{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 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 18 "Elementary Algebra" }}{PARA 256 "" 0 "" {TEXT -1 14 "(number04.mws)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 203 " In this exericse we study the \+ basic properties of polynomials and rational functions. Some revelant MAPLE commands will be factor, expand, solve, fsolve, denom, numer , simplify, subs, ........" }}{PARA 0 "" 0 "" {TEXT -1 320 " The first thing we need to learn is how to define a function. There are \+ two ways, each with its own advantages and disadvantages. First defin e the polynomial p1 as a symbolic expression in x in the following way : p1:=x^2-1; (Note the colon before the =, this is always requ ired when defining something.) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 105 " Now define the same pol ynomial as an operation on the variable x in the following way: p2:=x- >x^2-1;" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 257 "" 1 "" {TEXT -1 190 "Try the following command sequences to see some of \+ the differences in how MAPLE treats these two definitions. As always record your observations and comments on the worksheet in text mode. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "p1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "p2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "p2(x);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "p2(t-3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "p2(u^3-u^2+2);" }}}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 255 " This works just like we expect substituti on to work because p2 is an operation on whatever is in the ( ). How ever since p1 is a symbolic expression we have some difficulties with \+ substituting into it directly. Try the following command sequences. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "p1(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "p1(t-3);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "p1(u^3-u^2+2);" }}}{PARA 0 "" 0 "" {TEXT -1 148 " Notice the extra x in front. This is cu rious and not very reassuring. To substitute into a symbolic expressi on we must use the subs command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=t-3,p1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(x=6,p1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "p2(6);" }}}{PARA 0 "" 0 "" {TEXT -1 4 " " } }{PARA 0 "" 0 "" {TEXT -1 75 " Now let's do some simple operations on this polynomial. In each case " }{TEXT 256 46 "Explain what the c ommand appears to have done." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(p1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(p2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "factor(p2(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "fsolve(% %=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(p2=0);" }}} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(p2(x)=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "so lve(p1=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(p1=2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "fsolve(p1=2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(p2=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(p2(x)=2);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p3:=a*x^2+ b*x+c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(p3=0);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(p3=0,x);" }}}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "1. " }{TEXT 257 88 "Why did we need to include the , x in the solve command in order to find the roots of p3?" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "U se the solve command replacing x with a, b, c respectively. You shoul d be able to anticipate the results." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }{TEXT 258 58 "What's the famous formula we get when we so lve p3=0 for x?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "Of course we could have done the same exercise with the \+ polynomial defined as an operation. Execute the following:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p4:=x->a*x^2+b*x+c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "p4(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(p4(t)=0,t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 60 " Expand , then factor each of the fo llowing expressions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 10 "(x-3)(x+3)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 12 "(ax+1)(bx+1)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 10 "(x+I)(x-I)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 4 "3. " }{TEXT 259 72 "Look at the last result and determine what number I represents in MAPLE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 172 "4. See if MAPLE knows \+ a general formula for solving general cubic polynomials, i.e. is there a 3rd order formula comparable to the quadratic formula? If so, exhi bit it. " }}{PARA 0 "" 0 "" {TEXT -1 4 "5. " }{TEXT 260 111 "Why do y ou think high school students are never asked to remember the general \+ formula for the roots of a cubic?" }}{PARA 0 "" 0 "" {TEXT -1 24 "6. \+ For the polynomials " }{XPPEDIT 18 0 "p[5]" "6#&%\"pG6#\"\"&" }{TEXT -1 9 " through " }{XPPEDIT 18 0 "p[18]" "6#&%\"pG6#\"#=" }{TEXT -1 237 ", solve for all exact roots whereever possible, find all approxim ate roots to 10 decimal places, factor each into linear factors. You \+ will need to click on [> in the toolbar to obtain new execution lines just below the cursor position." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[5] := x^2-2;" "6#>&%\"pG6#\" \"&,&*$%\"xG\"\"#\"\"\"F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[6] := x^2+2;" "6#>&%\"pG6#\"\"',&*$%\"xG\"\"#\"\"\"F+F," }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[7] := 2*x^2+3*x-2;" "6#>&%\" pG6#\"\"(,(*&\"\"#\"\"\"*$%\"xGF*F+F+*&\"\"$F+F-F+F+F*!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[8] := x^2+x+1;" "6#>&%\"pG6# \"\"),(*$%\"xG\"\"#\"\"\"F*F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[9] := x^2+(sqrt(2)-2)*x-2*sqrt(2);" "6#>&%\"pG6#\"\"* ,(*$%\"xG\"\"#\"\"\"*&,&-%%sqrtG6#F+F,F+!\"\"F,F*F,F,*&F+F,-F06#F+F,F2 " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[10] := x^2+(1-Pi)*x-Pi; " "6#>&%\"pG6#\"#5,(*$%\"xG\"\"#\"\"\"*&,&F,F,%#PiG!\"\"F,F*F,F,F/F0" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[11] := x^2+(a+b)*x+a*b;" "6#>&%\"pG6#\"#6,(*$%\"xG\"\"#\"\"\"*&,&%\"aGF,%\"bGF,F,F*F,F,*&F/F,F0 F,F," }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[12] := x^3-x^2-2*x;" "6#>&%\"pG6#\"#7,(*$%\"xG\"\"$\" \"\"*$F*\"\"#!\"\"*&F.F,F*F,F/" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[13] := 6*x^3+11*x^2+6*x+1;" "6#>&%\"pG6#\"#8,**&\"\"'\"\"\"*$% \"xG\"\"$F+F+*&\"#6F+*$F-\"\"#F+F+*&F*F+F-F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[14] := x^3+x^2+x+1;" "6#>&%\"pG6#\"#9,**$% \"xG\"\"$\"\"\"*$F*\"\"#F,F*F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[15] := x^4-1;" "6#>&%\"pG6#\"#:,&*$%\"xG\"\"%\"\"\"F, !\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[16] := x^4+1;" "6# >&%\"pG6#\"#;,&*$%\"xG\"\"%\"\"\"F,F," }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[17] := 3*x^4+13*x^3+7*x^2-17*x-6;" "6#>&%\"pG6#\"#<,, *&\"\"$\"\"\"*$%\"xG\"\"%F+F+*&\"#8F+*$F-F*F+F+*&\"\"(F+*$F-\"\"#F+F+* &F'F+F-F+!\"\"\"\"'F7" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p[18 ] := 6*x^5-5*x^4+4*x^3-4*x^2-2*x+1;" "6#>&%\"pG6#\"#=,.*&\"\"'\"\"\"*$ %\"xG\"\"&F+F+*&F.F+*$F-\"\"%F+!\"\"*&F1F+*$F-\"\"$F+F+*&F1F+*$F-\"\"# F+F2*&F8F+F-F+F2F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "7. " }{TEXT 261 182 "Explain why every odd degree p olynomial with real coefficients must have at least one real root, whi le an even degree polynomial with real coefficients can have no real r oots at all." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " Perform the following MAPLE command sequence. Comment whe re appropriate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "p:=x->5*x^3-5*x+1;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "r:=solve(p(x)=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "si mplify(r[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(r[1] );evalf(r[2]);evalf(r[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p(r[1]);p(r[2]);p(r[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "simplify(p(r[1]));simplify(p(r[2]));simplify(p(r[3]));" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 251 " So \+ these are the roots of p. However, when we evalf them to get reasonab le numbers that are all complex. This seems to contradict our statemen t about all odd degree polynomials having at least one real root. Try \+ approximating the roots directly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fsolve(p(x)=0);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " Now we se em to have 3 real roots. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "8. " }{TEXT 262 30 "What appears to be going on? \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 181 "May be drawing the graph will help. Plot the graph of p with appropriate choices for the range of x-values and the range of y-values. The MAP LE plot command is a \"no brainer\" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "plot(expression,x=a..b, y=c..d);" "6#-%%plotG6%%+expressionG/%\"xG;%\"aG%\"bG/%\"yG;%\"cG%\"dG " }{TEXT -1 146 " where the choice of the y-range is optional. We co uld also include the optional command title=`name of the graph` \+ to label our graph. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 44 "9. Find all the zeros of the polyno mial 2" }{XPPEDIT 18 0 "x^4+x^3+x^2+1" "6#,**$%\"xG\"\"%\"\"\"*$F%\" \"$F'*$F%\"\"#F'F'F'" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "10. Factor the poly nomial 2" }{XPPEDIT 18 0 "x^4+x^3+x^2+1" "6#,**$%\"xG\"\"%\"\"\"*$ F%\"\"$F'*$F%\"\"#F'F'F'" }{TEXT -1 24 " into linear factors." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 63 "11. What are the x-intercepts of the graph of the polynomial 2 " }{XPPEDIT 18 0 "x^4+x^3+x^2+1" "6#,**$%\"xG\"\"%\"\"\"*$F%\"\"$F'*$F %\"\"#F'F'F'" }{TEXT -1 12 " ? " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 112 "A \"rational function\" is defined to be the quotient \+ of two polynomials. Define the following rational function:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f := (2*x^5+5*x^4+6*x^3+6*x^2+4*x+1)/(4*x^5+18*x^4+24*x^3+24*x^2+20*x+6); " "6#>%\"fG*&,.*&\"\"#\"\"\"*$%\"xG\"\"&F)F)*&F,F)*$F+\"\"%F)F)*&\"\"' F)*$F+\"\"$F)F)*&F1F)*$F+F(F)F)*&F/F)F+F)F)F)F)F),.*&F/F)*$F+F,F)F)*& \"#=F)*$F+F/F)F)*&\"#CF)*$F+F3F)F)*&F>F)*$F+F(F)F)*&\"#?F)F+F)F)F1F)! \"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 " Some importrant features of a rational function are its x and y \+ intercepts and its vertical asymptotes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "12. " }{TEXT 263 35 "What determin es the x-intercepts? " }}{PARA 0 "" 0 "" {TEXT -1 4 "13. " }{TEXT 264 44 " What determinres the vertical asymptotes? " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Execute the following M APLE command sequence:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n:=numer(f);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "d:=denom(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(d);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "14. " }{TEXT 265 32 "What are the x-intercepts of f? " } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "15. " }{TEXT 266 31 "Wha t is the y-intercept of f? " }}{PARA 0 "" 0 "" {TEXT -1 5 "16. " } {TEXT 267 40 "What are the vertical asymptotes of f? " }{TEXT -1 42 " (Try plotting the graph of f to see this.)" }}{PARA 0 "" 0 "" {TEXT -1 5 "17. " }{TEXT 268 82 "Why does f have only one vertical asymptot e when its denominator has 3 real roots?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(f);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 " Note that fact oring a rational function automatically simplifies it if possible. " } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "18. " } {TEXT 269 41 "What is the horizontal asymptote for f? " }{TEXT -1 39 "( Try to see this from the graph of f.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " Execute the following command seq uence." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Limit(f,x=-1)=limi t(f,x=-1);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(x=-1,f );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "19. " }{TEXT 270 50 "Explain the results of the above command sequence. " }}{PARA 0 "" 0 "" {TEXT -1 5 "20. " }{TEXT 271 51 "Use your brain t o compute the gcd(n,d) and lcm(n,d)" }{TEXT -1 122 ". (Hint: look at their factors.) Check your answer with MAPLE. (You may wish to use f actor to compare your answers.) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 " MAPLE can preform some neat operations on polynomia ls and rational functions. For some examples try the following comman d sequence:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "degree(d,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "coeff(d,x,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "co effs(d,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(d);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "op(4,d);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "op(d);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "f;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "d;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "n; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:=quo(n,d,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "r:=rem(n,d,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "d*q+r;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "21. " }{TEXT 272 41 "How does thi s last result compare to n? " }}{PARA 0 "" 0 "" {TEXT -1 5 "22. " } {TEXT 273 51 "What do you think quo and rem are abreviations for?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "pff:=convert(f,parfrac,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "This is really just another way to write f. To see that this equals \+ f try the following." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f-pff;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "23. " }{TEXT 274 49 "What do you think parfrac is an abreviation for? " }}{PARA 0 "" 0 "" {TEXT -1 5 "24. " }{TEXT 275 26 "Look at pff and determine " }{TEXT -1 3 " " }{XPPEDIT 18 0 "limit(f,x=infinity)" "6#-%&limitG6$%\"fG/% \"xG%)infinityG" }{TEXT -1 4 " ." }}{PARA 0 "" 0 "" {TEXT -1 36 "25. Use MAPLE to verify the result." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "For demonstration purposes it is very handy to be able t o construct polynomials with predetermined roots. " }}{PARA 0 "" 0 " " {TEXT -1 73 "26. Use the expand command to construct a polynomial w ith roots 1,2,2," }{XPPEDIT 18 0 "-1/2" "6#,$*&\"\"\"F%\"\"#!\"\"F' " }{TEXT -1 71 ", 1+I, and 1-I. (Here the \"2,2\" indicates that 2 i s a \"double root\".)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "27. Exp eriment with MAPLE to determine what number is represented by I. " } {TEXT 276 11 "What is it?" }}{PARA 0 "" 0 "" {TEXT -1 5 "28. " } {TEXT 277 69 "What was the effect of having both the numbers 1+I and 1 -I as roots? " }{TEXT -1 50 " (See what happens if only one of them is a root.)" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{MARK "192 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }