{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{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 }{PSTYLE "Headi ng 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 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" -1 257 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 }{PSTYLE "Nor mal" -1 258 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 }} {SECT 0 {PARA 258 "" 0 "" {TEXT 257 62 "Calculus Exploration 4A: Anal yze a Quotient of Two Polynomial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 258 493 "The following is a problem solver that \+ analyzes a function that is the quotient of two polynomials. It plots \+ the function, finds all x-intercepts, computes the derivative using th e quotient rule, finds the xy-coordinates of all bump points, and find s all vetical and horizontal asymptotes. In this exploration you shoul d experiment with changing the numerator and denominator to try to cre ate functions whose graphs have different numbers of x-intercepts, bum p points, and vertical asymptotes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 426 "Note that in order to view the entire g raph of a function, you will usually need to change the plot window by experimenting with different values of the end points on the x-axis a nd the y-axis. It is possible that not every feature of the function w ill be simultaneously visible in one plot window. Instead, you may hav e to zoom in on one part of the function, and then change the plot win dow to view another part separately. " }{TEXT -1 0 "" }{TEXT 259 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Analyze a Quotient of Two Polynomials" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 39 "Click in the red area and press [Enter]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2414 "restart:\nDigits:=5:\nf:=x->(3* x^2-5*x+4)/(2*x^2-1): # Change function here\nxmin:=-6: \+ # Change left end point on x-axis here\nxmax:=4: \+ # Change right end point on x-axis here\nymin:=-6: \+ # Change bottom end point on y-axis here\nymax:=6: \+ # Change top end point on y-axis here\n`f'`:=D(f):\n# Fin d x-intercepts\nxints:=fsolve(numer(f(x))):\nxintsList:=[]:\nfor i fro m 1 to nops([xints]) do\nxint[i]:=op(i,[xints]):\nif i = 1 then\nxints List:=[op(xintsList),` x-intercept(s): `*x=xint[i]]:\nelse\nxintsList: =[op(xintsList),xint[i]]:\nfi:\nod:\n# Find x-bumps\nxbumps:=fsolve(nu mer(`f'`(x))):\nxbumpsList:=[]:\nbumpsList:=[]:\nfor i from 1 to nops( [xbumps]) do\nxbump[i]:=op(i,[xbumps]):\nybump[i]:=evalf(f(xbump[i])): \nif i = 1 then\nxbumpsList:=[op(xbumpsList),` x-bump(s): `*x=xbump[i] ]:\nbumpsList:=[op(bumpsList),` bump point(s) `=[xbump[i],ybump[i]]]: \nelse\nxbumpsList:=[op(xbumpsList),xbump[i]]:\nbumpsList:=[op(bumpsLi st),[xbump[i],ybump[i]]]:\nfi:\nod:\n# Find x-vertical asymptotes\nxve rtasyms:=fsolve(denom(f(x))):\nxvertasymsList:=[]:\nfor i from 1 to no ps([xvertasyms]) do\nxvertasym[i]:=op(i,[xvertasyms]):\nif i = 1 then \nxvertasymsList:=[op(xvertasymsList),` x-vertical asymptote(s): `*x=x vertasym[i]]:\nelse\nxvertasymsList:=[op(xvertasymsList),xvertasym[i]] :\nfi:\nod:\n# Begin ouput \n'f(x)'=f(x);``;\nplot(f(x),x=xmin..xmax,y =ymin..ymax,color=blue,labels=[``,``]);\n'f(x)'=f(x);\n`Find x-interce pts:`;\n'f(x)'=0;\nnumer(f(x))=0;\n# Print x-intercepts\nif [xints] <> [] then\nprint(op(xintsList));\nelse\nprint(`No x-intercepts`);\nfi: \n'`f '`(x)'=numer(`f'`(x))/denom(f(x))^2;\n`Find bumps:`;\n'`f '`(x)' =0;\nnumer(`f'`(x))=0;\n# Print x-bumps\nif [xbumps] <> [] then\nprint (op(xbumpsList));\nprint(op(bumpsList));\nelse\nprint(`No bump points` );\nfi:\n`Find vertical asymptotes:`;\ndenom(f(x))=0;\n# Print x-verti cal asymptotes\nif [xvertasyms] <> [] then\nprint(op(xvertasymsList)); \nelse\nprint(`No vertical asymptotes`);\nfi:\n`Find horizontal asympt ote:`;\nd:=degree(numer(f(x))):\nLimit(f(x),x=infinity)=Limit(expand(n umer(f(x))/x^d)/expand(denom(f(` x`))/` x`^d),x=infinity);\n``=limit(f (x),x=infinity);\nLimit(f(x),x=-infinity)=Limit(expand(numer(f(x))/x^d )/expand(denom(f(` x`))/` x`^d),x=-infinity);\n``=limit(f(x),x=-infini ty);\nif degree(numer(f(x))) <= degree(denom(f(x))) then \nprint(`hori zontal asymptote: `*y=limit(f(x),x=infinity));\nelse\nprint(`no horizo ntal asymptote`);\nfi:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "___ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 "Filename: ExploreCalc04A.mws" }}{PARA 0 " " 0 "" {TEXT -1 36 "Copyright 2005, All Rights Reserved." }}{PARA 0 " " 0 "" {TEXT -1 44 "Permission is granted to use and modify for " }} {PARA 0 "" 0 "" {TEXT -1 37 "academic and non-commercial purposes." }} {PARA 0 "" 0 "" {TEXT -1 13 "Dr. John Pais" }}{PARA 0 "" 0 "" {TEXT -1 28 "Mathematics Department-MICDS" }}{PARA 0 "" 0 "" {TEXT -1 45 "E- mail: pais@micds.org or pais@kinetigram.com" }}{PARA 0 "" 0 "" {TEXT -1 33 "URL: http://kinetigram.com/micds" }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}}{MARK "7 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }