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\"+D>0ziF_^l$\"+%Gn-)**F17$$!+m>0ziF_^lFgdm7$$!+YJ\"Q(=F1$\"+2D(G#)*F1 7$$!+U*p,4$F1$\"+j^c5&*F17$$!+8HzdUF1$\"+E0F[!*F17$$!+YzEe`F1$\"+d#zKW )F17$$!+$*)RUP'F1$\"+JC80xF17$$!+$G'o*G(F1$\"+]5ZXoF17$$!+]*p,4)F1$\"+ 9D&y(eF17$$!+0o1j()F1$\"+Kn`<[F17$$!+i[w(H*F1$\"+>bC\"o$F17$$!+8;$eo*F 1$\"+l))*o[#F17$$!+9q9@**F1$\"+IBL`7F17$$F-F]y$!+:w1-TFcal7$$!+8q9@**F 1$!+QBL`7F17$$!+6;$eo*F1$!+t))*o[#F17$$!+f[w(H*F1$!+FbC\"o$F17$$!+,o1j ()F1$!+Sn`<[F17$$!+Y*p,4)F1$!+?D&y(eF17$$!+xio*G(F1$!+c5ZXoF17$$!+())R UP'F1$!+PC80xF17$$!+RzEe`F1$!+i#zKW)F17$$!+1HzdUF1$!+H0F[!*F17$$!+M*p, 4$F1$!+m^c5&*F17$$!+QJ\"Q(=F1$!+4D(G#)*F17$$!+%)=0ziF_^l$!+&Gn-)**F17$ $\"+2?0ziF_^l$!+%Gn-)**F17$$\"+]J\"Q(=F1$!+2D(G#)*F17$$\"+Y*p,4$F1$!+i ^c5&*F17$$\"+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Exact numer ical calculations: " }{TEXT 259 16 "*,^,ifactor,sqrt" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 15 "Maple \+ produces " }{TEXT 266 12 "exact values" }{TEXT -1 32 ", where possible or appropriate." }}{PARA 15 "" 0 "" {TEXT -1 58 "The multiplication o peration is represented by the symbol " }{TEXT 259 1 "*" }{TEXT -1 1 " ," }{TEXT 259 1 " " }{TEXT -1 31 "and must be entered explicitly." }} {PARA 15 "" 0 "" {TEXT -1 13 "A semi-colon " }{TEXT 259 1 ";" }{TEXT -1 73 " is used to indicate the end of the input associated with a com mand line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "(-3*4 + 5/2)/7;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 48 "The power operation is designated by the symbol " }{TEXT 259 1 "^" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "5^2;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "If you type something \+ which Maple does not understand, you will receive an error message." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "5^-1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Parentheses are needed around the -1." }}{PARA 0 "" 0 "" {TEXT -1 71 "Place the cursor on the following command line, and then press \+ [Enter]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "5^(-1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 42 "Very large numbers numbers can be handled." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "2^1000;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Maple has many built-in" }{TEXT 258 1 " " }{TEXT 266 10 "proced ures" }{TEXT -1 4 " or " }{TEXT 266 8 "routines" }{TEXT -1 58 ", which will perform mathematical operations very quickly." }}{PARA 0 "" 0 " " {TEXT -1 56 "The name of the procedure usually suggests what it does ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "For \+ example, it would require a huge amount of work to find the integer fa ctors of the number" }}{PARA 0 "" 0 "" {TEXT -1 63 " 2398232343243249 984321312321 by hand with pencil and paper." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Maple can find the factor s in a second or two with the procedure " }{TEXT 0 7 "ifactor" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ifactor(2398232343243249984321312321);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "If you don't trus t Maple's answer, you can check this by multiplication (assuming that \+ you trust Maple's multiplication)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "3 * 733 * 72140687161773179 \+ * 15117701;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Maple can calculate square roots using the procedure " }{TEXT 0 4 "sqrt" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "273*273;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sqrt(74529);" }}} {PARA 0 "" 0 "" {TEXT -1 63 "\nMaple will give simplified \"exact\" an swers involving radicals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sqrt(74529*8);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Try this" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " }{TEXT 0 7 "ifactor" }{TEXT -1 33 " to find the factors of 32936117." }}{PARA 0 "" 0 "" {TEXT -1 74 " By squaring each of the factors show that one of the factors is less t han " }{XPPEDIT 18 0 "sqrt(32936117)" "6#-%%sqrtG6#\") " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "ifactor(32936117);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%!G6#\"%Pd\"\"\"-F%6#\"%TdF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "5737^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\")pJ\"H$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "5741^2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\")\"3fH$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "57 37^2<32936117" "6#2*$\"%Pd\"\"#\") \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 317 10 "EXPERIMENT" }{TEXT 266 1 " " }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Floating point arithmetic with Maple and assignment: " }{TEXT 259 15 "evalf,:=,Digits" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Maple can work with decimals." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "1.234567890/2 + 0.3333333333*3;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Fractions can be converted to decimal a pproximations by using " }{TEXT 259 5 "evalf" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ev alf(2/3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "evalf" }{TEXT -1 15 " is short for \"" }{TEXT 267 35 "evaluate \+ as a floating point number" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The meaning of the term " } {TEXT 266 21 "floating point number" }{TEXT -1 161 " is connected with the way that the numbers are stored internally on the computer.\nA fl oating point number is similar to a number in scientific notation such as:" }}{PARA 0 "" 0 "" {TEXT -1 21 " 0.1234567891 * " }{XPPEDIT 18 0 "10^(-3);" "6#)\"#5,$\"\"$!\"\"" }{TEXT -1 39 ", where the * indi cates multiplication." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Maple stores this number in two parts:" }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 266 8 "mantissa" }{TEXT -1 102 ": 0.123 4567891\nThis is a decimal having its first non-zero digit immediately after the decimal point." }}{PARA 15 "" 0 "" {TEXT -1 4 "The " } {TEXT 266 8 "exponent" }{TEXT -1 57 ": -3\nThis is a positive or nega tive whole number (or 0)." }}{PARA 0 "" 0 "" {TEXT -1 75 "Large and sm all decimal numbers are printed out automatically in this form." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(2^100);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "evalf(2^(-100));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Maple can " }{TEXT 266 22 "stor e numerical values" }{TEXT -1 19 " by the process of " }{TEXT 266 10 " assignment" }{TEXT -1 18 " using the symbol " }{TEXT 0 2 ":=" }{TEXT -1 57 " (which must be typed with no space between the : and =)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can as sign the previous two numerical values to symbols or " }{TEXT 266 9 "v ariables" }{TEXT -1 1 " " }{TEXT 267 1 "a" }{TEXT -1 5 " and " }{TEXT 267 1 "b" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "a := evalf(2^100);\nb := e valf(2^(-100));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "You can move to a new line by typing [Shift-Return]." }} {PARA 0 "" 0 "" {TEXT -1 73 "As you can see, Maple performs these two \+ storing operations sequentially." }}{PARA 0 "" 0 "" {TEXT -1 34 "These values are now \"remembered\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a;\nb;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Notice that if you multip ly these numbers together, you don't get exactly 1." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a*b;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "This has \+ happened because the floating point numbers held by " }{TEXT 367 1 "a " }{TEXT -1 5 " and " }{TEXT 369 1 "b" }{TEXT -1 53 " are only approxi mate representations of the numbers " }{XPPEDIT 18 0 "2^100;" "6#*$\" \"#\"$+\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2^(-100) = 1/(2^100);" "6#/)\"\"#,$\"$+\"!\"\"*&\"\"\"F**$F%F'F(" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "If you mix fracti ons and decimals, the fractions are converted to decimals, and the ans wer is given as a decimal.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "1/3+0.5;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "evalf" }{TEXT -1 65 " can be used to obtain a decimal approxima tion for a square root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sqrt(40);\nevalf(%);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Here the symbol " }{TEXT 0 1 "%" }{TEXT -1 57 " has beeen used to refer to Maple's last \+ computed result." }}{PARA 0 "" 0 "" {TEXT -1 15 "It is like the " } {TEXT 258 3 "ans" }{TEXT -1 30 " function on some calculators." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "You must \+ be careful when using the last answer symbol " }{TEXT 0 1 "%" }{TEXT -1 10 ", because " }{TEXT 0 1 "%" }{TEXT -1 39 " refers to the last an swer in terms of " }{TEXT 266 4 "time" }{TEXT -1 173 " and not in term s of position on the worksheet.\nIt is probably good practice to only \+ use this symbol when commands have been typed sequentially following a single prompt [>." }}{PARA 0 "" 0 "" {TEXT -1 76 "The two sequential \+ commands above are equivalent to the single command . . ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf (sqrt(40));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 ". . except that you don't see the intermediate answer." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "Another \+ way to obtain an approximate decimal value for the square root is to f orce Maple to give a decimal output by making the input a decimal with a decimal point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sqrt(40.);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Unless you specify otherwise, such resu lts are given to 10 decimal significant figures." }}{PARA 0 "" 0 "" {TEXT -1 99 "You can overide this for a single calculation by adding t he number of figures required as a second " }{TEXT 266 8 "argument" } {TEXT -1 4 " or " }{TEXT 266 9 "parameter" }{TEXT -1 25 " for the Mapl e procedure " }{TEXT 259 5 "evalf" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(sqrt(4 0),20);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The following command gives the same result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf[20](sq rt(40));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The number of digits given in answers such as this is controlled b y the " }{TEXT 266 20 "environment variable" }{TEXT -1 1 " " }{TEXT 0 6 "Digits" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 0 6 "Digits" } {TEXT -1 82 " has a default value of 10, which is the value when Maple first starts or after a " }{TEXT 0 7 "restart" }{TEXT -1 9 " command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "If yo u are a programmer, you will know what is meant by a " }{TEXT 266 15 " global variable" }{TEXT -1 222 ". Maple environment variables behave l ike global variables except that if an assignment is made to an enviro nment variable in the body of a procedure, the assignment will automat ically be undone on exit from the procedure." }}{PARA 0 "" 0 "" {TEXT -1 41 "For more information click the hyperlink " }{HYPERLNK 17 "envva r" 2 "envvar" "" }{TEXT -1 57 " to obtain the help page concerning env ironment variables" }{HYPERLNK 17 "" 2 "envar" "" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "If you w ant to perform a sequence of computations using a different number of \+ digits than 10, you can set " }{TEXT 0 6 "Digits" }{TEXT -1 15 " appro priately." }}{PARA 0 "" 0 "" {TEXT -1 29 "It is good practice to reset " }{TEXT 0 6 "Digits" }{TEXT -1 45 " to the default value when you ha ve finished." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Digits := 15;\na := evalf(1/3);\nb := evalf(sqrt(2 ));\nc := a + b;\nDigits := 10;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "The variable " }{TEXT 267 1 "c" }{TEXT -1 56 " retains the full precision of 15 digits when recalled.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "c;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The assigned value can be remov ed by typing . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "c := 'c':\nc;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The number " }{XPPEDIT 18 0 "Pi;" "6 #%#PiG" }{TEXT -1 40 " is represented in Maple by the symbol " } {TEXT 259 2 "Pi" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(Pi,50);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sin(P i/6);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 2 "Pi" }{TEXT -1 14 " must have an " }{TEXT 266 12 "upper case P" } {TEXT -1 49 ". If you type \"pi\" Maple still prints the symbol " } {XPPEDIT 18 0 "pi" "6#%#piG" }{TEXT -1 62 ", but treats it as a variab le to which you can assign a value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "pi := 22/7;\nPi;\npi;\neval f(Pi-pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Try this" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " } {TEXT 0 7 "ifactor" }{TEXT -1 35 " to find the factors of 1276632899. " }}{PARA 0 "" 0 "" {TEXT -1 39 "By finding a decimal approximation fo r " }{XPPEDIT 18 0 "sqrt(1276632899)" "6#-%%sqrtG6#\"+**Gjw7" }{TEXT -1 43 " show that one of the factors is less than " }{XPPEDIT 18 0 "sq rt(1276632899)" "6#-%%sqrtG6#\"+**Gjw7" }{TEXT -1 37 ", and that the o ther is greater than " }{XPPEDIT 18 0 "sqrt(1276632899)" "6#-%%sqrtG6# \"+**Gjw7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "Which of th e factors is closer to " }{XPPEDIT 18 0 "sqrt(1276632899)" "6#-%%sqrtG 6#\"+**Gjw7" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" } {TEXT -1 96 ": It is probably a good idea to use more than 10 digits i n any numerical (decimal) calculations." }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ifactor(1276632899);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%!G6# \"&Jd$\"\"\"-F%6#\"&Hd$F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a := 35729;\nb := 35731;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"&Hd$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"&Jd$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sqrt(1276632899);\nc := evalf(%,15);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"+**Gjw7#\"\"\"\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"cG$\"0i+')****Hd$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(c-a,15 );\nevalf(b-c,15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+i+')****!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\",Q*R,+5!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The factor 35729 is very \+ slightly closer to " }{XPPEDIT 18 0 "sqrt(1276632899)" "6#-%%sqrtG6#\" +**Gjw7" }{TEXT -1 12 " than 35731." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 10 "EXPERIMENT" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Plo tting graphs: " }{TEXT 0 8 "plot,seq" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 4 "plot" }{TEXT -1 128 " can be used to plot graphs.\nIt is v ery versatile in the sense that it can handle many different forms for the input parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The following command plots the graph of the equatio n " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 6 " fro m " }{XPPEDIT 18 0 "x=-2" "6#/%\"xG,$\"\"#!\"\"" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 91 "The interval of numbers for the x values is typed in the \+ special form shown using two dots " }{TEXT 0 2 ".." }{TEXT -1 49 " and constitutes a Maple data structure called a " }{TEXT 266 5 "range" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(x^2,x=-2..2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "If you want Maple to label the ver tical axis with a " }{TEXT 272 1 "y" }{TEXT -1 12 ", just type:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(x^2,x=-2..2,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 87 "You can restrict or extend the graph in the vertic al direction by typing in a range of " }{TEXT 273 1 "y" }{TEXT -1 8 " \+ values:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(x^2,x=-2..2,y=-1..3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "You can choose the colour used \+ to plot the graph by adding an " }{TEXT 266 6 "option" }{TEXT -1 28 " \+ in the form of an equation " }{TEXT 0 8 "color=??" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }{TEXT 0 9 "colour=??" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(x^2,x=-2..2,y,colour=blue);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The ch oices for the colour option are: " }{TEXT 266 171 "aquamarine, black, \+ blue, coral, cyan, brown, gold, green, grey, khaki, magenta, maroon, n avy, orange, pink, plum, red, sienna, tan, turquoise, violet, wheat, w hite, yellow" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "You can specify your own colour by defining it \+ in terms of red, blue and green (RGB) components." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(x^2,x=- 4..4,color=COLOR(RGB,.7,0,1));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 105 "You can give a name to the colour if you wish,which can then be used elsewhere in the same Maple session." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "purple := COLOR(RGB,.7,0,1);\nplot(x^2,x=-4..4,color=purple);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Alternati vely you can use a " }{TEXT 0 5 "macro" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "macro(d arkgreen=COLOR(RGB,0,0.6,0));\nplot(x^2,x=-4..4,color=darkgreen);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "A number \+ of graphs can be drawn on the same set of axes by constucting a Maple \+ " }{TEXT 266 4 "list" }{TEXT -1 45 " of the expressions for the differ ent graphs." }}{PARA 0 "" 0 "" {TEXT -1 98 "A list is formed by arrang ing the members of the list between square brackets separated by comma s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "[x^2,x+1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot([x^2,x+1],x=-2..2,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "You ca n choose your own colours if you wish by forming a corresponding colou r list. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot([x^2,x+1],x=-2..2,y,color=[blue,green]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "When perf orming a plot using a variable such as " }{TEXT 274 1 "x" }{TEXT -1 6 ", the " }{TEXT 275 1 "x" }{TEXT -1 62 " must have no pre-assigned val ue, otherwise an error results.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x := 5;\nplot(2*x-x^2,x=-1..3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "This can be corrected by \+ removing the assigned value from " }{TEXT 276 1 "x" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "x := 'x':\nplot(2*x-x^2,x=-1..3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 108 "The graphs are drawn with a thicknes s of one pixel unless a thickness greater than 1 is specified with the \"" }{TEXT 267 9 "thickness" }{TEXT -1 9 "\" option." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(2*x- x^2,x=-1..3,thickness=2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The coordinates of a point may be represented in \+ Maple by means of a list with two members." }}{PARA 0 "" 0 "" {TEXT -1 22 "For example, the point" }{XPPEDIT 18 0 "``(1,2)" "6#-%!G6$\"\" \"\"\"#" }{TEXT -1 15 " is written as " }{XPPEDIT 18 0 "[1,2]" "6#7$\" \"\"\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 56 "The following Maple command sets up the list of po ints: " }{XPPEDIT 18 0 "``(-2,2), ``(-1,-2), ``(1,2), ``(2,3)" "6&-%!G 6$,$\"\"#!\"\"F'-F$6$,$\"\"\"F(,$F'F(-F$6$F,F'-F$6$F'\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "mypoints := [[-2,2],[-1,-2],[1,2],[2,3]];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We can plot the se points on a graph by using " }{TEXT 0 4 "plot" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 21 "Since, the variable \"" }{TEXT 267 8 "myp oints" }{TEXT -1 52 "\" now holds the list, it is sufficient to type j ust:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(mypoints,style=point);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The colour choices are th e same as for plotting graphs, and the symbol to be used can chosen fr om: " }{TEXT 259 3 "box" }{TEXT -1 2 ", " }{TEXT 259 5 "cross" }{TEXT -1 2 ", " }{TEXT 259 6 "circle" }{TEXT -1 2 ", " }{TEXT 0 5 "point" } {TEXT -1 6 ", and " }{TEXT 259 7 "diamond" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 31 "You may also type these in the " }{TEXT 266 10 "up per case" }{TEXT -1 7 " form: " }{TEXT 259 30 "BOX,CROSS,CIRCLE,POINT, DIAMOND" }{TEXT -1 50 ", but don't mix lower case and upper case lette rs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot(mypoints,style=point,colour=blue,symbol=circle); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "More information concerning the options for the procedure " }{TEXT 0 4 "pl ot" }{TEXT -1 47 " can be obtained from the associated help page." }} {PARA 0 "" 0 "" {TEXT -1 50 "Three ways to access the help page on opt ions for " }{TEXT 0 4 "plot" }{TEXT -1 6 " are: " }}{PARA 15 "" 0 "" {TEXT -1 21 "Click the hyperlink: " }{HYPERLNK 17 "plot[options]" 2 "p lot[options]" "" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 28 "High light (select) the word " }{TEXT 258 13 "plot[options]" }{TEXT -1 27 " with the mouse and choose " }{TEXT 325 23 "Help on \"plot[options]\" " }{TEXT -1 58 " which should appear in the second line of the help me nu. " }}{PARA 15 "" 0 "" {TEXT -1 5 "Type " }{TEXT 259 1 "?" }{TEXT 0 13 "plot[options]" }{TEXT -1 79 " on a command line (a semi-colon is n ot needed) and then execute this command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "?plot[options]" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The sequ ence of numbers " }{XPPEDIT 18 0 "1,4,9,16,25,` . . . `,100" "6)\"\"\" \"\"%\"\"*\"#;\"#D%(~.~.~.~G\"$+\"" }{TEXT -1 87 " of all the squares \+ of the numbers from 1 to 10 can be constructed using the procedure " } {TEXT 0 3 "seq" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "seq(i^2,i=1..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "This sequence c an be made into a list by putting square brackets around the command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "[seq(i^2,i=1..10)];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The following code plot the points" }{XPPEDIT 18 0 "``(0,1),``(1,3),``(2,5),``(3,7),``(4,9),``(5,11),``(6,13)" "6)-% !G6$\"\"!\"\"\"-F$6$F'\"\"$-F$6$\"\"#\"\"&-F$6$F*\"\"(-F$6$\"\"%\"\"*- F$6$F.\"#6-F$6$\"\"'\"#8" }{TEXT -1 28 ". These points have the form" }{XPPEDIT 18 0 " ``(k,2*k+1)" "6#-%!G6$%\"kG,&*&\"\"#\"\"\"F&F*F*F*F* " }{TEXT -1 21 " where the parameter " }{TEXT 326 1 "k" }{TEXT -1 70 " takes all integer values from 0 to 6 inclusive so we can make use of \+ " }{TEXT 259 3 "seq" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 76 "Th e view option can be used to specify the vertical range used in the pl ot. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "point_list := [seq([k,2*k+1],k=0..6)];\nplot(point_l ist,style=point,symbol=box,color=brown,view=0..13);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Maple can also plot 3 d imensional graphs. The surface can be manipulated with the mouse. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot3d(x^2+y^2,x=-2..2,y=-2..2,axes=boxed);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 11 "polygonplot" }{TEXT -1 63 ", which is used to plot the (filled-in) square with vertices at" }{XPPEDIT 18 0 " ``(0,0),``(0,1),``(1,1) " " 6%-%!G6$\"\"!F&-F$6$F&\"\"\"-F$6$F)F)" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 80 " in the following \+ commands, is not immediately available. It belongs to a Maple " } {TEXT 266 7 "package" }{TEXT -1 15 " with the name " }{TEXT 0 5 "plots " }{TEXT -1 110 " which contains various supplementary plotting proced ures. This package can be loaded by means of the command " }{TEXT 267 12 "with(plots);" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "You can avoid displa ying the list of all the procedures in the " }{TEXT 0 5 "plots" } {TEXT -1 37 " package by replacing the semi-colon " }{TEXT 0 1 ";" } {TEXT -1 16 " in the command " }{TEXT 267 12 "with(plots);" }{TEXT -1 14 " with a colon " }{TEXT 0 1 ":" }{TEXT -1 11 " by typing " }{TEXT 267 12 "with(plots):" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 81 "U sing a colon in place of a semi-colon suppresses the output of a Maple command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "The idea of the following commands is to illustrate some of th e standard plot colours. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "square := [[0,0],[0,1],[1,1],[1,0]] :\npolygonplot(square,axes=none,color=aquamarine);" }}{PARA 13 "" 1 " " {GLPLOT2D 162 149 149 {PLOTDATA 2 "6%-%)POLYGONSG6#7&7$$\"\"!F)F(7$F ($\"\"\"F)7$F+F+7$F+F(-%'COLOURG6&%$RGBG$\")p:#R%!\")$\")`B)e)F5$\")fq kdF5-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 6 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "polygonplot(square,axes=none,color= plum);" }}{PARA 13 "" 1 "" {GLPLOT2D 160 142 142 {PLOTDATA 2 "6%-%)POL YGONSG6#7&7$$\"\"!F)F(7$F($\"\"\"F)7$F+F+7$F+F(-%'COLOURG6&%$RGBG$\")1 Zw\"*!\")$\")PJ%y'F5F3-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 6 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "polygonplot(square ,axes=none,color=turquoise);" }}{PARA 13 "" 1 "" {GLPLOT2D 163 146 146 {PLOTDATA 2 "6%-%)POLYGONSG6#7&7$$\"\"!F)F(7$F($\"\"\"F)7$F+F+7$F+ F(-%'COLOURG6&%$RGBG$\")PJ%y'!\")$\")1Zw\"*F5F6-%*AXESSTYLEG6#%%NONEG " 1 2 0 1 10 0 2 6 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "polygonplot(square,axes=none,color=gold);" }}{PARA 13 "" 1 "" {GLPLOT2D 163 151 151 {PLOTDATA 2 "6%-%)POLYGONSG6#7&7$$\"\"!F)F(7$F($ \"\"\"F)7$F+F+7$F+F(-%'COLOURG6&%$RGBG$\")+++!)!\")$\")AR!)\\F5$\")Vyg >F5-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 6 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 0 5 "plots" }{TEXT -1 37 " package als o contains the procedure " }{TEXT 0 13 "polygonplot3d" }{TEXT -1 55 " \+ which is used in the following command to draw a cube." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "plots[ polygonplot3d]([[[0,0,0],[1,0,0],[1,1,0],[0,1,0]],\n[[0,0,1],[1,0,1],[ 1,1,1],[0,1,1]],\n[[0,0,0],[0,1,0],[0,1,1],[0,0,1]],\n[[1,0,0],[1,1,0] ,[1,1,1],[1,0,1]],\n[[0,0,0],[1,0,0],[1,0,1],[0,0,1]],\n[[0,1,0],[1,1, 0],[1,1,1],[0,1,1]]],\nlightmodel=light4,scaling=constrained,style=pat chnogrid);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Try this \+ " }}{PARA 0 "" 0 "" {TEXT -1 15 "Plot the curve " }{XPPEDIT 18 0 "y=x ^4-4*x^3-x^2+5*x-1" "6#/%\"yG,,*$%\"xG\"\"%\"\"\"*&F(F)*$F'\"\"$F)!\" \"*$F'\"\"#F-*&\"\"&F)F'F)F)F)F-" }{TEXT -1 145 " to show the main fea tures of the graph. In particular ensure that all intercepts and turni ng points are shown in the picture. Does the equation " }{XPPEDIT 18 0 "x^4-4*x^3-x^2+5*x-1=0" "6#/,,*$%\"xG\"\"%\"\"\"*&F'F(*$F&\"\"$F(!\" \"*$F&\"\"#F,*&\"\"&F(F&F(F(F(F,\"\"!" }{TEXT -1 37 " have any integer solutions? Explain." }}{PARA 0 "" 0 "" {TEXT -1 39 "_________________ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(x^4-4*x^3-x^2+5*x-1,x=-2..4.4, y=-25..30);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "By looking at the " }{TEXT 357 1 "x" }{TEXT -1 55 "-intercepts \+ of this graph it appears that the equation " }{XPPEDIT 18 0 "x^4-4*x^3 -x^2+5*x-1=0" "6#/,,*$%\"xG\"\"%\"\"\"*&F'F(*$F&\"\"$F(!\"\"*$F&\"\"#F ,*&\"\"&F(F&F(F(F(F,\"\"!" }{TEXT -1 26 " has the integer solution " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 26 "This can be checked using " }{TEXT 0 4 "subs" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs(x=1,x^4-4*x^3-x^2+5*x-1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Alternatively, the proced ure " }{TEXT 0 6 "factor" }{TEXT -1 27 " shows that the polynomial " } {XPPEDIT 18 0 "x^4-4*x^3-x^2+5*x-1" "6#,,*$%\"xG\"\"%\"\"\"*&F&F'*$F% \"\"$F'!\"\"*$F%\"\"#F+*&\"\"&F'F%F'F'F'F+" }{TEXT -1 15 " has the fac tor" }{XPPEDIT 18 0 " ``(x-1)" "6#-%!G6#,&%\"xG\"\"\"F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "factor(x^4-4*x^3-x^2+5*x-1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 350 9 "How does " }{TEXT 259 4 "plot" }{TEXT 352 52 " draw curves? -- adaptive and no n-adaptive plotting " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 17 "When executing a " }{TEXT 0 4 "plot" }{TEXT -1 21 " com mand of the form " }{TEXT 267 18 "plot(expr,x=a..b);" }{TEXT -1 7 " wh ere " }{TEXT 267 4 "expr" }{TEXT -1 41 " is an expression in the singl e variable " }{TEXT 267 1 "x" }{TEXT -1 17 ", the expression " }{TEXT 267 4 "expr" }{TEXT -1 36 " is evaluated at specific values of " } {TEXT 351 1 "x" }{TEXT -1 301 " (which are not necessarily equally spa ced horizontally) and straight lines are used to join consecutive pair s of points. Usually sufficiently many points are used to give the app earance of a smooth curve. More points are used where the curve has a \+ \"tight bend\". This is achieved by a proceess called " }{TEXT 266 17 "adaptive plotting" }{TEXT -1 9 " whereby " }{TEXT 259 4 "plot" } {TEXT -1 78 " \"discovers\" where more points are needed in order to o btain a smooth curve. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " }{XPPEDIT 18 0 "y = 1/((x-1)^2+1/36);" " 6#/%\"yG*&\"\"\"F&,&*$,&%\"xGF&F&!\"\"\"\"#F&*&F&F&\"#OF+F&F+" }{TEXT -1 27 " has a sharp \"spike\" where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\" \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(1/((x-1)^2+1/36),x=0..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 434 332 332 {PLOTDATA 2 "6%-%'CURVESG6$7ao7$$\"\" !F)$\"3FI(H(H(H(H(*!#=7$$\"39LLLL3VfV!#>$\"3;.+TP'>51\"!#<7$$\"3'pmm;H [D:)F0$\"3#*oKk#)QhZ6F37$$\"3LLLLe0$=C\"F,$\"3UR>^md7e7F37$$\"3ILLL3RB r;F,$\"3S5a$*ye2'Q\"F37$$\"3Ymm;zjf)4#F,$\"3\"*\\Z-V\"3N`\"F37$$\"3=LL $e4;[\\#F,$\"39&fsh6))=p\"F37$$\"3p****\\i'y]!HF,$\"3_QUj(*Ho#)=F37$$ \"3,LL$ezs$HLF,$\"3(*o!eA!4G:@F37$$\"3_****\\7iI_PF,$\"3%pdLm/*o\"R#F3 7$$\"3#pmmm@Xt=%F,$\"3w4T3wc([t#F37$$\"3QLLL3y_qXF,$\"3JTlc([2,5$F37$$ \"3i******\\1!>+&F,$\"3\\s2P#R\"Fcq7$$\"3_KLekX0<\")F,$\"353 $RPSi9e\"Fcq7$$\"3V++]i.tK$)F,$\"3;7zH4*[$*z\"Fcq7$$\"3t*****\\s/\"Q&) F,$\"3ML)fb%HiM?Fcq7$$\"39++](3zMu)F,$\"3_Cz3]gN&H#Fcq7$$\"3_LLe*olx&* )F,$\"3Mo!oY`rze#Fcq7$$\"3#pmm;H_?<*F,$\"3AYGk,-W()GFcq7$$\"3InmT&GM)o $*F,$\"3I#>i\\&zY[JFcq7$$\"3emm;zihl&*F,$\"3C`\"QMS85P$Fcq7$$\"3mK$e9E WBk'R([f$Fcq7$$\"39LLL3#G,***F,$\"3Y**yG1P()*f$Fc q7$$\"3N$3x\"yY_/5F3$\"3%3930p[tf$Fcq7$$\"3RL3_Nl.55F3$\"3m%[juH#*pe$F cq7$$\"3V$ekGR[b,\"F3$\"3k#z'H)>R*oNFcq7$$\"3CL$3-Dg5-\"F3$\"33'>)Rz8U VNFcq7$$\"34Le*['R3K5F3$\"3K#)=su$f8Z$Fcq7$$\"3HE8TD\"Fcq 7$$\"3%)***\\i&p@[7F3$\"3m9j%F37$$\"3Ymm\"H!o-*\\\"F3$\"3WrR/JF37$$\"3emmmT9C#e\"F3$\"3$fn!yZ'3ks#F37$$\"3\"**** \\i!*3`i\"F3$\"3]`:exs$yQ#F37$$\"3QLLL$*zym;F3$\"3=%yy,!=#p6#F37$$\"3G LL$3N1#4F3$\"3-V[\\2SO`6F37$ $\"3/++v.Uac>F3$\"35**QlJ9sg5F37$$\"\"#F)F*-%'COLOURG6&%$RGBG$\"#5!\" \"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fgal-%%VIEWG6$;F(Fj`l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The following \+ commands construct a picture which shows the points used to obtain the previous curve." }}{PARA 0 "" 0 "" {TEXT -1 48 "The plot structure us ed is assigned to the name " }{TEXT 267 5 "plot1" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot1 := plot(1/((x-1)^2+1/36),x=0..2,style=point):\nplot1;\n" }} {PARA 13 "" 1 "" {GLPLOT2D 400 339 339 {PLOTDATA 2 "6&-%'CURVESG6$7ao7 $$\"\"!F)$\"3FI(H(H(H(H(*!#=7$$\"39LLLL3VfV!#>$\"3;.+TP'>51\"!#<7$$\"3 'pmm;H[D:)F0$\"3#*oKk#)QhZ6F37$$\"3LLLLe0$=C\"F,$\"3UR>^md7e7F37$$\"3I LLL3RBr;F,$\"3S5a$*ye2'Q\"F37$$\"3Ymm;zjf)4#F,$\"3\"*\\Z-V\"3N`\"F37$$ \"3=LL$e4;[\\#F,$\"39&fsh6))=p\"F37$$\"3p****\\i'y]!HF,$\"3_QUj(*Ho#)= F37$$\"3,LL$ezs$HLF,$\"3(*o!eA!4G:@F37$$\"3_****\\7iI_PF,$\"3%pdLm/*o \"R#F37$$\"3#pmmm@Xt=%F,$\"3w4T3wc([t#F37$$\"3QLLL3y_qXF,$\"3JTlc([2,5 $F37$$\"3i******\\1!>+&F,$\"3\\s2P#R\"Fcq7$$\"3_KLekX0<\")F, $\"353$RPSi9e\"Fcq7$$\"3V++]i.tK$)F,$\"3;7zH4*[$*z\"Fcq7$$\"3t*****\\s /\"Q&)F,$\"3ML)fb%HiM?Fcq7$$\"39++](3zMu)F,$\"3_Cz3]gN&H#Fcq7$$\"3_LLe *olx&*)F,$\"3Mo!oY`rze#Fcq7$$\"3#pmm;H_?<*F,$\"3AYGk,-W()GFcq7$$\"3Inm T&GM)o$*F,$\"3I#>i\\&zY[JFcq7$$\"3emm;zihl&*F,$\"3C`\"QMS85P$Fcq7$$\"3 mK$e9EWBk'R([f$Fcq7$$\"39LLL3#G,***F,$\"3Y**yG1P() *f$Fcq7$$\"3N$3x\"yY_/5F3$\"3%3930p[tf$Fcq7$$\"3RL3_Nl.55F3$\"3m%[juH# *pe$Fcq7$$\"3V$ekGR[b,\"F3$\"3k#z'H)>R*oNFcq7$$\"3CL$3-Dg5-\"F3$\"33'> )Rz8UVNFcq7$$\"34Le*['R3K5F3$\"3K#)=su$f8Z$Fcq7$$\"3HE8TD \"Fcq7$$\"3%)***\\i&p@[7F3$\"3m9j%F37$$\"3Ymm\"H!o-*\\\"F3$\"3WrR/JF37$$\"3emmmT9C#e\"F3$\"3$fn!yZ'3ks#F37$$\"3 \"****\\i!*3`i\"F3$\"3]`:exs$yQ#F37$$\"3QLLL$*zym;F3$\"3=%yy,!=#p6#F37 $$\"3GLL$3N1#4F3$\"3-V[\\2S O`6F37$$\"3/++v.Uac>F3$\"35**QlJ9sg5F37$$\"\"#F)F*-%'COLOURG6&%$RGBG$ \"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fgal-%&STYLEG6#%&POINTG-%%VIEWG6 $;F(Fj`l%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "The following commands (which involve a standard programming loop) collec t the points used for the two sections of the graph where " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<1/2" "6#2%!G*&\"\"\"F& \"\"#!\"\"" }{TEXT -1 11 " and where " }{XPPEDIT 18 0 "1/2 " 0 "" {MPLTEXT 1 0 277 "pts1 := op(1,op(1,plot 1)):\nleft_list1 := []: right_list1 := []:\nfor i to nops(pts1) do\n \+ if pts1[i,1]<=.5 then left_list1 := [op(left_list1),pts1[i]];\n eli f pts1[i,1]<=1 then right_list1 := [op(right_list1),pts1[i]] end if;\n end do:\nleft_list1;\nnops(%);\nright_list1;\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7.7$$\"\"!F&$\"3FI(H(H(H(H(*!#=7$$\"39LLLL3VfV!#>$ \"3;.+TP'>51\"!#<7$$\"3'pmm;H[D:)F-$\"3#*oKk#)QhZ6F07$$\"3LLLLe0$=C\"F )$\"3UR>^md7e7F07$$\"3ILLL3RBr;F)$\"3S5a$*ye2'Q\"F07$$\"3Ymm;zjf)4#F)$ \"3\"*\\Z-V\"3N`\"F07$$\"3=LL$e4;[\\#F)$\"39&fsh6))=p\"F07$$\"3p****\\ i'y]!HF)$\"3_QUj(*Ho#)=F07$$\"3,LL$ezs$HLF)$\"3(*o!eA!4G:@F07$$\"3_*** *\\7iI_PF)$\"3%pdLm/*o\"R#F07$$\"3#pmmm@Xt=%F)$\"3w4T3wc([t#F07$$\"3QL LL3y_qXF)$\"3JTlc([2,5$F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#797$$\"3i******\\1!>+&!#=$\"3\\s2P#R\"FI7$$\"3_KLekX0<\")F'$\"353$RPSi9e\"FI7$$\"3V++]i.tK$)F'$\"3 ;7zH4*[$*z\"FI7$$\"3t*****\\s/\"Q&)F'$\"3ML)fb%HiM?FI7$$\"39++](3zMu)F '$\"3_Cz3]gN&H#FI7$$\"3_LLe*olx&*)F'$\"3Mo!oY`rze#FI7$$\"3#pmm;H_?<*F' $\"3AYGk,-W()GFI7$$\"3InmT&GM)o$*F'$\"3I#>i\\&zY[JFI7$$\"3emm;zihl&*F' $\"3C`\"QMS85P$FI7$$\"3mK$e9EWBk'R([f$FI7$$\"39LLL3#G, ***F'$\"3Y**yG1P()*f$FI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "If the gr aph is drawn using " }{TEXT 259 4 "plot" }{TEXT -1 59 " with non-adapt ive plotting (obtained by using the option \"" }{TEXT 267 14 "adaptive =false" }{TEXT -1 26 "\" instead of the default \"" }{TEXT 267 13 "ada ptive=true" }{TEXT -1 22 "\") the spike appears \"" }{TEXT 266 9 "too \+ sharp" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(1/((x-1)^2+1/36),x=0..2,adapti ve=false);" }}{PARA 13 "" 1 "" {GLPLOT2D 377 307 307 {PLOTDATA 2 "6%-% 'CURVESG6$7S7$$\"\"!F)$\"3FI(H(H(H(H(*!#=7$$\"39LLLL3VfV!#>$\"3;.+TP'> 51\"!#<7$$\"3'pmm;H[D:)F0$\"3#*oKk#)QhZ6F37$$\"3LLLLe0$=C\"F,$\"3UR>^m d7e7F37$$\"3ILLL3RBr;F,$\"3S5a$*ye2'Q\"F37$$\"3Ymm;zjf)4#F,$\"3\"*\\Z- V\"3N`\"F37$$\"3=LL$e4;[\\#F,$\"39&fsh6))=p\"F37$$\"3p****\\i'y]!HF,$ \"3_QUj(*Ho#)=F37$$\"3,LL$ezs$HLF,$\"3(*o!eA!4G:@F37$$\"3_****\\7iI_PF ,$\"3%pdLm/*o\"R#F37$$\"3#pmmm@Xt=%F,$\"3w4T3wc([t#F37$$\"3QLLL3y_qXF, $\"3JTlc([2,5$F37$$\"3i******\\1!>+&F,$\"3\\s2P#R\"Fcq7$$\"3V++]i.tK$)F,$\"3;7zH4*[$*z\"Fcq7$$\"3 9++](3zMu)F,$\"3_Cz3]gN&H#Fcq7$$\"3#pmm;H_?<*F,$\"3AYGk,-W()GFcq7$$\"3 emm;zihl&*F,$\"3C`\"QMS85P$Fcq7$$\"39LLL3#G,***F,$\"3Y**yG1P()*f$Fcq7$ $\"3%F37$$\"3Ym m\"H!o-*\\\"F3$\"3WrR/JF37$ $\"3emmmT9C#e\"F3$\"3$fn!yZ'3ks#F37$$\"3\"****\\i!*3`i\"F3$\"3]`:exs$y Q#F37$$\"3QLLL$*zym;F3$\"3=%yy,!=#p6#F37$$\"3GLL$3N1#4F3$\"3-V[\\2SO`6F37$$\"3/++v.Uac>F3$\"35**Ql J9sg5F37$$\"\"#F)F*-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\" x6\"Q!Fc[l-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot2 := plot(1/((x-1)^2+1/3 6),x=0..2,adaptive=false,style=point):\nplot2;" }}{PARA 13 "" 1 "" {GLPLOT2D 384 314 314 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)$\"3FI(H( H(H(H(*!#=7$$\"39LLLL3VfV!#>$\"3;.+TP'>51\"!#<7$$\"3'pmm;H[D:)F0$\"3#* oKk#)QhZ6F37$$\"3LLLLe0$=C\"F,$\"3UR>^md7e7F37$$\"3ILLL3RBr;F,$\"3S5a$ *ye2'Q\"F37$$\"3Ymm;zjf)4#F,$\"3\"*\\Z-V\"3N`\"F37$$\"3=LL$e4;[\\#F,$ \"39&fsh6))=p\"F37$$\"3p****\\i'y]!HF,$\"3_QUj(*Ho#)=F37$$\"3,LL$ezs$H LF,$\"3(*o!eA!4G:@F37$$\"3_****\\7iI_PF,$\"3%pdLm/*o\"R#F37$$\"3#pmmm@ Xt=%F,$\"3w4T3wc([t#F37$$\"3QLLL3y_qXF,$\"3JTlc([2,5$F37$$\"3i******\\ 1!>+&F,$\"3\\s2P#R\"Fcq7$$ \"3V++]i.tK$)F,$\"3;7zH4*[$*z\"Fcq7$$\"39++](3zMu)F,$\"3_Cz3]gN&H#Fcq7 $$\"3#pmm;H_?<*F,$\"3AYGk,-W()GFcq7$$\"3emm;zihl&*F,$\"3C`\"QMS85P$Fcq 7$$\"39LLL3#G,***F,$\"3Y**yG1P()*f$Fcq7$$\"3%F37$$\"3Ymm\"H!o-*\\\"F3$\"3WrR/JF37$$\"3emmmT9C#e\"F3$\"3$fn!yZ'3 ks#F37$$\"3\"****\\i!*3`i\"F3$\"3]`:exs$yQ#F37$$\"3QLLL$*zym;F3$\"3=%y y,!=#p6#F37$$\"3GLL$3N1#4F3$ \"3-V[\\2SO`6F37$$\"3/++v.Uac>F3$\"35**QlJ9sg5F37$$\"\"#F)F*-%'COLOURG 6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fc[l-%&STYLEG6#%&POINTG -%%VIEWG6$;F(Ffz%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 165 "With non-adaptive plotting the points are approxi mately evenly spaced horizontally so that intervals of equal width con tain approximately the same number of points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "pts2 := op( 1,op(1,plot2)):\nleft_list2 := []: right_list2 := []:\nfor i to nops(p ts2) do\n if pts2[i,1]<=.5 then left_list2 := [op(left_list2),pts1[i ]];\n elif pts2[i,1]<=1 then right_list2 := [op(right_list2),pts1[i] ] end if;\nend do:\nleft_list2;\nnops(%);\nright_list2;\nnops(%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7.7$$\"\"!F&$\"3FI(H(H(H(H(*!#=7$$\"39 LLLL3VfV!#>$\"3;.+TP'>51\"!#<7$$\"3'pmm;H[D:)F-$\"3#*oKk#)QhZ6F07$$\"3 LLLLe0$=C\"F)$\"3UR>^md7e7F07$$\"3ILLL3RBr;F)$\"3S5a$*ye2'Q\"F07$$\"3Y mm;zjf)4#F)$\"3\"*\\Z-V\"3N`\"F07$$\"3=LL$e4;[\\#F)$\"39&fsh6))=p\"F07 $$\"3p****\\i'y]!HF)$\"3_QUj(*Ho#)=F07$$\"3,LL$ezs$HLF)$\"3(*o!eA!4G:@ F07$$\"3_****\\7iI_PF)$\"3%pdLm/*o\"R#F07$$\"3#pmmm@Xt=%F)$\"3w4T3wc([ t#F07$$\"3QLLL3y_qXF)$\"3JTlc([2,5$F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7/7$$\"3i******\\1!>+&!#=$ \"3\\s2P#R\"FI7$$\"3_KLekX0<\")F'$\"353$RPSi9e\"FI7$$\"3V++ ]i.tK$)F'$\"3;7zH4*[$*z\"FI7$$\"3t*****\\s/\"Q&)F'$\"3ML)fb%HiM?FI7$$ \"39++](3zMu)F'$\"3_Cz3]gN&H#FI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"# 8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 353 8 "Forcing " }{TEXT 259 4 "plot" }{TEXT 354 30 " to use more points with the \"" } {TEXT 267 9 "numpoints" }{TEXT 356 30 "\" option when plotting a curve " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " } {XPPEDIT 18 0 "y = sin(x)*sin(30*x);" "6#/%\"yG*&-%$sinG6#%\"xG\"\"\"- F'6#*&\"#IF*F)F*F*" }{TEXT -1 34 " can be thought of as a sine wave " }{XPPEDIT 18 0 "y = a*sin(30*x);" "6#/%\"yG*&%\"aG\"\"\"-%$sinG6#*&\"# IF'%\"xGF'F'" }{TEXT -1 13 " with period " }{XPPEDIT 18 0 "Pi/15" "6#* &%#PiG\"\"\"\"#:!\"\"" }{TEXT -1 49 " and (high) angular frequency 30 \+ whose amplitude " }{TEXT 355 1 "a" }{TEXT -1 52 " varies between 0 and 1 as determined by the factor " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#% \"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Maple's plotting procedure " }{TEXT 259 4 "plot" }{TEXT -1 68 " (with Maple 9) can pro duce a satisfactory picture for the graph of " }{XPPEDIT 18 0 "y = sin (x)*sin(30*x);" "6#/%\"yG*&-%$sinG6#%\"xG\"\"\"-F'6#*&\"#IF*F)F*F*" } {TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[0,4*Pi]" "6#7$\"\"! *&\"\"%\"\"\"%#PiGF'" }{TEXT -1 2 ". 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}{TEXT 267 9 "numpoints" }{TEXT -1 45 "\" \+ option if necessary to get a good picture. " }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) Plot the curves " }{XPPEDIT 18 0 "y = sqrt(x)*sin(60*x)" "6 #/%\"yG*&-%%sqrtG6#%\"xG\"\"\"-%$sinG6#*&\"#gF*F)F*F*" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "y=sqrt(x)" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "y=-sqrt(x)" "6#/%\"yG,$-%%sqrtG6#%\"xG!\"\"" } {TEXT -1 43 " in the same picture to show how the curve " }{XPPEDIT 18 0 "y = sqrt(x)*sin(60*x)" "6#/%\"yG*&-%%sqrtG6#%\"xG\"\"\"-%$sinG6# *&\"#gF*F)F*F*" }{TEXT -1 33 " is \"trapped\" between the curves " } {XPPEDIT 18 0 "y=sqrt(x)" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "y=-sqrt(x)" "6#/%\"yG,$-%%sqrtG6#%\"xG!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "____________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(sqrt(x)*sin(60*x),x=0..3*Pi,numpoints=60); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "plot([sqrt(x)*sin(60*x) ,sqrt(x),-sqrt(x)],x=0..3*Pi,\n numpoints=60,color=[green,coral,cora l],linestyle=[1,3,3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 269 10 "EXPERIMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Expressions and functions: " } {TEXT 0 7 "->,eval" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "When using Maple it is necessary to disti nguish carefully between " }{TEXT 266 11 "expressions" }{TEXT -1 5 " a nd " }{TEXT 266 9 "functions" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2-4*x+3;" "6#,(*$%\"xG\"\"#\"\"\"*&\"\"%F'F%F'!\"\"\"\"$F'" }{TEXT -1 6 " and \+ " }{XPPEDIT 18 0 "sqrt(x)+2;" "6#,&-%%sqrtG6#%\"xG\"\"\"\"\"#F(" } {TEXT -1 46 " are examples of expressions in the variable " }{TEXT 277 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 59 "These are entered in Maple commands or input state ments as " }{TEXT 0 9 "x^2-4*x+3" }{TEXT -1 5 " and " }{TEXT 0 10 "sqr t(x)+2 " }{TEXT -1 13 "respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "In mathematics, we can use such express ions to define functions, say " }{XPPEDIT 18 0 "f(x) = x^2-4*x+3;" "6 #/-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"*&\"\"%F+F'F+!\"\"\"\"$F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = sqrt(x)+2;" "6#/-%\"gG6#%\"xG,&-%%sqr tG6#F'\"\"\"\"\"#F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "f \+ and g are now " }{TEXT 266 21 "names for the actions" }{TEXT -1 4 " or " }{TEXT 266 10 "procedures" }{TEXT -1 80 " associated with computing values of these expressions for a given input number " }{TEXT 296 1 " x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 69 "In Maple functions c an most easily be defined by using the following " }{TEXT 266 14 "arro w notation" }{TEXT -1 58 ". The Maple command which defines the functi on f above is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> x^2-4*x+3;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "Once you have executed this co mmand, you can use the function f in your current Maple session just a s you would use any of Maple's built-in mathematical functions." }} {PARA 0 "" 0 "" {TEXT -1 49 "You can evaluate it at a given input numb er . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f (3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " Symbolic inputs can also be handled. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(s+1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "You can plot the graph of the function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(f(x),x=0..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 59 "Other functions can be defined in terms of the fun ction f. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g := x -> f(x-1);\nh := x -> f(x-2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot([f (x),g(x),h(x)],x=0..6,y=-1..3,color=[red,aquamarine,gold],thickness=2) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 184 "On ce you have defined a function, Maple will remember that function duri ng your entire working session. It is possible to overwrite the defini tion of a function with a new definition. " }}{PARA 0 "" 0 "" {TEXT -1 76 "For example, if you want to change the definition of the functi on f so that " }{XPPEDIT 18 0 "f(x)=x/(x^2+1)" "6#/-%\"fG6#%\"xG*&F'\" \"\",&*$F'\"\"#F)F)F)!\"\"" }{TEXT -1 31 ", type and execute the comma nd:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> x/(x^2+1); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The \+ current (new) definition for the function f can be confirmed by typing " }{TEXT 267 5 "f(x);" }{TEXT -1 19 " on a command line." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Alte rnatively, the definition of a function can be obtained by using " } {TEXT 0 4 "eval" }{TEXT -1 61 " with the name of the function as a sin gle input parameter. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(f);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Simply typing " }{TEXT 267 2 "f ;" }{TEXT -1 24 " on a command line does " }{TEXT 266 10 "not recall" }{TEXT -1 16 " the definition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "f;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 84 ": If the de finition of f is changed then g and h are defined in terms of the new \+ f. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(x );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "App lying " }{TEXT 0 4 "eval" }{TEXT -1 76 " to g confirms the definitions of g in terms of f, but says nothing about f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(g);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "f := x -> x/(x^2+1);\nplot([f(x),g(x),h(x)],x=-5..7,color=[red,aqu amarine,gold],thickness=2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "If you want to clear the function f without red efining it, type and execute:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f := 'f';" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "A number of functions, \+ say, f, g and h, can cleared at the same time using the " }{TEXT 0 8 " unassign" }{TEXT -1 12 " procedure. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "unassign('f','g','h');" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "In orde r to save memory it is always a good idea to clear your functions, as \+ well as any other variables, when you start a new problem. Alternative ly, you can use " }{TEXT 370 8 "restart;" }{TEXT -1 10 " to clear " } {TEXT 266 10 "everything" }{TEXT -1 13 " from memory." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "In many situations it \+ is possible to work with " }{TEXT 266 11 "expressions" }{TEXT -1 23 " \+ instead of functions. " }}{PARA 0 "" 0 "" {TEXT -1 50 "An expression c an be assigned to a named variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "expr := x^2-4*x+3;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "The vari able can then be used to refer to the expression. For example an expre ssion in a single variable can be used in a " }{TEXT 0 4 "plot" } {TEXT -1 10 " command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(expr,x=0..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure \+ " }{TEXT 0 4 "eval" }{TEXT -1 60 " can be used to evaluate an expressi on in a single variable " }{TEXT 368 1 "x" }{TEXT -1 21 " at a specifi c value " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 29 ". For exa mple, the values of " }{XPPEDIT 18 0 "f(1/2)" "6#-%\"fG6#*&\"\"\"F'\" \"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(3)" "6#-%\"fG6#\"\"$" }{TEXT -1 75 " obtained previously may also be obtained by evaluating \+ the expression for " }{XPPEDIT 18 0 "x=1/2" "6#/%\"xG*&\"\"\"F&\"\"#! \"\"" }{TEXT -1 9 " and for " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eval(expr,x=1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(expr,x=3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 266 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 " " {TEXT -1 35 "The standard syntax in this use of " }{TEXT 0 4 "eval" }{TEXT -1 4 " is " }{TEXT 267 15 "eval(expr,x=a);" }{TEXT -1 6 " with \+ " }{TEXT 266 13 "two arguments" }{TEXT -1 49 ": the expression to be e valuated and an equation " }{XPPEDIT 18 0 "x=` . . . `" "6#/%\"xG%(~.~ .~.~G" }{TEXT -1 91 " to specify where the evaluation is to be perform ed. This differs from the previous use of " }{TEXT 0 4 "eval" }{TEXT -1 73 " to obtain the definition of a function where just one argument is used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 50 "The evaluation can also be achieved by means of a " }{TEXT 266 16 "two-step process" }{TEXT -1 19 " making use of the " }{TEXT 266 10 "inert form" }{TEXT -1 18 " of the procedure " }{TEXT 0 4 "eval" } {TEXT -1 24 " designated by the name " }{TEXT 0 4 "Eval" }{TEXT -1 20 " with an upper case " }{TEXT 0 1 "E" }{TEXT -1 16 ". The procedure " }{TEXT 0 4 "Eval" }{TEXT -1 109 " justs sets-up the evaluation command without actually performing the desired action. In the current exampl e " }{TEXT 0 4 "Eval" }{TEXT -1 22 " generates the output " }{XPPEDIT 18 0 "Eval(x^2-4*x+3,x=a)" "6#-%%EvalG6$,(*$%\"xG\"\"#\"\"\"*&\"\"%F*F (F*!\"\"\"\"$F*/F(%\"aG" }{TEXT -1 59 " which designates the pending e valuation of the expression " }{XPPEDIT 18 0 "x^2-4*x+3" "6#,(*$%\"xG \"\"#\"\"\"*&\"\"%F'F%F'!\"\"\"\"$F'" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 88 " without performing the necessar y actions to achieve the desired result. \nThe procedure " }{TEXT 0 5 "value" }{TEXT -1 84 " can then be used to actually perform the requir ed action. \nFor more information on " }{TEXT 0 4 "eval" }{TEXT -1 4 " or " }{TEXT 0 4 "Eval" }{TEXT -1 18 " see the relevant " }{TEXT 266 10 "help pages" }{TEXT -1 14 " by clicking " }{HYPERLNK 17 "eval" 2 " eval" "" }{TEXT -1 4 " or " }{HYPERLNK 17 "Eval" 2 "Eval" "" }{TEXT -1 15 " respectively. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Eval(expr_in_x,x=1/2);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Eval(expr_in_x,x=3);\nvalue(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "It is a matter of personal preference whether you want t o work with an expression or a function in a particular situation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 4 "plot" }{TEXT -1 146 " will accept either expr essions or functions (procedures) as input. Note that if f is a functi on, defined in Maple using the arrow notation, then " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " is an expression. " }}{PARA 0 " " 0 "" {TEXT -1 184 "If we just use the name of the function, the plot ting range must be given without mention of an independent variable.\n We should not expect the axes to be labelled at all if we do this." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := x -> 3-sqrt(x^2+4);\nplot(f,0..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 3 "map" }{TEXT -1 82 " is an example of a procedure which works only w ith functions and not expressions." }}{PARA 0 "" 0 "" {TEXT 259 3 "map " }{TEXT -1 78 " can be used to evaluate a function at all members of \+ a list of input numbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "g := x -> x^2;\nmap(g,[1,2,3,4,5,6] );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "In working with Maple we deal with what are primarily" }{TEXT 256 1 " " }{TEXT 266 19 "numerical functions" }{TEXT -1 78 " such as the functio n f just defined, or built-in mathematical functions like " }{TEXT 259 4 "sqrt" }{TEXT -1 2 ", " }{TEXT 259 3 "sin" }{TEXT -1 5 " and " } {TEXT 259 3 "cos" }{TEXT -1 32 ". We also use functions such as " } {TEXT 259 5 "solve" }{TEXT -1 4 " or " }{TEXT 259 4 "plot" }{TEXT -1 43 " which have algebraic or graphical outputs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "In the latter context, fu nctions are often called " }{TEXT 266 8 "routines" }{TEXT -1 4 " or " }{TEXT 266 10 "procedures" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 159 "Maple handles all functions (p rocedures) in the same way. For example, we could define a function \" evalf15\" to evaluate an expression to 15 digits as follows. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalf15 := x -> evalf(x,15);\nevalf15(Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 7 "WARNING" }{TEXT -1 118 ": An error which many students make initially is to try to define a fun ction by means of an assignment statement like " }{TEXT 267 17 "f(x) : = x^2-4*x+3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 82 "This fails to define the desired function because of the assignment to the symbo l " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 65 ". In genera l, this kind of assignment should be avoided in Maple." }}{PARA 0 "" 0 "" {TEXT -1 40 "For more information see the subsection " } {HYPERLNK 17 "Assigning specific values to a function, remember tables and a common error made in defining functions" 1 "" "Assigning specif ic values to a function, remember tables and a common error made in de fining functions" }{TEXT -1 8 " below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "More information concerning function s is given in the following subsections. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 292 78 "A function defined using the arrow notation is technically a Maple procedure " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The fun ction f with the mathematical definition " }{XPPEDIT 18 0 "f(x)=(x+2)/ (x^3+5*x+2)" "6#/-%\"fG6#%\"xG*&,&F'\"\"\"\"\"#F*F*,(*$F'\"\"$F**&\"\" &F*F'F*F*F+F*!\"\"" }{TEXT -1 48 " can be set-up in Maple with the com mand . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := x -> (x+2)/(x^3+5*x+2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "As before standard functi on notation can now be used to evluate the function f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(1/2); \nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(0.5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(a);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f(exp(x));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "If you select (highlight with t he mouse) the output" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"x G6\"6$%)operatorG%&arrowGF(*&,&9$\"\"\"\"\"#F/F/,(*$)F.\"\"$F/F/*&\"\" &F/F.F/F/F0F/!\"\"F(F(F(" }}{PARA 0 "" 0 "" {TEXT -1 99 "you will see \+ the following expression in the window in the context bar at the top o f the worksheet." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 291 65 "f \+ := proc (x) options operator, arrow; (x+2)/(x^3+5*x+2) end proc" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "Maple has set up a " } {TEXT 266 9 "procedure" }{TEXT -1 70 " to handle evaluation of the fun ction f as indicated by the key-words " }{TEXT 0 4 "proc" }{TEXT -1 7 " . . . " }{TEXT 0 8 "end proc" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "This code can be used on \+ a command line to set up the function f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f := proc(x) options \+ operator,arrow; (x+2)/(x^3+5*x+2) end proc;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "If the " }{TEXT 267 23 "options \+ operator,arrow;" }{TEXT -1 226 " statement is omitted from the body of the procedure. This prevents Maple from using the arrow notation in t he output form but otherwise the procedure still works in the same way . I'll use a g instead of f to illustrate this. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "g := proc(x) (x+2)/(x^3+5*x+2) end proc;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g(1/2);\nevalf(%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g(0.5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "g(a);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "g(exp(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 290 102 "Assigning specific values to a functi on, remember tables and a common error made in defining functions" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "Assigni ng specific values to a function, remember tables and a common error m ade in defining functions" {TEXT -1 18 "It is possible to " }{TEXT 258 22 "assign specific values" }{TEXT -1 15 " to a function " }{TEXT 258 10 "overriding" }{TEXT -1 70 " the value which would be given by a ny previous procedural definition." }}{PARA 0 "" 0 "" {TEXT -1 68 "To \+ illustrate this first define the function f by the Maple command " } {TEXT 267 19 "f := x -> sqrt(x+4)" }{TEXT -1 47 ", which corresponds t o mathematical definition " }{XPPEDIT 18 0 "f(x)=sqrt(x+4)" "6#/-%\"fG 6#%\"xG-%%sqrtG6#,&F'\"\"\"\"\"%F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> \+ sqrt(x+4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We obtain the expected values for " }{XPPEDIT 18 0 "f(0)" "6#-% \"fG6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(3)" "6#-%\"fG6#\"\"$" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "f(5)" "6#-%\"fG6#\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'f(0)'=f(0);\n'f(3)'=f(3);\n'f(5)'=f(5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 13 ": The quotes " }{TEXT 259 3 "' '" }{TEXT -1 55 " around f(0) ensur e that the left side of the equation " }{TEXT 267 11 "'f(0)'=f(0)" } {TEXT -1 58 " remains unevaluated, so that the corresponding output is " }{TEXT 256 8 "f(0) = 2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The following commands change the \+ values of the function f at the input values " }{XPPEDIT 18 0 "x=0" "6 #/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=5" "6#/%\"xG\"\"& " }{TEXT -1 60 " to 4 and 8 respectively. Values elsewhere remain unch anged." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f(0) := 4;\nf(5) := 8;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'f(0)'=f(0);\n'f(3)'= f(3);\n'f(5)'=f(5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 31 "The new values are stored in a " }{TEXT 266 14 "remembe r table" }{TEXT -1 54 " for the function f, which can be accessed as f ollows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "op(4,eval(f));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 154 "The fact that the values of f have been \+ changed at finite number of specific numbers does not change the appea rance of the graph of f since the procedure " }{TEXT 0 4 "plot" } {TEXT -1 58 " does not \"discover\" that these values have been change d. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-5..6);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 133 "If you execute the following command immediately after plotting the graph, you will see the points which h ave been used for the plot." }}{PARA 0 "" 0 "" {TEXT -1 67 "(See a lat er section for more details concerning plotting points.) " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "op(1, op(1,%));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 7 "WARNING" }{TEXT -1 118 ": An error which many students make ini tially is to try to define a function by means of an assignment statem ent like " }{TEXT 267 17 "f(x) := sqrt(x+4)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "This command simply assigns the specific symbol ic value given by the expression " }{TEXT 267 9 "sqrt(x+4)" }{TEXT -1 29 " at the symbolic input value " }{TEXT 267 1 "x" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := 'f';\nf(x) := sqrt(x+4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 65 "The remember table for f shows that the v alue of f at the symbol " }{TEXT 289 1 "x" }{TEXT -1 19 " is the expre ssion " }{XPPEDIT 18 0 "sqrt(x+4)" "6#-%%sqrtG6#,&%\"xG\"\"\"\"\"%F(" }{TEXT -1 77 ". However, since the earlier procedural definition is re moved by the command " }{TEXT 267 8 "f := 'f'" }{TEXT -1 54 ", the fun ction f has no value for any numerical input." }}{PARA 0 "" 0 "" {TEXT 30 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "op(4,eval(f) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(0);\nf(3);\nf(5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "Entering " }{TEXT 267 4 "f(x)" }{TEXT -1 46 " on a command line certainly gives the output " }{XPPEDIT 18 0 "sqrt(x+4)" "6#-%%sqrtG6#,&%\"xG\"\"\"\"\"%F(" }{TEXT -1 83 ", just li ke the output you would obtain if you defined f using the arrow notati on. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "However, entering " }{TEXT 267 4 "f(a)" }{TEXT -1 15 " does not give " }{XPPEDIT 18 0 "sqrt(a+4)" "6#-%%sqrtG6#,&%\"aG\" \"\"\"\"%F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(a);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Unfortunately the \"error \" will not be discovered in the " }{TEXT 0 4 "plot" }{TEXT -1 15 " co mmand . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-5..6);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " . . . but the following " }{TEXT 266 11 "do not work" }{TEXT -1 35 " with the (\"incorrect\") definitio n " }{TEXT 267 17 "f(x) := sqrt(x+4)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "plot(f,-5 ..6);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(s),s=-5..6);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 55 "The following command deletes the rem ember table of f. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 10 "forget(f);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Try this " }}{PARA 0 "" 0 "" {TEXT -1 24 "(a) Set up the function " }{XPPEDIT 18 0 "f(x)=(x^3+3*x^2-1)/(x^4+ 4)" "6#/-%\"fG6#%\"xG*&,(*$F'\"\"$\"\"\"*&F+F,*$F'\"\"#F,F,F,!\"\"F,,& *$F'\"\"%F,F3F,F0" }{TEXT -1 40 " in Maple form using the arrow notati on." }}{PARA 0 "" 0 "" {TEXT -1 9 "(b) Find " }{XPPEDIT 18 0 "f(-1),f( 0)" "6$-%\"fG6#,$\"\"\"!\"\"-F$6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(1)" "6#-%\"fG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "(c) Plot the graph of f over the interval from " } {XPPEDIT 18 0 "x=-4" "6#/%\"xG,$\"\"%!\"\"" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 22 "(d) Use the procedure " }{TEXT 0 4 "eval" }{TEXT -1 28 " to evaluate the expression " }{XPPEDIT 18 0 "(x^3+3*x^2-1)/(x^4+4)" "6#*&,(*$%\"xG\"\"$\"\"\"*&F'F(*$F&\"\"#F(F(F(!\"\"F(,&*$F&\"\"%F(F/F( F," }{TEXT -1 5 " for " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f := x -> (x^3+3*x^2-1)/(x^4+4);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f(-1);\nf(0);\nf(1);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-4..4);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "eval((x^3+3*x^2-1)/(x^4+4),x =-1);\neval((x^3+3*x^2-1)/(x^4+4),x=0);\neval((x^3+3*x^2-1)/(x^4+4),x= 1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 10 "EXPERIM ENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 19 "Solving equations: " }{TEXT 259 5 "solve" }{TEXT 0 7 ", factor" }{TEXT -1 5 " and " }{TEXT 0 4 "subs" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure \+ " }{TEXT 259 5 "solve" }{TEXT -1 32 " can be used to solve equations. " }}{PARA 0 "" 0 "" {TEXT -1 27 "Make sure that the unknown " }{TEXT 323 1 "x" }{TEXT -1 33 " does not have an assigned value." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x := \+ 'x':\nsolve(x^2=25);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 213 "Maple can solve an equation in terms of variables whi ch appear in the equation.\nIf there is more than one variable in the \+ equation to be solved, you need to specify the variable you wish to so lve for as a second \"" }{TEXT 258 8 "argument" }{TEXT -1 6 "\" or \" " }{TEXT 258 15 "input parameter" }{TEXT -1 10 "\" for the " }{TEXT 0 5 "solve" }{TEXT -1 11 " procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a := 'a': b := 'b': c := 'c ':\nsolve(a*x+b=c,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(x^2=a+1,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The following command \+ solves the equation " }{XPPEDIT 18 0 "x^2-2*x-3 = 0;" "6#/,(*$%\"xG\" \"#\"\"\"*&F'F(F&F(!\"\"\"\"$F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 29 ": The second input parameter " } {TEXT 278 1 "x" }{TEXT -1 16 " may be omitted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x^2-2* x-3=0,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The solutions of the equation " }{XPPEDIT 18 0 "x^2-2*x-3=0" "6 #/,(*$%\"xG\"\"#\"\"\"*&F'F(F&F(!\"\"\"\"$F*\"\"!" }{TEXT -1 5 " are \+ " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 3" "6#/%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The next command solve s the equation " }{XPPEDIT 18 0 "x^2-2*x-4 = 0;" "6#/,(*$%\"xG\"\"#\" \"\"*&F'F(F&F(!\"\"\"\"%F*\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x^2-2* x-4=0,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "As usual, approximate numerical values for the solutions can be obtained using " }{TEXT 259 5 "evalf" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(x^2 -2*x-4=0,x);\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 37 "Maple can solve systems of equations." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "x := \+ 'x': y:= 'y':\nsolve(\{y=x+1,x+y=5\});" }}}{PARA 0 "" 0 "" {TEXT -1 107 "\nAgain, if there are other variables in the equation, you must s pecify the variables you wish to solve for." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(\{y=x+a,x+y= 5\},\{x,y\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The general quadratic equation " }{XPPEDIT 18 0 "a*x^2+b* x+c=0" "6#/,(*&%\"aG\"\"\"*$%\"xG\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF'\"\"! " }{TEXT -1 6 " with " }{XPPEDIT 18 0 "a<>0" "6#0%\"aG\"\"!" }{TEXT -1 19 " has the solutions:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x=-b/(2*a)" "6#/%\"xG,$*&%\"bG\"\"\"*&\"\"#F(%\"aGF(!\" \"F," }{TEXT -1 1 " " }{TEXT 318 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(b^2-4*a*c)/(2*a)" "6#*&-%%sqrtG6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F+% \"aGF+%\"cGF+!\"\"F+*&F*F+F.F+F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 82 "Maple can obtain these solutions which constitute the sta ndard quadratic formula: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(a*x^2+b*x+c=0,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "If you want mor e information concerning the procedure " }{TEXT 259 5 "solve" }{TEXT -1 52 " this can be obtained from the associated help page." }}{PARA 0 "" 0 "" {TEXT -1 38 "Three ways to access the help page on " }{TEXT 0 5 "solve" }{TEXT -1 6 " are: " }}{PARA 15 "" 0 "" {TEXT -1 21 "Click the hyperlink: " }{HYPERLNK 17 "solve" 2 "solve" "" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 28 "Highlight (select) the word " }{TEXT 258 5 "solve" }{TEXT -1 27 " with the mouse and choose " }{TEXT 319 15 "Help on \"solve\"" }{TEXT -1 58 " which should appear in the secon d line of the help menu. " }}{PARA 15 "" 0 "" {TEXT -1 5 "Type " } {TEXT 259 6 "?solve" }{TEXT -1 79 " on a command line (a semi-colon is not needed) and then execute this command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "?solve" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "It may be possibl e to solve a quadratic equation by using the procedure " }{TEXT 259 6 "factor" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(6*x^2+x-2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The solutions of the equation " }{XPPEDIT 18 0 "6*x^2+x-2=0" "6#/,(*&\"\"'\"\"\"*$%\"xG\" \"#F'F'F)F'F*!\"\"\"\"!" }{TEXT -1 29 " therefore has the solutions " }{XPPEDIT 18 0 "x = 1/2;" "6#/%\"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = -2/3;" "6#/%\"xG,$*&\"\"#\"\"\"\"\"$!\"\"F* " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "We could solve the eq uation " }{XPPEDIT 18 0 "6*x^2+x-2 = 0" "6#/,(*&\"\"'\"\"\"*$%\"xG\"\" #F'F'F)F'F*!\"\"\"\"!" }{TEXT -1 95 " in two steps as follows to imita te steps which could be used to solve this equation \"by hand\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "6*x^2+x-2=0;\nfactor(%);\nsolve(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{TEXT 0 5 "solve" }{TEXT -1 123 " gives the solutions as a sequence with two members, it is pos sible to save the solutions for future reference by making a " }{TEXT 266 17 "double assignment" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eq := 6*x^2+x-2=0; \nx1,x2 := solve(eq);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The solutions can be checked in the original equatio n with the aid of the procedure " }{TEXT 0 4 "subs" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(x=x1,eq);\nsubs(x=x2,eq);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 47 ": Of course, if w e try subsituting a value for " }{TEXT 327 1 "x" }{TEXT -1 16 " differ ent from " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-2/3" "6#,$*&\"\"#\"\"\"\"\"$!\"\"F(" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "6*x^2+x-2 = 0" "6#/,(*&\"\"' \"\"\"*$%\"xG\"\"#F'F'F)F'F*!\"\"\"\"!" }{TEXT -1 41 ", we find that t he left side is not zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=3/4,eq);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 4 "subs" }{TEXT -1 15 " is similar to " }{TEXT 0 4 "eval" }{TEXT -1 57 ", but the order of the two input parameters is reversed. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(eq,x=3/4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 99 "In a case where the factorization cannot be performed u sing rational numbers, the second argument \"" }{TEXT 0 4 "real" } {TEXT -1 14 "\" can be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(2*x^2-x-4,real);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Try this " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 27 " be the quartic polynomial " }{XPPEDIT 18 0 "4*x^4+8*x^3-15*x^2 -14*x+14" "6#,,*&\"\"%\"\"\"*$%\"xGF%F&F&*&\"\")F&*$F(\"\"$F&F&*&\"#:F &*$F(\"\"#F&!\"\"*&\"#9F&F(F&F1F3F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 14 "(a) Show that " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG " }{TEXT -1 30 " has the two quadratic factors" }{XPPEDIT 18 0 " ``(4* x^2-7)" "6#-%!G6#,&*&\"\"%\"\"\"*$%\"xG\"\"#F)F)\"\"(!\"\"" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(x^2+2*x-2)" "6#-%!G6#,(*$%\"xG\"\"#\"\"\" *&F)F*F(F*F*F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "(b ) Solve the equation " }{XPPEDIT 18 0 "p(x)=0" "6#/-%\"pG6#%\"xG\"\"! " }{TEXT -1 101 " giving exact expressions for the solutions and also \+ obtain decimal approximations for the solutions." }}{PARA 0 "" 0 "" {TEXT -1 22 "(c) Plot the graph of " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6# %\"xG" }{TEXT -1 18 " to show that the " }{TEXT 322 1 "x" }{TEXT -1 57 "-intercepts correspond to the solutions obtained in (b). " }} {PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "p := x -> 4*x^4+8*x^3-15*x^2-14*x+1 4;\nfactor(p(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(p(x )=0);\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(p(x) ,x=-3..2,y=-20..35);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Clicking the " }{TEXT 320 1 "x" }{TEXT -1 55 "-interce pts with the mouse shows that they occur where " }{TEXT 321 1 "x" } {TEXT -1 56 " is approximately equal to -2.74, -1.32, 0.72 and 1.32. \+ " }}{PARA 0 "" 0 "" {TEXT -1 85 "These values correspond roughly to th e values calculated in (b) for the solutions of " }{XPPEDIT 18 0 "p(x) =0" "6#/-%\"pG6#%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 271 10 "EXPERIMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Sample graphs" }} {SECT 1 {PARA 4 "" 0 "" {TEXT 260 9 "Graph of " }{XPPEDIT 18 0 "y = x^ 2-2*x-3;" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"*&F(F)F'F)!\"\"\"\"$F+" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "If you wa nt to make repeated reference to an expression, you can give it a name , using " }{TEXT 259 2 ":=" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "x := 'x':\nexpr1 : = x^2-2*x-3;\nplot(expr1,x=-3..5,color=blue);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Specific values can be co mputed if desired." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "subs(x=4,expr1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 279 1 "x" }{TEXT -1 85 " intercepts can easily be read from the graph, but they can als o be calculated using " }{TEXT 259 5 "solve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "so lve(expr1,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 282 11 "Grap hs of " }{XPPEDIT 18 0 "y = x+1;" "6#/%\"yG,&%\"xG\"\"\"F'F'" }{TEXT 283 7 " and " }{XPPEDIT 18 0 "x+y = 5;" "6#/,&%\"xG\"\"\"%\"yGF&\"\" &" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "x := 'x':\nplot([x+1,5-x],x=-2..6,color=[red,blue]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 261 11 "Graphs of " }{XPPEDIT 18 0 "y = sin(x);" "6#/%\"yG-%$sinG6#%\"xG" }{TEXT 262 7 " and " }{XPPEDIT 18 0 "y = cos(x);" "6#/%\"yG-%$cosG6#%\"xG" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "plot([sin(x),cos(x)],x=-Pi..4*Pi,color=[red,blue],thickness=2,\n \+ legend=[`y=sin(x)`,`y=cos(x)`],\n title=`The graphs of the sine \+ and cosine functions`);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 264 10 "Graph of " }{XPPEDIT 18 0 "x^2+y^2 = 4; " "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(\"\"%" }{TEXT 265 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "I n order to graph an equation in which " }{TEXT 280 1 "y" }{TEXT -1 12 " is defined " }{TEXT 266 10 "implicitly" }{TEXT -1 18 " as a function of " }{TEXT 281 1 "x" }{TEXT -1 16 ", the procedure " }{TEXT 259 12 " implicitplot" }{TEXT -1 14 " can be used. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 259 12 "implicitplot" }{TEXT -1 102 " is not immediately available as it belongs to a collection of supplement ary plotting procedures in a " }{TEXT 266 13 "Maple package" }{TEXT -1 15 " with the name " }{TEXT 259 5 "plots" }{TEXT -1 8 ". The " } {TEXT 259 5 "plots" }{TEXT -1 48 " package can be loaded by executing \+ the command " }{TEXT 267 12 "with(plots);" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 82 "If you want to see the various procedures which t he package contains, replace the " }{TEXT 259 1 ":" }{TEXT -1 3 "by " }{TEXT 259 1 ";" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "Using \+ a colon " }{TEXT 259 1 ":" }{TEXT -1 55 " at the end of a Maple comman d instead of a semi-colon " }{TEXT 259 1 ";" }{TEXT -1 37 " causes the output to be suppressed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 74 "x := 'x': y := 'y':\nimplicitplot(x^2+y^2=4,x= -2..2,y=-2..2,color=magenta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 191 "To see the graph as a genuine circle, select the graph by clickin g it, and click the 1:1 button on the context bar, or go to the projec tion menu and choose the projection to be 'constrained'." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 45 ": An alternative way to access the procedure " }{TEXT 259 12 "impl icitplot" }{TEXT -1 27 " without loading the whole " }{TEXT 259 5 "plo ts" }{TEXT -1 67 " package containing various plotting procedures is t o use the name " }{TEXT 259 19 "plots[implicitplot]" }{TEXT -1 21 ", w hich has the form " }{TEXT 267 53 " .. name of package .. [ .. procedu re in package .. ]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plots[implicitplot](x^2+y^2=4,x=-2..2,y=-2. .2,color=magenta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT 263 11 "Graph of " }{XPPEDIT 18 0 "y \+ = 1/(1-x^2);" "6#/%\"yG*&\"\"\"F&,&F&F&*$%\"xG\"\"#!\"\"F+" }}{PARA 0 "" 0 "" {TEXT -1 121 "Since this graph has vertical asymptotes, it is \+ necessary to specify a restriction on the values which are to be plott ed." }}{PARA 0 "" 0 "" {TEXT -1 40 "In the following only points with \+ their " }{TEXT 358 1 "y" }{TEXT -1 42 " coordinates between -5 and 5 a re plotted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "plot(1/(1-x^2),x=-3..3,y=-5..5,color=green);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Vertical \+ green lines appear where " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\" \"" }{TEXT -1 11 " and where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" } {TEXT -1 116 " so that it looks as though the asympotes are part of th e graph. This happens because of the way that the procedure " }{TEXT 259 4 "plot" }{TEXT -1 172 " constructs the graph by joining points wi th straight line segments. It is as if you tried to draw the graph wit h pencil and paper without taking the pencil off the paper. " }}{PARA 0 "" 0 "" {TEXT -1 19 "Adding the option \"" }{TEXT 267 12 "discont=tr ue" }{TEXT -1 39 "\" avoids getting the vertical lines at " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 76 " by breaking up the graph \+ into three separate pieces at the discontinuities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(1/(1-x^ 2),x=-3..3,y=-5..5,color=green,discont=true);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "The asymptotes may be ad ded back to the graph in different colour and also drawn in some form \+ of broken line via the \"" }{TEXT 267 9 "linestyle" }{TEXT -1 9 "\" op tion." }}{PARA 0 "" 0 "" {TEXT -1 12 "The option \"" }{TEXT 267 13 "li nestyle = 3" }{TEXT -1 35 "\" makes the asymptotes appear as a " } {TEXT 349 11 "dashed line" }{TEXT -1 51 ". The other choices for the l inestyle option are: \"" }{TEXT 267 13 "linestyle = 1" }{TEXT -1 17 " \", which gives a " }{TEXT 348 10 "solid line" }{TEXT -1 25 " (This is the default), \"" }{TEXT 267 13 "linestyle = 2" }{TEXT -1 17 "\", whi ch gives a " }{TEXT 346 11 "dotted line" }{TEXT -1 7 ", and \"" } {TEXT 267 14 "linestyle = 4" }{TEXT -1 15 "\", which gives " }{TEXT 347 25 "alternate dashes and dots" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "It is assumed that you have loaded the " }{TEXT 259 5 "pl ots" }{TEXT -1 18 " package, so that " }{TEXT 259 12 "implicitplot" } {TEXT -1 5 " and " }{TEXT 259 7 "display" }{TEXT -1 13 " can be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "cu rve := plot(1/(1-x^2),x=-3..3,y=-5..5,color=red,discont=true):\nasympt otes := implicitplot(\{x=-1,x=1\},x=-1..1,y=-5..5,color=grey,linestyle =3):\ndisplay([curve,asymptotes]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "For more infor mation concerning the " }{TEXT 259 4 "plot" }{TEXT -1 17 " procedure, \+ type " }{TEXT 259 5 "?plot" }{TEXT -1 29 ", on a command line or click " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Type " }{TEXT 259 14 "?plot[options]" }{TEXT -1 28 " on a \+ command line or click " }{HYPERLNK 17 "plot[options]" 2 "plot[options] " "" }{TEXT -1 58 " to obtain information on options which can be adde d to a " }{TEXT 0 4 "plot" }{TEXT -1 10 " command \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 " Try this " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "2^x=x^4" "6#/)\"\"#%\"xG*$F&\"\"%" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "x=16" "6#/%\"xG\"#;" }{TEXT -1 37 " as one of its solutions sinc e, when " }{XPPEDIT 18 0 "x=16" "6#/%\"xG\"#;" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "x^4=16^4" "6#/*$%\"xG\"\"%*$\"#;F&" }{XPPEDIT 18 0 "``=(2^4)^4" "6#/%!G*$*$\"\"#\"\"%F(" }{XPPEDIT 18 0 "``=(2^4)*(2^4)*( 2^4)*(2^4)" "6#/%!G**\"\"#\"\"%F&F'F&F'F&F'" }{XPPEDIT 18 0 "``=2^16" "6#/%!G*$\"\"#\"#;" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "Do es this equation have any other solutions? " }}{PARA 0 "" 0 "" {TEXT -1 33 "Plot the graphs of the equations " }{XPPEDIT 18 0 "y=x^4" "6#/% \"yG*$%\"xG\"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=2^x" "6#/%\"yG )\"\"#%\"xG" }{TEXT -1 35 " in the same diagram with suitable " } {TEXT 345 1 "x" }{TEXT -1 5 " and " }{TEXT 328 1 "y" }{TEXT -1 57 " ra nges to show points of intersection of the two graphs." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 329 1 "x" }{TEXT -1 78 " coordinates of \+ any points of intersection provide solutions for the equation " } {XPPEDIT 18 0 "2^x=x^4" "6#/)\"\"#%\"xG*$F&\"\"%" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Try to \" zoom in\" close to any points of intersection so that you can estimate the solutions of " }{XPPEDIT 18 0 "2^x = x^4" "6#/)\"\"#%\"xG*$F&\"\" %" }{TEXT -1 22 " correct to 3 figures." }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "There ar e two solutions other than " }{XPPEDIT 18 0 "x=16" "6#/%\"xG\"#;" } {TEXT -1 40 " as indicated by the following picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([x^4 ,2^x],x=-2..2,y=0..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "By clicking at the points of intersection we see tha t there are solutions with " }{TEXT 333 1 "x" }{TEXT -1 1 " " }{TEXT 332 1 "~" }{TEXT -1 11 " -0.86 and " }{TEXT 340 1 "x" }{TEXT -1 1 " " }{TEXT 339 1 "~" }{TEXT -1 25 " 1.25. The corresponding " }{TEXT 337 1 "y" }{TEXT -1 12 " values are " }{TEXT 342 1 "y" }{TEXT -1 1 " " } {TEXT 341 1 "~" }{TEXT -1 10 " 0.55 and " }{TEXT 338 1 "y" }{TEXT -1 1 " " }{TEXT 336 1 "~" }{TEXT -1 19 " 2.36 respectively." }}{PARA 0 " " 0 "" {TEXT -1 33 "By constructing plots with small " }{TEXT 334 1 "x " }{TEXT -1 5 " and " }{TEXT 335 1 "y" }{TEXT -1 104 " ranges which in clude these values we see that the solutions are closer to -0.861 and \+ 1.24 respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 45 "plot([x^4,2^x],x=-0.865..-0.86,y=0.54..0.56) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot([x^4,2^x],x=1.23..1.25,y=2.3..2.4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "It is impossible \+ to choose suitable scales for " }{TEXT 330 1 "x" }{TEXT -1 5 " and " } {TEXT 331 1 "y" }{TEXT -1 59 " to show all three points of intersectio n of the graphs of " }{XPPEDIT 18 0 "y=x^4" "6#/%\"yG*$%\"xG\"\"%" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y=2^x" "6#/%\"yG)\"\"#%\"xG" } {TEXT -1 21 " in the one picture. " }}{PARA 0 "" 0 "" {TEXT -1 36 "How ever, by converting the equation " }{XPPEDIT 18 0 "2^x=x^4" "6#/)\"\"# %\"xG*$F&\"\"%" }{TEXT -1 23 " to the logarithm form " }{XPPEDIT 18 0 "x=4*log[2](x)" "6#/%\"xG*&\"\"%\"\"\"-&%$logG6#\"\"#6#F$F'" }{TEXT -1 46 " we can illustrate the two positive solutions " }{TEXT 344 1 "x " }{TEXT -1 1 " " }{TEXT 343 1 "~" }{TEXT -1 10 " 1.24 and " } {XPPEDIT 18 0 "x=16" "6#/%\"xG\"#;" }{TEXT -1 45 " as points of inters ection of the two graphs " }{XPPEDIT 18 0 "y=log[4](x)" "6#/%\"yG-&%$l ogG6#\"\"%6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=x" "6#/%\"yG% \"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot([4*log[2](x),x],x=0..17,y=-1..17);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 53 ": More accurate numerical solutions for the equation " } {XPPEDIT 18 0 "2^x=x^4" "6#/)\"\"#%\"xG*$F&\"\"%" }{TEXT -1 46 " can b e obtained by using the Maple procedure " }{TEXT 0 6 "fsolve" }{TEXT -1 63 ". The approximate solutions found above can be refined by this \+ " }{TEXT 266 32 "numerical root-finding procedure" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fsolve(x^4=2^x,x=1.24);\nfsolve(x^4=2^x,x=-0.861);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "For more information on " }{TEXT 0 6 "fsolve" }{TEXT -1 7 " click " }{HYPERLNK 17 "fsolve " 2 "fsolve" "" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Some algebraic manipulations: " }{TEXT 259 22 "expand,nor mal,simplify" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 259 6 "expand" }{TEXT -1 44 " will \"multiply out\" bracketed expressions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "expr1 := \+ (2*x-3)*(x+5);\nexpr2 := expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&expr1G*&,&*&\"\"#\"\"\"%\"xGF)F)\"\"$!\"\"F),&F*F)\"\"&F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr2G,(*&\"\"#\"\"\")%\"xGF'F(F(*& \"\"(F(F*F(F(\"#:!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Here the effect of " }{TEXT 259 6 "expand" }{TEXT -1 20 " can be reversed by " }{TEXT 259 6 "factor" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(expr2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&\"\"#\"\" \"%\"xGF'F'\"\"$!\"\"F',&F(F'\"\"&F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Try this " }}{PARA 0 "" 0 "" {TEXT -1 8 "(a) Use " }{TEXT 0 6 "factor" }{TEXT -1 29 " to factorise the polynomial " } {XPPEDIT 18 0 "x^5+x^4-2*x^3-2*x^2+x+1" "6#,.*$%\"xG\"\"&\"\"\"*$F%\" \"%F'*&\"\"#F'*$F%\"\"$F'!\"\"*&F+F'*$F%F+F'F.F%F'F'F'" }{TEXT -1 38 " and check the factorisation by using " }{TEXT 0 6 "expand" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "(b) Plot the graph of " } {XPPEDIT 18 0 "y=" "6#/%\"yG%#%?G" }{XPPEDIT 18 0 "" "6#%#%?G" } {XPPEDIT 18 0 "x^5+x^4-2*x^3-2*x^2+x+1" "6#,.*$%\"xG\"\"&\"\"\"*$F%\" \"%F'*&\"\"#F'*$F%\"\"$F'!\"\"*&F+F'*$F%F+F'F.F%F'F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "(c) Explain how you could have used \+ the result from part (a) to predict that the graph crosses the " } {TEXT 359 1 "x" }{TEXT -1 9 " axis at " }{XPPEDIT 18 0 "x=1" "6#/%\"xG \"\"\"" }{TEXT -1 22 " but just touches the " }{TEXT 360 1 "x" }{TEXT -1 9 " axis at " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 27 " \+ without crossing the axis." }}{PARA 0 "" 0 "" {TEXT -1 39 "___________ ____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "__ _____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "x^5+x^4-2*x^3-2*x^2+x+1;\nfactor(%);\nexpand(%);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(x^5+x^4-2*x^3-2*x^2+x+1,x=-2.. 2,y=-4..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{PARA 0 "" 0 "" {TEXT -1 21 "Since the polynomial " } {XPPEDIT 18 0 "x^5+x^4-2*x^3-2*x^2+x+1" "6#,.*$%\"xG\"\"&\"\"\"*$F%\" \"%F'*&\"\"#F'*$F%\"\"$F'!\"\"*&F+F'*$F%F+F'F.F%F'F'F'" }{TEXT -1 16 " has the factors" }{XPPEDIT 18 0 "``(x+1)" "6#-%!G6#,&%\"xG\"\"\"F(F( " }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(x-1)" "6#-%!G6#,&%\"xG\"\"\"F( !\"\"" }{TEXT -1 39 " the graph of the polynomial meets the " }{TEXT 361 1 "x" }{TEXT -1 12 " axis where " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG, $\"\"\"!\"\"" }{TEXT -1 11 " and where " }{XPPEDIT 18 0 "x=1" "6#/%\"x G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Since the fac tor" }{XPPEDIT 18 0 "``(x+1)" "6#-%!G6#,&%\"xG\"\"\"F(F(" }{TEXT -1 73 " is raised to the power 3 (which is an odd integer) in the factori sation " }{XPPEDIT 18 0 "(x+1)^3*(x-1)^2" "6#*&,&%\"xG\"\"\"F&F&\"\"$, &F%F&F&!\"\"\"\"#" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "x^5+x^4-2*x^3-2* x^2+x+1" "6#,.*$%\"xG\"\"&\"\"\"*$F%\"\"%F'*&\"\"#F'*$F%\"\"$F'!\"\"*& F+F'*$F%F+F'F.F%F'F'F'" }{TEXT -1 7 " while " }{XPPEDIT 18 0 "``(x-1) " "6#-%!G6#,&%\"xG\"\"\"F(!\"\"" }{TEXT -1 71 " is raised to the power 2 (which is even), the value of the polynomial " }{XPPEDIT 18 0 "x^5+ x^4-2*x^3-2*x^2+x+1" "6#,.*$%\"xG\"\"&\"\"\"*$F%\"\"%F'*&\"\"#F'*$F%\" \"$F'!\"\"*&F+F'*$F%F+F'F.F%F'F'F'" }{TEXT -1 17 " changes sign as " } {TEXT 363 1 "x" }{TEXT -1 29 " increases through the value " } {XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 29 " but does n ot change sign as " }{TEXT 362 1 "x" }{TEXT -1 29 " increases through \+ the value " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 14 "The graphs of " }{XPPEDIT 18 0 "y=(x+1)^3 " "6#/%\"yG*$,&%\"xG\"\"\"F(F(\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=(x-1)^2" "6#/%\"yG*$,&%\"xG\"\"\"F(!\"\"\"\"#" }{TEXT -1 34 " can be drawn to illustrate this. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot([(x+1)^3,(x-1)^2],x=-2. 5..2.5,y=-2..4,color=[blue,coral]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 14 "The procedure " }{TEXT 259 6 "normal" }{TEXT -1 90 " performs some basic simplifications putting an algebraic expressi on into a \"normal\" form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "2*x*(x^2-3)+x^2-x^3;\nnormal(%);\nf actor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"#\"\"\"%\"xGF&,&* $)F'F%F&F&\"\"$!\"\"F&F&F)F&*$)F'F+F&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"F(*&\"\"'F(F&F(!\"\"*$)F&\"\"#F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"xG\"\"\",&F$F%\"\"$F%F%,&F$F%\"\" #!\"\"F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Algebraic \"fractions\" are combined with a common denominator." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "1/x+4/(x-3);\nnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*& \"\"\"F%%\"xG!\"\"F%*&\"\"%F%,&F&F%\"\"$F'F'F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*&\"\"&\"\"\"%\"xGF'F'\"\"$!\"\"F'F(F*,&F(F'F)F*F* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Try this " } }{PARA 0 "" 0 "" {TEXT -1 12 "(a) Express " }{XPPEDIT 18 0 "1/x+1/(x+1 )+1/(x+2)+1/(x+3)" "6#,**&\"\"\"F%%\"xG!\"\"F%*&F%F%,&F&F%F%F%F'F%*&F% F%,&F&F%\"\"#F%F'F%*&F%F%,&F&F%\"\"$F%F'F%" }{TEXT -1 52 " as single r ational expression (algebraic fraction)." }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) Solve the equation " }{XPPEDIT 18 0 "1/x+1/(x+1)+1/(x+2)+1/(x+ 3)=0" "6#/,**&\"\"\"F&%\"xG!\"\"F&*&F&F&,&F'F&F&F&F(F&*&F&F&,&F'F&\"\" #F&F(F&*&F&F&,&F'F&\"\"$F&F(F&\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 106 "(c) Give a sequence of Maple commands to indicate how you could solve the equation in part (b) \"by hand\"." }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "1/x+1/(x+1)+1/(x+2)+1/(x+3);\nnorma l(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " (b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "1/x+1/(x+1)+1/(x+2)+ 1/(x+3);\nsolve(%);\nevalf(evalf(%,15));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The solutions are " }{XPPEDIT 18 0 "x=-3/2" "6#/%\"xG,$*&\"\"$\"\"\"\"\"#!\"\"F*" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``=-1.5" "6#/%!G,$-%&FloatG6$\"#:!\"\"F*" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "x=(sqrt(5)-3)/2" "6#/%\"xG*&,&-%%sqrtG6#\"\"&\"\" \"\"\"$!\"\"F+\"\"#F-" }{TEXT -1 1 " " }{TEXT 365 1 "~" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "-0" "6#,$\"\"!!\"\"" }{TEXT -1 15 ".381966013 and \+ " }{XPPEDIT 18 0 "x=-(sqrt(5)+3)/2" "6#/%\"xG,$*&,&-%%sqrtG6#\"\"&\"\" \"\"\"$F,F,\"\"#!\"\"F/" }{TEXT -1 1 " " }{TEXT 366 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-2.618033989;" "6#,$-%&FloatG6$\"+*)R.=E!\"*!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "1/x+1/(x +1)+1/(x+2)+1/(x+3)=0;\nnormal(%);\nfactor(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The factor" }{XPPEDIT 18 0 "``(2*x+3)" "6#-%!G6#,&*&\"\"#\"\"\"%\"xGF)F)\"\"$F)" }{TEXT -1 40 " of the numerator leads to the solution " }{XPPEDIT 18 0 "x=-3/2" "6#/ %\"xG,$*&\"\"$\"\"\"\"\"#!\"\"F*" }{TEXT -1 84 " while the quadratic f actor of the numerator gives rise to the other two solutions. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x^2+3*x+1=0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 20 "Using the procedure " }{TEXT 259 8 "simplify" }{TEXT -1 13 " in place of " }{TEXT 259 6 "normal" }{TEXT -1 53 " gives the s ame results in the previous two examples." }}{PARA 0 "" 0 "" {TEXT -1 8 "However " }{TEXT 259 8 "simplify" }{TEXT -1 79 " will also make use of rules involving specific functions such as the identity " }{TEXT 267 19 "cos(x)^2+sin(x)^2=1" }{TEXT -1 55 " and rules for exponents in the simplification process." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "3*sin(x)^2+cos(x)^2+exp(x)*e xp(2*x);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"$\" \"\")-%$sinG6#%\"xG\"\"#F&F&*$)-%$cosGF*F,F&F&*&-%$expGF*F&-F36#,$*&F, F&F+F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"$\"\"\"*&\"\"#F% )-%$cosG6#%\"xGF'F%!\"\"-%$expG6#,$*&F$F%F,F%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Try this " }}{PARA 0 "" 0 "" {TEXT -1 9 "Simplify " }{XPPEDIT 18 0 "exp(cos(x)^2)*exp(sin(x)^2);" "6#*&-%$expG 6#*$-%$cosG6#%\"xG\"\"#\"\"\"-F%6#*$-%$sinG6#F+F,F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "exp(cos(x)^2)/exp(sin(x)^2);" "6#*&-%$expG6#*$-%$co sG6#%\"xG\"\"#\"\"\"-F%6#*$-%$sinG6#F+F,!\"\"" }{TEXT -1 24 " by using the procedure " }{TEXT 0 8 "simplify" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "It is necessary to apply " }{TEXT 0 8 "simplify" }{TEXT -1 10 " twice to " }{XPPEDIT 18 0 "exp(cos(x)^2)*exp(sin(x)^2);" "6#*& -%$expG6#*$-%$cosG6#%\"xG\"\"#\"\"\"-F%6#*$-%$sinG6#F+F,F-" }{TEXT -1 46 " (with Maple 9) to obtain the expected result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "exp(cos(x)^2 )*exp(sin(x)^2);\nsimplify(%);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#*$)-%$sinG6#%\"xG\"\"#\"\"\"F.-F%6#*$)-%$cos GF+F-F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,&*$)-%$sinG6#% \"xG\"\"#\"\"\"F.*$)-%$cosGF+F-F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$expG6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Again two applications of " }{TEXT 0 8 "simplify" }{TEXT -1 61 " are needed to make a good job of simplifying the expression " }{XPPEDIT 18 0 "exp(cos(x)^2)/exp(sin(x)^2)" "6#*&-%$expG6#*$-%$cosG6# %\"xG\"\"#\"\"\"-F%6#*$-%$sinG6#F+F,!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "exp( cos(x)^2)/exp(sin(x)^2);\nsimplify(%);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#*$)-%$cosG6#%\"xG\"\"#\"\"\"F.-F%6#*$)-%$ sinGF+F-F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#*&,&-%$cos G6#%\"xG\"\"\"-%$sinGF*!\"\"F,,&F-F,F(F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,&*&\"\"#\"\"\")-%$cosG6#%\"xGF(F)F)F)!\"\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" } {TEXT -1 64 ": A simpler result can be obtained by using the Maple pro cedure " }{TEXT 0 7 "combine" }{TEXT -1 45 " which makes use of the do uble angle formula " }{XPPEDIT 18 0 "cos(x)^2-sin(x)^2=cos(2*x)" "6#/, &*$-%$cosG6#%\"xG\"\"#\"\"\"*$-%$sinG6#F)F*!\"\"-F'6#*&F*F+F)F+" } {TEXT -1 5 " or " }{XPPEDIT 18 0 "2*cos(x)^2-1=cos(2*x)" "6#/,&*&\"\" #\"\"\"*$-%$cosG6#%\"xGF&F'F'F'!\"\"-F*6#*&F&F'F,F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "For information on " }{TEXT 0 7 "combin e" }{TEXT -1 7 " click " }{HYPERLNK 17 "combine" 2 "combine" "" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "exp(cos(x)^2)/exp(sin(x)^2);\ncombine(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#*$)-%$cosG6#%\"xG\"\"#\"\" \"F.-F%6#*$)-%$sinGF+F-F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$e xpG6#-%$cosG6#,$*&\"\"#\"\"\"%\"xGF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Differentiation and int egration: " }{TEXT 259 23 "diff,int,Diff,Int,value" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Maple can perform symbolic differentiation and integration." }}{PARA 0 "" 0 "" {TEXT -1 15 "The derivative " }{XPPEDIT 18 0 "Diff(x^2*sin(x),x)" "6#- %%DiffG6$*&%\"xG\"\"#-%$sinG6#F'\"\"\"F'" }{TEXT -1 37 " can be found \+ by using the procedure " }{TEXT 259 4 "diff" }{TEXT -1 6 " with " } {TEXT 259 10 "x^2*sin(x)" }{TEXT -1 8 " as the " }{TEXT 258 14 "first \+ argument" }{TEXT -1 39 " (or input parameter) and the variable " } {TEXT 259 1 "x" }{TEXT -1 84 " (the variable with respect to which the differentiation is to be performed) as the " }{TEXT 258 15 "second ar gument" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x^2*sin(x);\ndiff(%,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "It is often con venient to use the procedure " }{TEXT 259 4 "Diff" }{TEXT -1 21 " (wit h an upper case " }{TEXT 259 1 "D" }{TEXT -1 35 ") initially. This is \+ the so-called " }{TEXT 266 10 "inert form" }{TEXT -1 118 " of the diff entiation procedure which simply \"sets up\" the symbolic form of the \+ unavaluated derivative. The procedure " }{TEXT 259 5 "value" }{TEXT -1 50 " can then be used to obtain the actual derivative." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Diff( x^2*sin(x),x);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 109 "You can ev en use one of the following input schemes to make the Maple output loo k like the written statement " }{XPPEDIT 18 0 "Diff(x^2*sin(x),x)=2*x* sin(x)+x^2*cos(x)" "6#/-%%DiffG6$*&%\"xG\"\"#-%$sinG6#F(\"\"\"F(,&*(F) F-F(F--F+6#F(F-F-*&F(F)-%$cosG6#F(F-F-" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Diff(x^2*sin(x),x)=diff(x^2*sin(x), x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(x^2*sin(x),x);\n``=value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 284 25 "Note on using the symbol " }{TEXT 259 2 "``" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 125 "Placing quotes \+ around any group of keyboard symbols enables Maple to use the resultin g expression for the name of a variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "`%^&*` := 5+3;\n`%^&* `;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Th e symbol " }{TEXT 259 2 "``" }{TEXT -1 52 " used above is consequently an \"invisible\" variable." }}{PARA 0 "" 0 "" {TEXT -1 55 "Maple make s use of this \"trick\" in the output given by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ifactor(77); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "If \+ you select this output and look at the window in the context bar you w ill see that this output is produced by the Maple expression " }{TEXT 259 12 "``(7)*``(11)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 " To show the " }{TEXT 258 14 "round brackets" }{TEXT -1 86 " Maple uses function notation in which the function has the \"invisible\" functio n name " }{TEXT 259 2 "``" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "``(7)*``(11);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Here is \+ a challenge for you. What is the purpose of the 2nd, 3rd and 4th comma nds in the following sequence of commands?" }}{PARA 0 "" 0 "" {TEXT 266 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 1 " " } {TEXT 259 2 "%%" }{TEXT -1 40 " refers to the last-but-one Maple resul t" }}{PARA 15 "" 0 "" {TEXT -1 14 "Using a colon " }{TEXT 259 1 ":" } {TEXT -1 26 " in place of a semi-colon " }{TEXT 259 1 ";" }{TEXT -1 49 " at the end of a command suppresses the output). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "``(7)*``( 11);\n``:=z->z:\nmap(%,%%);\n``:='``':" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "If the expession to be d ifferentiated involves more than one variable, then the output from th e procedure " }{TEXT 259 4 "Diff" }{TEXT -1 24 " is displayed using th e " }{TEXT 258 18 "partial derivative" }{TEXT -1 8 " symbol." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Di ff(a*x^2,x);\n``=value(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The partial derivative " }{XPPEDIT 18 0 "Diff(a *x^2,x)" "6#-%%DiffG6$*&%\"aG\"\"\"*$%\"xG\"\"#F(F*" }{TEXT -1 22 " is the derivative of " }{XPPEDIT 18 0 "a*x^2" "6#*&%\"aG\"\"\"*$%\"xG\" \"#F%" }{TEXT -1 17 " with respect to " }{TEXT 286 1 "x" }{TEXT -1 6 " with " }{TEXT 287 1 "a" }{TEXT -1 23 " treated as a constant." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 259 3 "int" }{TEXT -1 65 " performs integration. T o integrate with respect to the variable " }{TEXT 259 1 "x" }{TEXT -1 29 " provide int with the symbol " }{TEXT 259 1 "x" }{TEXT -1 28 " as \+ its 2nd input parameter." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x^3+cos(5*x);\nint(%,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Note that Maple does not provide a constant of integration. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The inert form of the pro cedure is " }{TEXT 259 3 "Int" }{TEXT -1 20 " with an upper case " } {TEXT 259 1 "I" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Int(x^3+cos(5*x),x);\nvalue( %);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "I f Maple does not \"know\" of a closed symbolic form for the integral i t is left " }{TEXT 258 11 "unevaluated" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Int(1/( x+exp(x)),x);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 34 "The interval of integration for a " }{TEXT 258 17 "definite integral" }{TEXT -1 41 " can be provided by using the same M aple " }{TEXT 258 5 "range" }{TEXT -1 29 " stucture is is used for the " }{TEXT 259 4 "plot" }{TEXT -1 12 " procedure. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Int(x^2,x=0. .2);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 99 "The following command s draw a picture to illustrate the region whose area is given by the i ntegral " }{XPPEDIT 18 0 "Int(x^2,x=0..2)" "6#-%$IntG6$*$%\"xG\"\"#/F' ;\"\"!F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "plot1 := plot(x^2,x=0..2.2,thickne ss=2,color=red):\nplot2 := plot(x^2,x=0..2,color=grey,filled=true):\np lot3 := plot([[2,0],[2,4]],color=black):\nplots[display]([plot1,plot2, plot3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Try this " } }{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x) = (4*x+1)/(3* x^2-x+1);" "6#/-%\"fG6#%\"xG*&,&*&\"\"%\"\"\"F'F,F,F,F,F,,(*&\"\"$F,*$ F'\"\"#F,F,F'!\"\"F,F,F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "(a) Find the derivative " }{XPPEDIT 18 0 "`f '`(x);" "6#-%$f~'G6#% \"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) Solve the eq uation " }{XPPEDIT 18 0 "`f '`(x) = 0;" "6#/-%$f~'G6#%\"xG\"\"!" } {TEXT -1 67 " and hence find the coordinates of the turning points on \+ the graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "(c) Plot the graph of " } {XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 28 " to show \+ the turning points." }}{PARA 0 "" 0 "" {TEXT -1 9 "(d) Find " } {XPPEDIT 18 0 "Int(f(x),x=0..2)" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!\" \"#" }{TEXT -1 96 " in exact (analytical) form and also find a numeric al (decimal) approximation for the integral. " }}{PARA 0 "" 0 "" {TEXT -1 63 "(e) Draw a picture to illustrate the region under the gra ph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 49 " which has area given by this definite integral. " }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 29 "First we set up the function " }{XPPEDIT 18 0 "f(x)=(4*x+ 1)/(3*x^2-x+1)" "6#/-%\"fG6#%\"xG*&,&*&\"\"%\"\"\"F'F,F,F,F,F,,(*&\"\" $F,*$F'\"\"#F,F,F'!\"\"F,F,F2" }{TEXT -1 34 " and then we find its der ivative. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "f := x -> (4*x+1)/(3*x^2-x+1);\nDiff(f(x),x);\nvalue( %);\nderiv := normal(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 324 1 "x" }{TEXT -1 63 " coordinates of the turning points can be obta ined as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "x1,x2 := solve(deriv=0);\nevalf(%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The coord inates of the turning points are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "[x1,f(x1)];\nevalf(%); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "[x2,f(x2)];\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-4..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "(d) We can evaluate " }{XPPEDIT 18 0 "Int(f(x),x=0..2)" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 13 " as follows. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(f(x ),x=0..2);\nvalue(%);\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 62 "(e) The area of the shaded region in the \+ following picture is " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 2)" "6#-%$Int G6$-%\"fG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 185 "plot1 := plot(f(x),x=0..2.2,thickness=2,color =red):\nplot2 := plot(f(x),x=0..2,color=wheat,filled=true):\nplot3 := \+ plot([[2,0],[2,f(2)]],color=black):\nplots[display]([plot1,plot2,plot3 ]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "The limit of a sequence and the limit of a function: \+ " }{TEXT 259 11 "limit,Limit" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Limits of sequences and limits \+ of functions can be found by using the Maple procedure " }{TEXT 0 5 "l imit" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Example 1: The limit of a sequence " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the sequ ence:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2,3/4,7/8, 15/16,31/32,` . . . `,1-1/(2^n),` . . . `;" "6**&\"\"\"F$\"\"#!\"\"*& \"\"$F$\"\"%F&*&\"\"(F$\"\")F&*&\"#:F$\"#;F&*&\"#JF$\"#KF&%(~.~.~.~G,& F$F$*&F$F$)F%%\"nGF&F&F3" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 304 1 "n" }{TEXT -1 9 " th term " }{XPPEDIT 18 0 "a [n]" "6#&%\"aG6#%\"nG" }{TEXT -1 29 " of the sequence is given by " } {XPPEDIT 18 0 "a[n] = 1-1/(2^n);" "6#/&%\"aG6#%\"nG,&\"\"\"F)*&F)F))\" \"#F'!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "This sequ ence converges to 1." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(a[n],n = infinity) = Limit(``(1-1/(2^n)),n = infinity);" "6#/ -%&LimitG6$&%\"aG6#%\"nG/F*%)infinityG-F%6$-%!G6#,&\"\"\"F3*&F3F3)\"\" #F*!\"\"F7/F*F," }{TEXT -1 6 " = 1. " }}{PARA 0 "" 0 "" {TEXT -1 14 "T he procedure " }{TEXT 0 5 "limit" }{TEXT -1 21 " can find this value. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "1-1/2^n;\nlimit(%,n=infinity);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The inert (unevaluated) form of the \+ limit procedure is " }{TEXT 0 5 "Limit" }{TEXT -1 20 " with an upper c ase " }{TEXT 267 1 "L" }{TEXT -1 63 ", so the previous calculation can be set up in Maple as . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Limit(1-1/2^n,n=infinity);\n value(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The following code draws a picture to illustrate the first few terms of this sequence by plotting the points" }{XPPEDIT 18 0 "``(n,1 -1/(2^n));" "6#-%!G6$%\"nG,&\"\"\"F(*&F(F()\"\"#F&!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "pts := [seq([n,1-1/2^n],n=1..8)];\nplot([1-1/2^n,1,p ts],n=0..8,0..1.1,style=[line,line,point],\n color=[grey,brown,red] ,linestyle=[2,3],symbol=circle);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 34 "Example 2: The limit of a function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x) = (x^3-8)/(x-2);" "6#/-%\"fG6#%\"xG*&,&*$F'\"\"$\"\"\"\"\") !\"\"F,,&F'F,\"\"#F.F." }{TEXT -1 19 " has no value when " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 9 ", but as " }{TEXT 307 1 "x" } {TEXT -1 29 " approaches 2, the values of " }{TEXT 306 1 "x" }{TEXT -1 34 " become progresively closer to 12." }}{PARA 0 "" 0 "" {TEXT -1 8 "Indeed, " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 72 " ca n attain a value which is arbitrarily close to 12 obtained by taking \+ " }{TEXT 308 1 "x" }{TEXT -1 47 " sufficiently close to 2 (on either s ide of 2)." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "Limi t(f(x),x=2)=12" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"#\"#7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Fo r example, given the sequence of " }{TEXT 309 1 "x" }{TEXT -1 8 " valu es " }{XPPEDIT 18 0 "2.1,2.01,2.001,2.0001,` . . . `;" "6'-%&FloatG6$ \"#@!\"\"-F$6$\"$,#!\"#-F$6$\"%,?!\"$-F$6$\"&,+#!\"%%(~.~.~.~G" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "seq(2.+10^(-n),n=1..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 " the corresponding sequen ce of values " }{XPPEDIT 18 0 "f(2.1),f(2.01),f(2.001),f(2.0001),` . . . `" "6'-%\"fG6#-%&FloatG6$\"#@!\"\"-F$6#-F'6$\"$,#!\"#-F$6#-F'6$\"%, ?!\"$-F$6#-F'6$\"&,+#!\"%%(~.~.~.~G" }{TEXT -1 5 " is: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f := x \+ -> (x^3-8)/(x-2);\nseq(f(2.+10^(-n)),n=1..4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Limit((x^3-8 )/(x-2),x=2);\nvalue(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The value of " }{XPPEDIT 18 0 "Limit((x^3-8)/(x-2),x=2)" "6#-%&LimitG6$*& ,&*$%\"xG\"\"$\"\"\"\"\")!\"\"F+,&F)F+\"\"#F-F-/F)F/" }{TEXT -1 54 " c an by obtained \"by hand\" by using the factorisation " }{XPPEDIT 18 0 "(x^3-8)=(x-2)*(x^2+2*x+4)" "6#/,&*$%\"xG\"\"$\"\"\"\"\")!\"\"*&,&F& F(\"\"#F*F(,(*$F&F-F(*&F-F(F&F(F(\"\"%F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "This means that " }{XPPEDIT 18 0 "f(x)=x^2+2*x+ 4" "6#/-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"*&F*F+F'F+F+\"\"%F+" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x<>2" "6#0%\"xG\"\"#" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "Limit((x^3-8)/(x-2),x = 2)" "6#-%&LimitG6$*&,&*$%\" xG\"\"$\"\"\"\"\")!\"\"F+,&F)F+\"\"#F-F-/F)F/" }{TEXT -1 20 " can be o btained as " }{XPPEDIT 18 0 "Eval(x^2+2*x+4,x=2)" "6#-%%EvalG6$,(*$%\" xG\"\"#\"\"\"*&F)F*F(F*F*\"\"%F*/F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y=(x^3-8)/(x-2)" "6#/% \"yG*&,&*$%\"xG\"\"$\"\"\"\"\")!\"\"F*,&F(F*\"\"#F,F," }{TEXT -1 17 " \+ is the parabola " }{XPPEDIT 18 0 "y=x^2+2*x+4" "6#/%\"yG,(*$%\"xG\"\"# \"\"\"*&F(F)F'F)F)\"\"%F)" }{TEXT -1 25 " with the \"missing point\"" }{XPPEDIT 18 0 "``(2,12)" "6#-%!G6$\"\"#\"#7" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plot([(x^3-8)/(x-2),[[2,12]]],x=-5..3,color=red,\nstyle=[line,poin t],symbol=circle,symbolsize=12);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 52 "Example 3: Finding derivatives from first principles" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The deri vative of the function " }{XPPEDIT 18 0 "f(x)=x^3" "6#/-%\"fG6#%\"xG*$ F'\"\"$" }{TEXT -1 43 " can be found from first principles as f '(" } {TEXT 305 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``=Limit((f(x+h)-f(x)) /h,h=0)" "6#/%!G-%&LimitG6$*&,&-%\"fG6#,&%\"xG\"\"\"%\"hGF/F/-F+6#F.! \"\"F/F0F3/F0\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "Th is can be achieved using Maple as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f := x -> x^3;\nLi mit((f(x+h)-f(x))/h,h=0);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "An intermediate step can be shown by \+ using " }{TEXT 0 6 "expand" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f := x -> x^3;\nLi mit((f(x+h)-f(x))/h,h=0);\nexpand(%);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The simplification " } {XPPEDIT 18 0 "(``(x^3+3*x^2*h+3*x*h^2+h^3)-x^3)/h = (3*x^2*h+3*x*h^2+ h^3)/h;" "6#/*&,&-%!G6#,**$%\"xG\"\"$\"\"\"*(F,F-*$F+\"\"#F-%\"hGF-F-* (F,F-F+F-F1F0F-*$F1F,F-F-*$F+F,!\"\"F-F1F5*&,(*(F,F-*$F+F0F-F1F-F-*(F, F-F+F-F1F0F-*$F1F,F-F-F1F5" }{XPPEDIT 18 0 "``=3*x^2+3*x*h+h^2" "6#/%! G,(*&\"\"$\"\"\"*$%\"xG\"\"#F(F(*(F'F(F*F(%\"hGF(F(*$F-F+F(" }{TEXT -1 28 " is performed automatically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The s um of a series: " }{TEXT 0 7 "sum,Sum" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The (infinite) " } {TEXT 266 6 "series" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[n],n=1..inf inity)=a[1]+a[2]+a[3]+a[4]+` . . . `+a[n]+` . . . `" "6#/-%$SumG6$&%\" aG6#%\"nG/F*;\"\"\"%)infinityG,0&F(6#F-F-&F(6#\"\"#F-&F(6#\"\"$F-&F(6# \"\"%F-%(~.~.~.~GF-&F(6#F*F-F;F-" }{TEXT -1 22 " converges to the sum \+ " }{TEXT 310 1 "S" }{TEXT -1 45 " provided that the sequence of partia l sums: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[1]=a[1]" "6#/&%\"sG6#\"\"\"&%\"aG6#F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[2]=a[1]+a[2]" "6#/&%\"sG6#\"\"#,&&%\" aG6#\"\"\"F,&F*6#F'F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "s[3]=a[1]+a[2]+a[3]" "6#/&%\"sG6#\"\"$,(&%\"aG6#\"\" \"F,&F*6#\"\"#F,&F*6#F'F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[4]=a[1]+a[2]+a[3]+a[4]" "6#/&%\"sG6#\"\"%,*&% \"aG6#\"\"\"F,&F*6#\"\"#F,&F*6#\"\"$F,&F*6#F'F," }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 7 " : " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[n]=a[1]+a[2]+` . . . `+a[n]" "6#/&%\"sG6#%\"nG,*&%\" aG6#\"\"\"F,&F*6#\"\"#F,%(~.~.~.~GF,&F*6#F'F," }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 7 " : " }}{PARA 0 "" 0 "" {TEXT -1 13 "co nverges to " }{TEXT 311 1 "S" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "We write " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity)=S" "6#/-% $SumG6$&%\"aG6#%\"nG/F*;\"\"\"%)infinityG%\"SG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 11 "The finite " }{TEXT 266 16 "geometric ser ies" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[n]=Sum(a*r^(k-1),k=1..n)" "6#/&%\"sG6#%\"nG-%$SumG6$*&%\"aG\"\"\" )%\"rG,&%\"kGF-F-!\"\"F-/F1;F-F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a+a*r+a*r^2+a*r^3+` . . . `+a*r^(n-1);" "6#/%!G,.%\"aG\"\"\"*&F&F'%\" rGF'F'*&F&F'*$F)\"\"#F'F'*&F&F'*$F)\"\"$F'F'%(~.~.~.~GF'*&F&F')F),&%\" nGF'F'!\"\"F'F'" }{TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 16 "with first term " }{TEXT 312 1 "a" }{TEXT -1 18 " and com mon ratio " }{TEXT 313 1 "r" }{TEXT -1 14 ", has the sum " }{XPPEDIT 18 0 "s[n]=a*(1-r^n)/(1-r)" "6#/&%\"sG6#%\"nG*(%\"aG\"\"\",&F*F*)%\"rG F'!\"\"F*,&F*F*F-F.F." }{TEXT -1 45 ", as can be seen by subtracting t he equation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r*s[n ]=a*r+a*r^2+a*r^3+` . . . `+a*r^n" "6#/*&%\"rG\"\"\"&%\"sG6#%\"nGF&,,* &%\"aGF&F%F&F&*&F-F&*$F%\"\"#F&F&*&F-F&*$F%\"\"$F&F&%(~.~.~.~GF&*&F-F& )F%F*F&F&" }{TEXT -1 14 " ------- (ii) " }}{PARA 0 "" 0 "" {TEXT -1 18 "from equation (i)." }}{PARA 0 "" 0 "" {TEXT -1 36 "Hence the infin ite geometric series " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a+a*r+a*r^2+a*r^3+` . . . `+a*r^(n-1)+` . . . `;" "6#,0%\"aG\"\" \"*&F$F%%\"rGF%F%*&F$F%*$F'\"\"#F%F%*&F$F%*$F'\"\"$F%F%%(~.~.~.~GF%*&F $F%)F',&%\"nGF%F%!\"\"F%F%F.F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "converges to " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(s[n],n=infinity) =Limit(a*(1-r^n)/(1-r),n=infinit y)" "6#/-%&LimitG6$&%\"sG6#%\"nG/F*%)infinityG-F%6$*(%\"aG\"\"\",&F1F1 )%\"rGF*!\"\"F1,&F1F1F4F5F5/F*F," }{XPPEDIT 18 0 "``=a/(1-r)" "6#/%!G* &%\"aG\"\"\",&F'F'%\"rG!\"\"F*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 "provided that " }{XPPEDIT 18 0 "abs(r)<1" "6#2-%$absG6#% \"rG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "In this ca se " }{XPPEDIT 18 0 "Sum(a*r^(n-1),n = 1 .. infinity)=a/(1-r)" "6#/-%$ SumG6$*&%\"aG\"\"\")%\"rG,&%\"nGF)F)!\"\"F)/F-;F)%)infinityG*&F(F),&F) F)F+F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 30 "The infinite geometr ic series " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+1/2+1 /4+1/8+` . . . `+1/(2^(n-1))+` . . . `" "6#,0\"\"\"F$*&F$F$\"\"#!\"\"F $*&F$F$\"\"%F'F$*&F$F$\"\")F'F$%(~.~.~.~GF$*&F$F$)F&,&%\"nGF$F$F'F'F$F ,F$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "with first term " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"" }{TEXT -1 18 " and common ratio \+ " }{XPPEDIT 18 0 "r=1/2" "6#/%\"rG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 22 " converges to the sum " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a/(1-r)=1/(1-1/2)" "6#/*&%\"aG\"\"\",&F&F&%\"rG!\"\"F)*&F&F&,&F& F&*&F&F&\"\"#F)F)F)" }{XPPEDIT 18 0 "``=2" "6#/%!G\"\"#" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 14 "In summatiion " }{XPPEDIT 18 0 "Sigm a" "6#%&SigmaG" }{TEXT -1 10 " notation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``(1/(2^(n-1))),n = 1 .. infinity) = 2;" "6#/-%$SumG6$-%!G6#*&\"\"\"F+)\"\"#,&%\"nGF+F+!\"\"F0/F/;F+%)infinityG F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Maple's procedure \+ " }{TEXT 0 3 "sum" }{TEXT -1 34 " can find the sum of this series. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sum(1/(2^(n-1)),n=1..infinity);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "The inert (unevaluated) form of the p rocedure " }{TEXT 0 3 "sum" }{TEXT -1 4 " is " }{TEXT 0 3 "Sum" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Sum(1/(2^(n-1)),n=1..infinity);\nvalue(%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 " " 0 "" {TEXT -1 77 "The infinite repeating decimal 0.333333 . . . repr esents the rational number " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "This infinite dec imal is essentially the infinite geometric series: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3/10+3/(10^2)+3/(10^3)+3/(10^4)+` . \+ . . `+3/(10^n)+` . . . `;" "6#,0*&\"\"$\"\"\"\"#5!\"\"F&*&F%F&*$F'\"\" #F(F&*&F%F&*$F'F%F(F&*&F%F&*$F'\"\"%F(F&%(~.~.~.~GF&*&F%F&)F'%\"nGF(F& F1F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "with first term \+ " }{XPPEDIT 18 0 "a = 3/10;" "6#/%\"aG*&\"\"$\"\"\"\"#5!\"\"" }{TEXT -1 18 " and common ratio " }{XPPEDIT 18 0 "r = 1/10;" "6#/%\"rG*&\"\" \"F&\"#5!\"\"" }{TEXT -1 22 " converges to the sum " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a/(1-r) = 3/10/(1-1/10);" "6#/*&%\"a G\"\"\",&F&F&%\"rG!\"\"F)*(\"\"$F&\"#5F),&F&F&*&F&F&F,F)F)F)" } {XPPEDIT 18 0 "`` = 3/9;" "6#/%!G*&\"\"$\"\"\"\"\"*!\"\"" }{XPPEDIT 18 0 "``=1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 14 "In summatiion " }{XPPEDIT 18 0 "Sigma" "6#%&SigmaG " }{TEXT -1 10 " notation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(``(3/(10^n)),n = 1 .. infinity) = 1/3;" "6#/-%$SumG 6$-%!G6#*&\"\"$\"\"\")\"#5%\"nG!\"\"/F/;F,%)infinityG*&F,F,F+F0" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 3 "su m" }{TEXT -1 35 " procedure can obtain this result. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Sum(3/(10^ n),n=1..infinity);\n``=value(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedur e " }{TEXT 0 3 "sum" }{TEXT -1 73 " can also be used to obtain formula s for finite sums such as the formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a+a*r+a*r^2+a*r^3+` . . . `+a*r^(n-1)=a*(1-r^n)/ (1-r)" "6#/,.%\"aG\"\"\"*&F%F&%\"rGF&F&*&F%F&*$F(\"\"#F&F&*&F%F&*$F(\" \"$F&F&%(~.~.~.~GF&*&F%F&)F(,&%\"nGF&F&!\"\"F&F&*(F%F&,&F&F&)F(F3F4F&, &F&F&F(F4F4" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "for the s um of a finite geometric series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Sum(a*r^(k-1),k=1..n);\nvalu e(%);\nsimplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 3 "su m" }{TEXT -1 39 " procedure \"knows\" the classic formula " }{XPPEDIT 18 0 "Sum(1/n^2,n=1..infinity)=Pi^2/6" "6#/-%$SumG6$*&\"\"\"F(*$%\"nG \"\"#!\"\"/F*;F(%)infinityG*&%#PiGF+\"\"'F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Su m(1/n^2,n=1..infinity);\nvalue(%);\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Approximate values for fi nite sums can be obtained rapidly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Sum(1/n^2,n=1..10000);\neval f(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Sum(1/n^2,n=1..100000); \nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Using " }{TEXT 0 3 "add" }{TEXT -1 13 " in place of " }{TEXT 0 3 "sum" }{TEXT -1 67 " forces Maple to perform the addition (using rat ional arithmetic). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "add(1/n^2,n=1..10000):\nevalf(%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "Maple ta kes a few seconds to evaluate the following. (Replacing the colon by a semi-colon in the first line will cause a rational number with huge n umerator and denominator to be displayed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "add(1/n^2,n=1..100 000):\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 71 "Maple is just a tool: it cannot replace an understanding of mathematics" }}{PARA 0 "" 0 "" {TEXT -1 86 "(This remark comes from a worksheet by A. Kluge at the Un iversity of Texas at Austin.)" }}{PARA 0 "" 0 "" {TEXT -1 322 "It is i mportant to keep the purpose of Maple in perspective. Maple is just a \+ tool. Maple cannot replace your understanding of mathematics, but it c an help you apply that understanding in new and creative ways. If you \+ don't exercise your judgment while using Maple, you may find yourself \+ led astray. Take the integral . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Int(x^n,x);\nvalue(%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 227 "Maple ap plies its rules for doing integrals and quickly produces a result. Ma ple lacks the insight to ask, \"Does this result make sense?\" If you ask yourself the same question, you find that it clearly does not mak e sense if " }{XPPEDIT 18 0 "n = -1" "6#/%\"nG,$\"\"\"!\"\"" }{TEXT -1 30 "? Maple can do that integral." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Int(x^(-1),x);\nvalue(%) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 200 "Th is looks nothing like the result Maple gave earlier. Similar situation s arise frequently with this type of software. So don't neglect your k nowledge and insight expecting Maple to do it all for you." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 " Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 60 "Try the following questions. Specimen answers are provi ded. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Q1: numerical calculatio ns" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Ass ign the value " }{XPPEDIT 18 0 "4*Pi/3;" "6#*(\"\"%\"\"\"%#PiGF%\"\"$! \"\"" }{TEXT -1 22 " to the symbol (name) " }{TEXT 298 1 "w" }{TEXT -1 20 ", and then find the " }{TEXT 266 11 "exact value" }{TEXT -1 7 " and a " }{TEXT 266 21 "decimal approximation" }{TEXT -1 6 " for: " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "w^2" "6#*$%\"wG\"\"# " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 " sqrt(w)" "6#-%%sqrtG6#%\"wG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "sin(w)" "6#-%$sinG6#%\"wG" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "w^2+sqrt(w)+sin( w)" "6#,(*$%\"wG\"\"#\"\"\"-%%sqrtG6#F%F'-%$sinG6#F%F'" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "(e) " }{XPPEDIT 18 0 "w^2*sqrt(w)*sin( w)" "6#*(%\"wG\"\"#-%%sqrtG6#F$\"\"\"-%$sinG6#F$F)" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w := 4*Pi/3;" }}}{PARA 0 "" 0 "" {TEXT -1 47 "It is convenient to assign the exact values of " }{XPPEDIT 18 0 "w^2, sqrt(w)" "6$*$%\"wG\"\"#-%%sqrtG6#F$" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "sin(w)" "6#-%$sinG6#%\"wG" }{TEXT -1 24 " to suitable v ariables. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a := w^2;\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "b := sqrt(w); \nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "c := sin(w);\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "a + b + \+ c;\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "(e) " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "a*b*c;\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Q2: functions" }}{PARA 0 "" 0 "" {TEXT -1 26 "(a) Define the function " }{XPPEDIT 18 0 "f(x) = 5*x-x^2;" "6#/-%\"fG6#%\"xG,&* &\"\"&\"\"\"F'F+F+*$F'\"\"#!\"\"" }{TEXT -1 13 " using the " }{TEXT 0 2 "->" }{TEXT -1 12 " notation. " }}{PARA 0 "" 0 "" {TEXT -1 24 "(b ) Plot the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 6 " from " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "x=6" "6#/%\"xG\"\"'" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "(c) Estimate the maximum value attai ned by the function." }}{PARA 0 "" 0 "" {TEXT -1 39 "_________________ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________ ________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := x -> 5*x-x^2;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-1..6);" }}} {PARA 0 "" 0 "" {TEXT -1 44 "(c) The line of symmetry of the parabola \+ is " }{XPPEDIT 18 0 "x=5/2" "6#/%\"xG*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 7 ". This " }{TEXT 299 1 "x" }{TEXT -1 30 " value is mid-way betwee n the " }{TEXT 300 1 "x" }{TEXT -1 12 "-intercepts " }{XPPEDIT 18 0 "x =0" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=5" "6#/%\"xG \"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Hence the maxim um point is" }{XPPEDIT 18 0 "``(5/2,f(5/2))" "6#-%!G6$*&\"\"&\"\"\"\" \"#!\"\"-%\"fG6#*&F'F(F)F*" }{TEXT -1 54 ". The maximum value attained by the function is . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f(5/2);\n``=evalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Alternatively, the " }{TEXT 301 1 "x" }{TEXT -1 65 "-coordinate of the maximum point occurs where the derivative f '( " }{TEXT 302 1 "x" }{TEXT -1 10 ") is zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Diff(f(x),x)=0;\nv alue(%);\nsolve(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Q3: plotting graphs " }}{PARA 0 "" 0 "" {TEXT -1 134 " Plot the graphs given by the follow ing equations. In each case choose a plotting interval which shows the main features of the graph." }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " } {XPPEDIT 18 0 "y = x^2-5*x-3" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"*&\"\"&F)F' F)!\"\"\"\"$F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(b) " } {XPPEDIT 18 0 "y = x^3-x;" "6#/%\"yG,&*$%\"xG\"\"$\"\"\"F'!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "y^ 2 = x;" "6#/*$%\"yG\"\"#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(d) " }{XPPEDIT 18 0 "x^2-y^2 = 2;" "6#/,&*$%\"xG\"\"#\" \"\"*$%\"yGF'!\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "F or the graph in (d), display the asymptotes " }{XPPEDIT 18 0 "y=x" "6# /%\"yG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=-x" "6#/%\"yG,$%\"x G!\"\"" }{TEXT -1 25 " together with the graph." }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "__ _____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(x^2-5*x-3,x=-2..7,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(x^3-x,x=-2..2,y=-4..4,color=blue);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plots[implicitplot](y^2=x,x=0..9,y= -3..3,grid=[30,30],color=coral);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 111 "Small \"gaps\" may appear in the curve c onstructed by implicit plot. A smoother curve may be obtained as follo ws." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([sqrt(x),-sqrt(x)],x=0..9,y,color=coral);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(d)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "darkgrey := COLOR(RGB,.4,.4 ,.4):\ndarkgreen := COLOR(RGB,0,.6,0):\nplots[implicitplot]([x^2-y^2=2 ,y=x,y=-x],x=-3..3,y=-3..3,\n color=[darkgreen,darkgrey,darkgre y],linestyle=[1,4,4]);" }}}{PARA 0 "" 0 "" {TEXT -1 3 "OR " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "darkgrey := COLOR(RGB,.4,.4,.4):\n darkgreen := COLOR(RGB,0,.6,0):\nplot([sqrt(x^2-2),-sqrt(x^2-2),x,-x], x=-3..3,y=-3..3,\n color=[darkgreen$2,darkgrey$2],linestyle=[ 1,1,4,4],numpoints=80);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Q4: solvin g equations analytically" }}{PARA 0 "" 0 "" {TEXT -1 54 "Solve the fol lowing equations, or system of equations." }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "6*x^2-7*x+2 = 0;" "6#/,(*&\"\"'\"\"\"*$%\"xG \"\"#F'F'*&\"\"(F'F)F'!\"\"F*F'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 5 "(b) " }{XPPEDIT 18 0 "4*x^2-7*x+2 = 0;" "6#/,(*&\"\"%\" \"\"*$%\"xG\"\"#F'F'*&\"\"(F'F)F'!\"\"F*F'\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "6*x^3-35*x^2+64*x-3 5 = 0" "6#/,**&\"\"'\"\"\"*$%\"xG\"\"$F'F'*&\"#NF'*$F)\"\"#F'!\"\"*&\" #kF'F)F'F'F,F/\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d ) " }{XPPEDIT 18 0 " ln(x^2+1)=1" "6#/-%#lnG6#,&*$%\"xG\"\"#\"\"\"F+F+ F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(e) " }{XPPEDIT 18 0 "2*x+5*y = 7,x-3*y = 2;" "6$/,&*&\"\"#\"\"\"%\"xGF'F'*&\"\"&F'%\"yGF 'F'\"\"(/,&F(F'*&\"\"$F'F+F'!\"\"F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 71 "Illustrate the solution of the system of equations in ( e) graphically. " }}{PARA 0 "" 0 "" {TEXT -1 39 "_____________________ __________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "____________ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(6* x^2-7*x+2=0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The following commands show the steps of the solution." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "6*x^2-7*x+2=0;\nfactor(%);\nsolve(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We can even arrange for the sol utions to be displayed in the form " }{XPPEDIT 18 0 "x=``" "6#/%\"xG%! G" }{TEXT -1 7 " . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "6*x^2-7*x+2=0;\nfactor(%);\nop(map( u->x=u,\{solve(%,x)\}));" }}}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "4*x^2-7*x+2=0;\nsolve(%);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The fol lowing commands can be used to show how the solutions are obtained by \+ substituting " }{XPPEDIT 18 0 "a=4, b=-7" "6$/%\"aG\"\"%/%\"bG,$\"\"(! \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c=2" "6#/%\"cG\"\"#" }{TEXT -1 26 " in the quadratic formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = -b/(2*a)" "6#/%\"xG,$*&%\"bG\"\"\"*&\"\"#F(%\"aGF( !\"\"F," }{TEXT -1 1 " " }{TEXT 303 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(b^2-4*a*c)/(2*a)" "6#*&-%%sqrtG6#,&*$%\"bG\"\"#\"\"\"*(\"\" %F+%\"aGF+%\"cGF+!\"\"F+*&F*F+F.F+F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 52 "for the solutions of the general quadratic equation " } {XPPEDIT 18 0 "a*x^2+b*x+c = 0;" "6#/,(*&%\"aG\"\"\"*$%\"xG\"\"#F'F'*& %\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "unassign('a','b','c' );\na*x^2+b*x+c=0;\nsolve(%,x);\nop(subs(\{a=4,b=-7,c=2\},[%]));" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "6*x^3-35*x^2+64*x-35=0;\nsol ve(%);" }}}{PARA 0 "" 0 "" {TEXT -1 3 "OR " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "6*x^3-35*x^2+64*x-35=0;\nfactor(%);\nsolve(%);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ln(x^2+1) = 1;\nsolve(%);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(e) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "\{2*x+5*y=7,x-3*y=2\};\nsolv e(%);\nevalf(%,8);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 97 "The solution is given by the coordinates of the point o f intersection of the two straight lines: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*x+5*y = 7, `or `*y = -2/5*x +7/5],[x-3*y = 2, `or `*y = x/3-2/3]);" "6#-%*PIECEWISEG6$7$/,& *&\"\"#\"\"\"%\"xGF+F+*&\"\"&F+%\"yGF+F+\"\"(/*&%)or~~~~~~GF+F/F+,&*(F *F+F.!\"\"F,F+F6*&F0F+F.F6F+7$/,&F,F+*&\"\"$F+F/F+F6F*/*&%+or~~~~~~~~G F+F/F+,&*&F,F+F " 0 "" {MPLTEXT 1 0 55 "solve(2*x+5* y=7,y),solve(x-3*y=2,y);\nplot([%],x=-1..5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 37 "Q5: changing the subject of a formula" }}{PARA 0 "" 0 " " {TEXT -1 6 "Solve " }{XPPEDIT 18 0 "T = 2*Pi*sqrt(L/g);" "6#/%\"TG*( \"\"#\"\"\"%#PiGF'-%%sqrtG6#*&%\"LGF'%\"gG!\"\"F'" }{TEXT -1 5 " for \+ " }{TEXT 285 1 "L" }{TEXT -1 33 ", in terms of the other variables" }} {PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "____________ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "T=2*Pi*sqrt( L/g);\nL=solve(%,L);\n" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Q6: differentiation and integration" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "Int(sqrt(x)+x*exp(x^2),x)" "6#-%$In tG6$,&-%%sqrtG6#%\"xG\"\"\"*&F*F+-%$expG6#*$F*\"\"#F+F+F*" }{TEXT -1 42 " and check the result by differentiation. " }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Int(sqrt(x)+x*exp(x^2),x);\nans := value( %);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Diff(ans,x);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 " Q7: the area under a graph" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " } {XPPEDIT 18 0 "Int(sqrt(x)+x*exp(x^2),x = 0 .. 1);" "6#-%$IntG6$,&-%%s qrtG6#%\"xG\"\"\"*&F*F+-%$expG6#*$F*\"\"#F+F+/F*;\"\"!F+" }{TEXT -1 85 " and draw a picture to illustrate the region for which this integr al gives the area. " }}{PARA 0 "" 0 "" {TEXT -1 39 "__________________ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________ ________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Int(sqrt(x)+x*exp(x^2),x=0.. 1);\nvalue(%);\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "f := x -> sqrt(x)+x*exp(x^2):\nplo t1 := plot(f(x),x=0..1.1,thickness=2,color=red):\nplot2 := plot(f(x),x =0..1,color=grey,filled=true):\nplot3 := plot([[1,0],[1,f(1)]],color=b lack):\nplots[display]([plot1,plot2,plot3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "Q8: differentiation and integration" }}{PARA 0 "" 0 "" {TEXT -1 40 "(a) Find the derivative with respect to " }{TEXT 294 1 "x " }{TEXT -1 4 " of " }{XPPEDIT 18 0 "(x^2+1)^4" "6#*$,&*$%\"xG\"\"#\" \"\"F(F(\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "(b) Int egrate the result from (a) with respect to " }{TEXT 295 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 75 "(c) The result from (b) diffe rs by a constant from the original expression." }}{PARA 0 "" 0 "" {TEXT -1 40 " Illustrate this fact by expanding " }{XPPEDIT 18 0 "(x^2+1)^4" "6#*$,&*$%\"xG\"\"#\"\"\"F(F(\"\"%" }{TEXT -1 64 " before \+ differentiating and integrating as in parts (a) and (b)." }}{PARA 0 " " 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " } }{PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Diff((x^2+1)^4,x);\nans := value(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "Int(ans,x);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "(x^2+1)^4;\nexpand(%);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 59 "This last expression is obtained by adding 1 to th e result " }{XPPEDIT 18 0 "x^8+4*x^6+6*x^4+4*x^2" "6#,**$%\"xG\"\")\" \"\"*&\"\"%F'*$F%\"\"'F'F'*&F+F'*$F%F)F'F'*&F)F'*$F%\"\"#F'F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "Int(8*(x^2+1)^3*x,x)" "6#-%$IntG6$*(\"\") \"\"\"*$,&*$%\"xG\"\"#F(F(F(\"\"$F(F,F(F," }{TEXT -1 15 " from part (b )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Not e" }{TEXT -1 3 ": " }{XPPEDIT 18 0 "Int(8*(x^2+1)^3*x,x)" "6#-%$IntG6 $*(\"\")\"\"\"*$,&*$%\"xG\"\"#F(F(F(\"\"$F(F,F(F," }{TEXT -1 40 " can \+ be found by using the substitution " }{XPPEDIT 18 0 "u=x^2+1" "6#/%\"u G,&*$%\"xG\"\"#\"\"\"F)F)" }{TEXT -1 9 " to give " }{XPPEDIT 18 0 "du/ dx=2*x" "6#/*&%#duG\"\"\"%#dxG!\"\"*&\"\"#F&%\"xGF&" }{TEXT -1 20 " or (symbolically) " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 1 " " }{TEXT 316 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "du=x*d x" "6#/%#duG*&%\"xG\"\"\"%#dxGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(8*(x^2+1)^3*x,x)=1/2" "6#/-%$IntG6$*(\"\")\"\"\"*$,&*$%\"xG\"\"#F )F)F)\"\"$F)F-F)F-*&F)F)F.!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(8 *u^3,u)=Int(4*u^3,u)" "6#/-%$IntG6$*&\"\")\"\"\"*$%\"uG\"\"$F)F+-F%6$* &\"\"%F)*$F+F,F)F+" }{XPPEDIT 18 0 "``=u^4+c" "6#/%!G,&*$%\"uG\"\"%\" \"\"%\"cGF)" }{XPPEDIT 18 0 "`` = (x^2+1)^4+c;" "6#/%!G,&*$,&*$%\"xG\" \"#\"\"\"F+F+\"\"%F+%\"cGF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Q9: plotting points and the limit of a sequence" }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Use the procedure " }{TEXT 0 3 "seq" } {TEXT -1 24 " to construct the list " }{XPPEDIT 18 0 "[2,4,8,16,32,` \+ . . . . `,1024]" "6#7)\"\"#\"\"%\"\")\"#;\"#K%*~.~.~.~.~G\"%C5" } {TEXT -1 44 " consisting of all powers of 2 of the form " }{XPPEDIT 18 0 "2^k;" "6#)\"\"#%\"kG" }{TEXT -1 8 ", where " }{TEXT 315 1 "k" } {TEXT -1 42 " goes from 1 to 10. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "(b) Construct the l ist " }{XPPEDIT 18 0 "[[0, 0], [1, 1/2], [2, 2/3], [3, 3/4], [4, 4/5] , ` . . . `, [20, 20/21]];" "6#7)7$\"\"!F%7$\"\"\"*&F'F'\"\"#!\"\"7$F) *&F)F'\"\"$F*7$F-*&F-F'\"\"%F*7$F0*&F0F'\"\"&F*%(~.~.~.~G7$\"#?*&F6F' \"#@F*" }{TEXT -1 35 ", consisting of points of the form " }{XPPEDIT 18 0 "[n, n/(n+1)];" "6#7$%\"nG*&F$\"\"\",&F$F&F&F&!\"\"" }{TEXT -1 8 ", where " }{TEXT 314 1 "n" }{TEXT -1 52 " goes from 0 to 20 and show \+ these points on a graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 8 "(c) Use " }{TEXT 0 5 "limit" }{TEXT -1 4 " or " } {TEXT 0 5 "Limit" }{TEXT -1 5 " and " }{TEXT 0 5 "value" }{TEXT -1 35 " to find the limit of the sequence " }{XPPEDIT 18 0 "0,1,1/2,2/3,3/4, 4/5,5/6,` . . . `,n/(n+1),` . . . `;" "6,\"\"!\"\"\"*&F$F$\"\"#!\"\"*& F&F$\"\"$F'*&F)F$\"\"%F'*&F+F$\"\"&F'*&F-F$\"\"'F'%(~.~.~.~G*&%\"nGF$, &F2F$F$F$F'F0" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "seq(2^k,k=1..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "pts := [seq([n,n/(n+1)],n=0..20)];\nplot(pts,n=0..20, 0..1,style=point,symbol=circle);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Limit(n/(n+1),n=infinity);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The picture from part (b ) can be enhanced to illustrate the convergence of the sequence to 1 m ore clearly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "pts := [seq([n,n/(n+1)],n=0..20)]:\nplot([n/(n+ 1),1,pts],n=0..20,0..1.1,style=[line,line,point],\n color=[grey,bro wn,red],linestyle=[2,3],symbol=circle);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 " Q10: the sum of a series" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 23 "The alternating series " }{XPPEDIT 18 0 "1-1/2+1/3 -1/4+1/5-1/6+1/7-1/8+` . . . `" "6#,4\"\"\"F$*&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'%(~.~.~.~GF$" }{TEXT -1 51 " can be given in summation n otation by the formula " }{XPPEDIT 18 0 "Sum((-1)^(n-1)/n,n=1..infinit y)" "6#-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF)F)F*F)F,F*/F,;F)%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "(a) Use " }{TEXT 0 3 "s eq" }{TEXT -1 39 " to check that the underlying sequence " }{XPPEDIT 18 0 "1, -1/2, 1/3, -1/4, 1/5, -1/6, 1/7, -1/8, 1/9, -1/10" "6,\"\"\", $*&F#F#\"\"#!\"\"F'*&F#F#\"\"$F',$*&F#F#\"\"%F'F'*&F#F#\"\"&F',$*&F#F# \"\"'F'F'*&F#F#\"\"(F',$*&F#F#\"\")F'F'*&F#F#\"\"*F',$*&F#F#\"#5F'F'" }{TEXT -1 57 " as far as the tenth term is generated by the expression " }{XPPEDIT 18 0 "(-1)^(n-1)/n" "6#*&),$\"\"\"!\"\",&%\"nGF&F&F'F&F)F '" }{TEXT -1 11 " used here." }}{PARA 0 "" 0 "" {TEXT -1 8 "(b) Use " }{TEXT 0 3 "sum" }{TEXT -1 4 " or " }{TEXT 0 3 "Sum" }{TEXT -1 5 " and " }{TEXT 0 5 "value" }{TEXT -1 22 " to find the value of " }{XPPEDIT 18 0 "Sum((-1)^(n-1)/n,n = 1 .. infinity)" "6#-%$SumG6$*&),$\"\"\"!\" \",&%\"nGF)F)F*F)F,F*/F,;F)%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "(c) Calculate a deci mal approximation for the finite sum " }{XPPEDIT 18 0 "Sum((-1)^(n-1)/ n,n=1..100000)" "6#-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF)F)F*F)F,F*/F,;F) \"'++5" }{TEXT -1 68 " and compare this with a decimal approximation f or the infinite sum " }{XPPEDIT 18 0 "Sum((-1)^(n-1)/n,n = 1 .. infini ty)" "6#-%$SumG6$*&),$\"\"\"!\"\",&%\"nGF)F)F*F)F,F*/F,;F)%)infinityG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "____________________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq((-1)^(n-1)/n,n=1..10);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Sum((-1)^(n-1)/n,n=1..infini ty);\nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Sum((-1) ^(n-1)/n,n=1..100000);\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 67 "This value agrees to 4 figures with the v alue of the infinite sum. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ln(2);\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 49 "Q11: making sense of results obtained using Maple" }} {PARA 0 "" 0 "" {TEXT -1 36 "(a) Use Maple to solve the equation " } {XPPEDIT 18 0 "ln(4-x^2)=0" "6#/-%#lnG6#,&\"\"%\"\"\"*$%\"xG\"\"#!\"\" \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "(b) Illustrate \+ the solutions of this equation as the " }{TEXT 297 1 "x" }{TEXT -1 41 "-intercepts of the graph of the equation " }{XPPEDIT 18 0 "y=ln(4-x^2 )" "6#/%\"yG-%#lnG6#,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 36 "(c) Use Maple to solve the equation " } {XPPEDIT 18 0 "ln(2+x)+ln(2-x)=0" "6#/,&-%#lnG6#,&\"\"#\"\"\"%\"xGF*F* -F&6#,&F)F*F+!\"\"F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 306 "(d) Using the multiplication property of logarithms you can se e that the equation in part (c) is equivalent to the equation in part \+ (a) so you would expect Maple to give the same solutions as before. Th is does not happen (with Maple 9). Are the solutions obtained in part \+ (c) correct? Justify your answer. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 266 4 "Hint" }{TEXT -1 16 ": the procedure " } {TEXT 0 11 "rationalize" }{TEXT -1 111 " can be used to rationalise th e denominator of a numerical fraction which contains radicals in the d enominator." }}{PARA 0 "" 0 "" {TEXT -1 19 "For example, . . . " }} {PARA 0 "" 0 "" {TEXT -1 4 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "1/(1+sqrt(2));\nrationalize(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&F$F$*$\"\"##F$F'F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"\"##\"\"\"F%F'F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " } }{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(ln(4-x^2)=0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "plot1 := plot(ln(4-x^2),x=-2.2..2.2,y=-2.5..1.5):\np lot2 := plots[implicitplot](\{x=-2,x=2\},x=-2.2..2.2,y=-2.5..1.5,\n \+ color=COLOR(RGB,.4,.4,.4),linestyle=3):\nplots[display]([plot1,plot 2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "( c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "x1,x2 := solve(ln(2+x )+ln(2-x)=0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }}{PARA 0 "" 0 "" {TEXT -1 121 "It is easy to check \+ that the solutions given in part (a) are correct. The following comman ds show two ways of doing this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Eval(ln(4-x^2)=0,x=sqrt(3)); \nvalue(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(x=-sqrt(3),ln(4-x^2)=0);\nsimplify(%);\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The solut ions in part (c) can be checked in a similar way. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Eval(ln(2+x) +ln(2-x)=0,x=x1);\nvalue(%);\nsimplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(x=x2,ln(2+x)+ ln(2-x)=0);\nsimplify(%);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 157 "The solutions in part (c) are the same as thos e given in part (a). This can be seen by rationalising the denominator s in the pair of solutions for part (c). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x1;\nrationalize(%); \nexpand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x2;\nrationalize(%);\nexpand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Alternatively, we c an evaluate each of the two pairs of solutions to obtain approximately the same numerical values. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sqrt(3),-sqrt(3);\nevalf(%); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x1,x2;\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 178 "Since one of these values exhibits rath er severe rounding error (Why?), we could perform the evaluations with a few extra (guard) digits, and then round the results to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x1,x2;\nevalf(%,15);\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Now we get exact agreement between the10 digit numerical values for the two pairs of solutions. 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