{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 " INTRODUCTION TO MAPLE" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 " This \+ Maple worksheet introduces arithmetic operations, algebraic manipulat ions and" }}{PARA 0 "" 0 "" {TEXT -1 89 "plotting. The material is ta ken from selected portions of Chapters 1 and 2 of the manual" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 " \+ Single Variable CalcLabs with Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "which is a companion text for \"Single Variable Calculus\" by James Stewart. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 " The actual Maple co mmands are in \"red\" print, with commentary given in \"black\"" }} {PARA 0 "" 0 "" {TEXT -1 84 "print. You can execute the Maple command s by placing the cursor anywhere within the" }}{PARA 0 "" 0 "" {TEXT -1 56 "command expression and then pressing the \"enter\" key. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 " After \+ executing a set of commands, you should type in your own variations fo r practice." }}{PARA 0 "" 0 "" {TEXT -1 86 "If you place the cursor wi thin any given Maple command and click on the \"[>\" button on" }} {PARA 0 "" 0 "" {TEXT -1 88 "the tool bar, then a new line headed by t he prompt will appear under the line containing" }}{PARA 0 "" 0 "" {TEXT -1 49 "the cursor. Enter your own command on this line." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "================================== ===============================" }}{PARA 0 "" 0 "" {TEXT -1 27 "1.1 N UMERICAL CALCULATIONS" }}{PARA 0 "" 0 "" {TEXT -1 65 "================ =================================================" }}{PARA 0 "" 0 "" {TEXT -1 91 " To make numerical calculations enter the appropriate s ymbols, terminate the command with" }}{PARA 0 "" 0 "" {TEXT -1 34 "the semicolon \";\" and press enter:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "3+4;" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "11.2 - 7.1;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "12*13;" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "18/6;" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2^4;" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "3*%;" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Th e ditto symbol is %. It always has the value of the last previous comp utation. Thus" }}{PARA 0 "" 0 "" {TEXT -1 44 "% had the value 16 in th e computation above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "26*26;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "sqrt(%);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The function sqrt \+ is one of many standard mathematical functions found in Maple." }} {PARA 0 "" 0 "" {TEXT -1 92 "For a list of some of the others (eg: exp , ln, sin, abs, .. etc) see page two of the manual." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sin(Pi/2) ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ln(1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "exp(3);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 75 "Help on any Maple command, for example \"exp\", can be \+ obtained by use of the" }}{PARA 0 "" 0 "" {TEXT -1 9 "? symbol." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?exp;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "?%;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Ma ple usually tries to provide an exact rational or symbolic answer to m ost" }}{PARA 0 "" 0 "" {TEXT -1 74 "computations. This sometimes prod uces unusual results. A decimal version" }}{PARA 0 "" 0 "" {TEXT -1 56 "of an answer can be obtained with the \"evalf\" function." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(4+7)/3;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "exp(3);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "exp(3.0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "exp(1.0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 79 "A decimal answer also can be generated b y including a decimal point in at least" }}{PARA 0 "" 0 "" {TEXT -1 38 "one of the arguments of the operation." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(4+7)/3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(4+7.)/3;" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "== ============================================================" }}{PARA 0 "" 0 "" {TEXT -1 30 "1.2 VARIABLES AND EXPRESSIONS" }}{PARA 0 "" 0 "" {TEXT -1 62 "====================================================== ========" }}{PARA 0 "" 0 "" {TEXT -1 80 " The variable name assignme nt operator is := (colon followed by equal sign)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "a:=3; b:= 2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c:=a+b; " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a:= sin( 3) + exp(2) +sqrt(7);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s:=evalf(a);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "s^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "area:=Pi*r^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "r:=2; area;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Clear the value of the variable r:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "r:='r';" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "When the subs comman d is used to evaluate a function for specific values of a" }}{PARA 0 " " 0 "" {TEXT -1 85 "variable, it is unnecessary to assign values to th e variables in separate statements." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "volume:=(4/3)*Pi*r^3;" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(r=2,vo lume);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "su bs(r=2.0,volume);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a:=(1/2)*b*h;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "val:=subs(b=3,h=5,a);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(val);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "== ============================================================" }}{PARA 0 "" 0 "" {TEXT -1 21 "1.3 ALGEBRA COMMANDS" }}{PARA 0 "" 0 "" {TEXT -1 62 "============================================================== " }}{PARA 0 "" 0 "" {TEXT -1 53 " (1) expand (2) factor \+ (3) simplify" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=(3*x-2)^2*(x^3+2*x);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(f);" }{TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f:=x^6-1;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(f);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=(x^2-x)/ (x^3-x) - (x^2-1)/(x^2+x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "simplify(f);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "==================================== =========================" }}{PARA 0 "" 0 "" {TEXT -1 11 "1.4 PLOTS " }}{PARA 0 "" 0 "" {TEXT -1 61 "===================================== ========================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:=x^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plot(f,x);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Th e default x range is -10 " 0 "" {MPLTEXT 1 0 16 "plot(f ,x=-3..4);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "New f:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=2*x^3-5*x^2+x+2;" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plot(f,x); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Specify both an x range and a y range to give a wind ow that more clearly displays the" }}{PARA 0 "" 0 "" {TEXT -1 26 "func tion near the origin.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " plot(f,x=-4..4,y=-10..10);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Several graphs can be sho wn on the same axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g:=x^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(\{f,g\},x=-4..4,y=-10..10);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "========================================================= =====" }}{PARA 0 "" 0 "" {TEXT -1 49 " SOLVING EQUATIONS Some items from Ch 2." }}{PARA 0 "" 0 "" {TEXT -1 62 "==================== ==========================================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=x^2+2*x-1;" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(f=0,x );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "One can name the solutions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "r:=solve(f= 0,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r[1] ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r[2];" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "r:=evalf(r) ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eq:= x^2-3*x+2=0;" }{TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "t:=solve(eq,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "You can solve systems of equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eq1:=3*x+2* y=1; eq2:=x+2*y=3;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(\{eq1,eq2\},\{x,y\});" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Th e \"exact\" and/or \"symbolic\" answers often provided by a Maple c omputation can" }}{PARA 0 "" 0 "" {TEXT -1 77 "be very cumbersome. Yo u can always use the \"evalf\" command to obtain decimal" }}{PARA 0 " " 0 "" {TEXT -1 85 "answers (also called floating point numbers). Als o recall the use of decimal numbers" }}{PARA 0 "" 0 "" {TEXT -1 27 "in the original expression." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x^3-2*x^2+x-3;" }{TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eq:= f=0;" }{TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s:=solve(eq,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(s);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "f:=x^3-2.0*x^2+x-3.0; " }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(f=0,x);" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Sometimes Maple finds it impossible to get an exact solution to an equation, " }}{PARA 0 "" 0 "" {TEXT -1 71 "in which case the response is usually something like the example below:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:= x^7+3*x ^4+2*x-1;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(f=0,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "When this happens, use the fsolve c ommand, which tells Maple to find" }}{PARA 0 "" 0 "" {TEXT -1 39 " app roximate solutions in decimal form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(f=0,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Since a 7th degree polynomial equation has 7 roots, it is likel y that there " }}{PARA 0 "" 0 "" {TEXT -1 76 "are 6 complex roots. We \+ can see these by using the complex form of fsolve :" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fsolve(f= 0,x,complex);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "For some equations, you need to give Ma ple some help in finding all the" }}{PARA 0 "" 0 "" {TEXT -1 16 " real solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:= x^2 + 1/x - 1/x^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(f=0,x);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We see that Maple can't find exact solutions, so we use fsolve :" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(f=0,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We can see if the above is the \+ only real solution by graphing the" }}{PARA 0 "" 0 "" {TEXT -1 69 " ex pression f and checking the number of times it crosses the x-axis:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plot(f,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Change the scale to get better defin ition near the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(f,x=-2..2,y=-20..20);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fsolve(f=0,x,comp lex);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plo t(f,x=-3..3,y=-10..10);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We see another solution b etween 0 and 1 so we tell Maple to look for it there:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve(f =0,x=0..1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 64 "========================================= =======================" }}{PARA 0 "" 0 "" {TEXT -1 64 "============== ==================================================" }}}}{MARK "11 2 0 " 76 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }