{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 "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 "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 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 }{PSTYLE "Heading 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 "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Tim es" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 } } {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 51 "multiple length scale beh avior in a single function" }}{PARA 0 "" 0 "" {TEXT -1 247 "In this ex ample interesting behavior occurs on many different scales (which can \+ only be found by using derivative information), so many windows are ne eded to see all these details, but we can exaggerate them to make our \+ \"road map\" of the function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=x->x^2*(x+1)^3/((x-2)^2*(x-4)^4);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "simplify(D(f)(x)):\nsimplify((D@@2)(f)(x)):" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 "stewart 4-6 example 3: rational f unction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Here is the default gra phing calculator window:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(f(x),x=-10..10,-10..10,numpoints=100);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 137 "If we let maple show all the points in this domain we \+ get something which is practically useless because of the two vertical asymptotes::" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(f(x), x=-10..10,numpoints=100);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "You can see that the right vertical asymptote is squeezing together at a \+ much larger length scale than the first one:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(f(x),x=0..10,y=0..4000,numpoints=200);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "However, trying to capture these \+ vertical asymptotes squashes all the remaining interesting behavior of the function." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "symbolic deriv atives, critical numbers, possible inflection points" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The normal simplifying command also factors the result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "D(f)(x); normal (%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "By setting only the nume rator to zero we can find all roots, since fsolve on a polynomial will find all roots:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fsolve( x*(x+1)*(x^3+18*x^2-44*x-16)): evalf(%,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "\"map\" is a way to apply a function to each input in a l ist of numbers, to get the corresponding " }{XPPEDIT 18 0 "y;" "6#%\"y G" }{TEXT -1 35 " values for these critical numbers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "map(f,[%]): evalf(%,3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Notice the 3 different scales of the " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 38 "-values in this list, on th e order of " }{XPPEDIT 18 0 "10^(-2);" "6#)\"#5,$\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "10^(-5);" "6#)\"#5,$\"\"&!\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "10^3;" "6#*$\"#5\"\"$" }{TEXT -1 5 ", so " } {TEXT 256 48 "no one window can capture all 3 features at once" } {TEXT -1 6 ". The " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 40 "-value s are also on 3 different scales: " }{XPPEDIT 18 0 "10;" "6#\"#5" } {TEXT -1 5 ", 1, " }{XPPEDIT 18 0 "10^(-1);" "6#)\"#5,$\"\"\"!\"\"" } {TEXT -1 35 ", though not as exaggerated as the " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 8 "-values." }}{PARA 0 "" 0 "" {TEXT -1 31 "Also n otice that the factor of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 88 " obviously changes sign in the derivative as it crosses through zero \+ but the factor and " }{XPPEDIT 18 0 "(x+1)^2;" "6#*$,&%\"xG\"\"\"F&F& \"\"#" }{TEXT -1 184 " does not; the remaining 3 critical numbers are \+ zeros of a cubic, so since they are all distinct, the cubic must chang e sign as one crosses these zeros, so all critical numbers except " } {XPPEDIT 18 0 "x = -1;" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 32 " corresp ond to relative extrema." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(D@@2)(f)(x); normal(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Now \+ we use the same procedure to determine the coordinates of the possible points of inflection" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "fs olve((x+1)*(64+672*x+684*x^2-628*x^3+6*x^4+x^6+36*x^5)): evalf(%,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "map(f,[%]): evalf(%,3);" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 28 "really points of inflection?" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "while the factor of " }{XPPEDIT 18 0 "x+1;" "6#,&%\"xG\"\"\"F%F%" }{TEXT -1 147 " clearly changes sign when it vanishes, we don't know about the 4 roots of the 6th degree f actor, so we can plot to see if the sign always changes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(64+672*x+36*x^5+x^6+684*x^2-62 8*x^3+6*x^4,x=-40..0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "p lot(64+672*x+36*x^5+x^6+684*x^2-628*x^3+6*x^4,x=-5.5..-4.5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(64+672*x+36*x^5+x^6+684 *x^2-628*x^3+6*x^4,x=-0.8..-0.3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot(64+672*x+36*x^5+x^6+684*x^2-628*x^3+6*x^4,x=-0.3 ..0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "So yes, they all change \+ sign, so all correspond to points of inflection." }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(f(x),x=-10..10,-10..10); #not very use ful" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(f(x),x=-10000.. 0); #shows left hor asympt" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(f(x),x=-40..0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(f(x),x=-1.5..0.2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot(f(x),x=0..10,0..500,numpoints=100); #shows right hor asympt star t" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(f(x),x=10..500,nu mpoints=100); # shows right hor asympt " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }