2+z^2)\n" "6#>-%\"h G6%%\"xG%\"yG%\"zG*&F)\"\"\",(*$F'\"\"#F+*$F(\"\"#F+*$F)\"\"#F+!\"\"" }{TEXT 369 1 "\n" }{TEXT 555 333 "(a) Determine all partial derivative s of f of order 2.\n\n(b) Check that g is a solution of the Laplace di ffeential equation, i.e., \n (diff(``, x,x)+diff (``, y,y)+diff(``,z,z))*g=0.\n\n(c) Check that h is a solution of the \+ differenial equation\n diff(h,y,x)+(4*x/(x^ 2+y^2+z^2)*diff(h,y))=0" }{TEXT 553 1 "." }{TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 370 1 " " }{TEXT 556 1 "2" }{TEXT 557 54 ". Comp are the result of the following Maple commands:\n" }{TEXT 0 64 " > d iff( f(x), x );\n > convert( \", D);\n > unapply( \", x );" }}} {SECT 0 {PARA 277 "" 0 "" {TEXT -1 1 " " }{TEXT 558 2 "3." }{TEXT -1 40 " Compute the derivative of the function " }{XPPEDIT 18 0 "f(x) := \+ max(x^3,x)" "6#>-%\"fG6#%\"xG-%$maxG6$*$F'\"\"$F'" }{TEXT -1 1 "." }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 4 " 4. " }{TEXT 371 47 "Let the funct ion y(x) be implicitly defined by " }{XPPEDIT 18 0 "sqrt(x)+sqrt(y)=1 " "6#/,&-%%sqrtG6#%\"xG\"\"\"-F&6#%\"yGF)\"\"\"" }{TEXT 372 25 ". Comp ute the derivative " }{XPPEDIT 18 0 "`y'`" "6#%#y'G" }{TEXT 373 26 " a nd the second derivative" }{XPPEDIT 18 0 "y'';" "6#%#%?G" }}}{SECT 0 {PARA 278 "" 0 "" {TEXT -1 1 " " }{TEXT 559 3 "5. " }{TEXT -1 27 "Let \+ the bivariate function " }{XPPEDIT 18 0 "z(x,y)" "6#-%\"zG6$%\"xG%\"yG " }{TEXT -1 26 " be implicitly defined by " }{XPPEDIT 18 0 "h(x,y,z)=0 " "6#/-%\"hG6%%\"xG%\"yG%\"zG\"\"!" }{TEXT -1 57 ", for some trivariat e function h. Determine formulae for " }{XPPEDIT 18 0 "diff(z,x)" "6#- %%diffG6$%\"zG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "diff(z,y,x)" "6#-%%diffG6%%\"zG%\"yG%\"xG" }{TEXT -1 26 ". What are the result for \+ " }{XPPEDIT 18 0 "h=sqrt(x)+sqrt(y)+sqrt(z)-1" "6#/%\"hG,*-%%sqrtG6#% \"xG\"\"\"-F'6#%\"yGF*-F'6#%\"zGF*\"\"\"!\"\"" }{TEXT -1 2 "?\n" }}} {SECT 1 {PARA 279 "" 0 "" {TEXT -1 1 " " }{TEXT 560 2 "6." }{TEXT -1 23 " Consider the function " }{XPPEDIT 18 0 "f[n]" "6#&%\"fG6#%\"nG" } {TEXT -1 27 " recursively defined by \n " }{XPPEDIT 18 0 "f[0]=0, f[1 ]=x, f[n]=f[n-1]+sin(f[n-2]) " "6%/&%\"fG6#\"\"!F'/&F%6#\"\"\"%\"xG/&F %6#%\"nG,&&F%6#,&F0\"\"\"\"\"\"!\"\"F5-%$sinG6#&F%6#,&F0F5\"\"#F7F5" } {TEXT -1 4 "for " }{XPPEDIT 18 0 "n*`>`*1" "6#*(%\"nG\"\"\"%\">GF%\"\" \"F%" }{TEXT -1 91 ". \nDetermine (by automatic differentation) a proc edure to compute the first derivative of " }{XPPEDIT 18 0 "f[n]" "6#& %\"fG6#%\"nG" }{TEXT -1 2 ".\n" }}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 1 " " }{TEXT 561 2 "7." }{TEXT -1 22 " Define the functions " } {XPPEDIT 18 0 "f[n]" "6#&%\"fG6#%\"nG" }{TEXT -1 25 ", n = 1,2.., by \+ setting " }{XPPEDIT 18 0 "f[1]=x" "6#/&%\"fG6#\"\"\"%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f[2](x)= sin(x)" "6#/-&%\"fG6#\"\"#6#%\"xG-%$si nG6#F*" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "f[n](x) = sin(f[n-1](x) + f[n-2](x))" "6#/-&%\"fG6#%\"nG6#%\"xG-%$sinG6#,&-&F&6#,&F(\"\"\"\"\" \"!\"\"6#F*F3-&F&6#,&F(F3\"\"#F56#F*F3" }{TEXT -1 59 " Write a proced ure that computes values of the functions " }{XPPEDIT 18 0 "f" "6#%\" fG" }{TEXT -1 14 ", and compute " }{XPPEDIT 18 0 "D(f[10])(0)" "6#--% \"DG6#&%\"fG6#\"#56#\"\"!" }{TEXT -1 2 ".\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 " 22. Integration" }}{SECT 0 {PARA 280 "" 0 "" {TEXT -1 1 " " }{TEXT 563 2 "1." }{TEXT -1 113 " Compute the following indef inite integrals and check the answers by differentiation and simplific ation.\n\n(a) " }{XPPEDIT 18 0 "int(sqrt(exp(x)-1),x) \n" "6#-%$int G6$-%%sqrtG6#,&-%$expG6#%\"xG\"\"\"\"\"\"!\"\"F-" }{TEXT -1 6 "\n(b) \+ " }{XPPEDIT 18 0 " Int(x/(2*a*x-x^2)^(3/2),x )\n" "6#-%$IntG6$*&%\"xG \"\"\"),&*(\"\"#F(%\"aGF(F'F(F(*$F'\"\"#!\"\"*&\"\"$F(\"\"#F0F0F'" } {TEXT -1 7 "\n(c) " }{XPPEDIT 18 0 "Int(sqrt(x^2-a^2),x) " "6#-%$Int G6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"aG\"\"#!\"\"F+" }}}{SECT 1 {PARA 281 "" 0 "" {TEXT -1 1 " " }{TEXT 564 2 "2." }{TEXT -1 9 " Compu te " }{XPPEDIT 18 0 "Int(x^n*exp(x), x)" "6#-%$IntG6$*&)%\"xG%\"nG\"\" \"-%$expG6#F(F*F(" }{TEXT -1 21 " for general integer " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 44 " and check the result for distinct value of " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 2 ".\n" }}}{SECT 0 {PARA 282 "" 0 "" {TEXT -1 1 " " }{TEXT 565 2 "3." }{TEXT -1 9 " Compute " } {XPPEDIT 18 0 "Int(ln(x^2+1)/(x^2+1),x);\n" "6#-%$IntG6$*&-%#lnG6#,&*$ %\"xG\"\"#\"\"\"\"\"\"F.F.,&*$F,\"\"#F.\"\"\"F.!\"\"F," }{TEXT -1 75 " Hint. Use the changevar command of the student package, \nand substi tute " }{XPPEDIT 18 0 "x = tan(y)" "6#/%\"xG-%$tanG6#%\"yG" }{TEXT -1 2 ".\n" }}}{SECT 0 {PARA 283 "" 0 "" {TEXT -1 1 " " }{TEXT 566 2 "4 ." }{TEXT -1 1 " " }{TEXT 393 3 "(a)" }{TEXT -1 9 " Compute " } {XPPEDIT 18 0 "int(int((x-y)/(x+y)^3,y=0..1),x=0..1)" "6#-%$intG6$-F$6 $*&,&%\"xG\"\"\"%\"yG!\"\"F+*$,&F*F+F,F+\"\"$F-/F,;\"\"!\"\"\"/F*;F3\" \"\"" }{TEXT -1 5 "\n\n " }{TEXT 394 3 "(b)" }{TEXT -1 9 " Compute \+ " }{XPPEDIT 18 0 "int(int((x-y)/(x+y)^3,x=0..1),y=0..1)" "6#-%$intG6$- F$6$*&,&%\"xG\"\"\"%\"yG!\"\"F+*$,&F*F+F,F+\"\"$F-/F*;\"\"!\"\"\"/F,;F 3\"\"\"" }{TEXT -1 5 "\n\n " }{TEXT 395 3 "(c)" }{TEXT -1 100 " Comp are the results of (a) and (b). Does Maple make a mistake or is there \+ something else going on?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 567 2 "5." }{TEXT -1 44 " Com pute the following definite integrals.\n\n" }{TEXT 374 3 "(a)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int((4*x^4+4*x^3-2*x^2-10*x+6)/(x^5+7*x^4+1 6*x^3+10*x^2),x=1..10)" "6#-%$IntG6$*&,,*&\"\"%\"\"\"*$%\"xG\"\"%F*F** &\"\"%F**$F,\"\"$F*F**&\"\"#F**$F,\"\"#F*!\"\"*&\"#5F*F,F*F6\"\"'F*F*, **$F,\"\"&F**&\"\"(F**$F,\"\"%F*F**&\"#;F**$F,\"\"$F*F**&\"#5F**$F,\" \"#F*F*F6/F,;\"\"\"\"#5" }{TEXT -1 2 "\n\n" }{TEXT 375 3 "(b)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(x^4*sin(x)*cos(x),x=0..Pi/2)" "6#-%$Int G6$*(%\"xG\"\"%-%$sinG6#F'\"\"\"-%$cosG6#F'F,/F';\"\"!*&%#PiGF,\"\"#! \"\"" }{TEXT -1 2 "\n\n" }{TEXT 376 3 "(c)" }{TEXT -1 3 " " } {XPPEDIT 18 0 "Int(1/(x*sqrt(5*x^2-6*x+1)),x=1/7..1/5)" "6#-%$IntG6$*& \"\"\"\"\"\"*&%\"xGF(-%%sqrtG6#,(*&\"\"&F(*$F*\"\"#F(F(*&\"\"'F(F*F(! \"\"\"\"\"F(F(F5/F*;*&\"\"\"F(\"\"(F5*&\"\"\"F(\"\"&F5" }{TEXT -1 2 " \n\n" }{TEXT 377 3 "(d)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/x,x=- 2..-1)" "6#-%$IntG6$*&\"\"\"\"\"\"%\"xG!\"\"/F);,$\"\"#F*,$\"\"\"F*" } {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 568 1 "6" }{TEXT -1 45 ". Compute the following definite integrals.\n \n" }{TEXT 378 3 "(a)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/sqrt(1- x^2), x=0..1)" "6#-%$IntG6$*&\"\"\"\"\"\"-%%sqrtG6#,&\"\"\"F(*$%\"xG\" \"#!\"\"F1/F/;\"\"!\"\"\"" }{TEXT -1 2 "\n\n" }{TEXT 379 3 "(b)" } {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(x*arctan(x),x=0..1)" "6#-%$IntG6$ *&%\"xG\"\"\"-%'arctanG6#F'F(/F';\"\"!\"\"\"" }{TEXT -1 2 "\n\n" } {TEXT 380 3 "(c)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(exp(-a*x)*cos( b*x)^2,x=0..infinity)" "6#-%$IntG6$*&-%$expG6#,$*&%\"aG\"\"\"%\"xGF-! \"\"F-*$-%$cosG6#*&%\"bGF-F.F-\"\"#F-/F.;\"\"!%)infinityG" }{TEXT -1 24 ", for a positive number " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 3 ".\n\n" }{TEXT 381 3 "(d)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sin (x)/x,x=0..infinity)" "6#-%$IntG6$*&-%$sinG6#%\"xG\"\"\"F*!\"\"/F*;\" \"!%)infinityG" }{TEXT -1 2 "\n\n" }{TEXT 382 3 "(e)" }{TEXT -1 3 " \+ " }{XPPEDIT 18 0 "Int(exp(-x)*ln(x),x=0..infinity)" "6#-%$IntG6$*&-%$e xpG6#,$%\"xG!\"\"\"\"\"-%#lnG6#F+F-/F+;\"\"!%)infinityG" }{TEXT -1 2 " \n\n" }{TEXT 383 3 "(f)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(exp(-a* x)*ln(x)/sqrt(x), x=0..infinity)" "6#-%$IntG6$*(-%$expG6#,$*&%\"aG\"\" \"%\"xGF-!\"\"F--%#lnG6#F.F--%%sqrtG6#F.F//F.;\"\"!%)infinityG" } {TEXT -1 29 ", for a positive real number " }{XPPEDIT 18 0 "a" "6#%\"a G" }{TEXT -1 3 ".\n\n" }{TEXT 384 3 "(g)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(exp(-sqrt(t))/(t^(1/4)*(1-exp(-sqrt(t)))), t=0..infinity)" " 6#-%$IntG6$*&-%$expG6#,$-%%sqrtG6#%\"tG!\"\"\"\"\"*&)F.*&\"\"\"F0\"\"% F/F0,&\"\"\"F0-F(6#,$-F,6#F.F/F/F0F//F.;\"\"!%)infinityG" }{TEXT -1 2 "\n\n" }{TEXT 385 3 "(h)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/sqrt (x^4-1),x=1..infinity)" "6#-%$IntG6$*&\"\"\"\"\"\"-%%sqrtG6#,&*$%\"xG \"\"%F(\"\"\"!\"\"F1/F.;\"\"\"%)infinityG" }{TEXT -1 2 "\n\n" }{TEXT 386 3 "(i)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sqrt(cos(x)), x=0..P i/2)" "6#-%$IntG6$-%%sqrtG6#-%$cosG6#%\"xG/F,;\"\"!*&%#PiG\"\"\"\"\"#! \"\"" }{TEXT -1 2 "\n\n" }{TEXT 387 3 "(j)" }{TEXT -1 3 " " } {XPPEDIT 18 0 "Int(1/sqrt(x^4+4*x^2+3), x=1..3)" "6#-%$IntG6$*&\"\"\" \"\"\"-%%sqrtG6#,(*$%\"xG\"\"%F(*&\"\"%F(*$F.\"\"#F(F(\"\"$F(!\"\"/F.; \"\"\"\"\"$" }{TEXT -1 2 "\n\n" }{TEXT 388 3 "(k)" }{TEXT -1 3 " " } {XPPEDIT 18 0 "Int(sqrt(tan(x)),x=0..Pi/2)" "6#-%$IntG6$-%%sqrtG6#-%$t anG6#%\"xG/F,;\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 "\n\n" }{TEXT 389 3 "(l)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sin(a*x)^2*sin(b*x)/ x,x=0..infinity)" "6#-%$IntG6$*(-%$sinG6#*&%\"aG\"\"\"%\"xGF,\"\"#-F(6 #*&%\"bGF,F-F,F,F-!\"\"/F-;\"\"!%)infinityG" }{TEXT -1 2 "\n\n" } {TEXT 390 3 "(m)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/cosh(a*x),x= 0..infinity)" "6#-%$IntG6$*&\"\"\"\"\"\"-%%coshG6#*&%\"aGF(%\"xGF(!\" \"/F.;\"\"!%)infinityG" }{TEXT -1 29 ", for a positive real number " } {XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 2 " " } {TEXT 569 2 "7." }{TEXT -1 5 " Let " }{XPPEDIT 18 0 "F" "6#%\"FG" } {TEXT -1 28 " be the function defined by " }{XPPEDIT 18 0 "F(T):=Int(e xp(-u^2*T)/u,u=1..T)" "6#>-%\"FG6#%\"TG-%$IntG6$*&-%$expG6#,$*&%\"uG\" \"#F'\"\"\"!\"\"F3F1F4/F1;\"\"\"F'" }{TEXT -1 3 ".\n\n" }{TEXT 391 3 " (a)" }{TEXT -1 41 " Define the corresponding Maple function " } {XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 42 " and determine a numeric app roximation of " }{XPPEDIT 18 0 "F(2)" "6#-%\"FG6#\"\"#" }{TEXT -1 3 ". \n\n" }{TEXT 392 3 "(b)" }{TEXT -1 25 " Compute the derivative " } {XPPEDIT 18 0 "`F'`" "6#%#F'G" }{TEXT -1 5 " (by " }{TEXT 0 1 "D" } {TEXT -1 14 ") and compute " }{XPPEDIT 18 0 "`F'`(2)" "6#-%#F'G6#\"\"# " }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT 396 1 " " }{TEXT 570 2 "8." }{TEXT 571 23 " Compute the integral " }{XPPEDIT 562 0 "int(x^(1/3),x=-1..2)" "6 #-%$intG6$)%\"xG*&\"\"\"\"\"\"\"\"$!\"\"/F';,$\"\"\"F,\"\"#" }{TEXT 397 2 ".\n" }}}}}{MARK "0 0 1" 12 }{VIEWOPTS 1 1 0 1 1 1803 }