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{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 32 "f:=x->(x+4)*(x-3)^2/(x^4*(x-1));" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 201 "From the sign chart for f plus the vertical and horizont
al asymptote information, we expect at least 4 critical points and 3 p
oints of inflection, which is what we find [see the hand drawn \"road \+
map\"]." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "stewart 4-6-15: ratio
nal function" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Here is the defaul
t graphing 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 -1 136 "If we let maple show all the points in this domai
n we get something which is practically useless because of the two ver
tical asymptotes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(f
(x),x=-10..10,numpoints=100);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118
"You can see that the right vertical asymptote is squeezing together a
t a much smaller length scale than the first one:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 45 "plot(f(x),x=-5..3,y=-200..100,numpoints=200)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "However, trying to capture \+
these vertical asymptotes squashes all the remaining interesting behav
ior of the function." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "symbolic
derivatives, critical numbers, possible inflection points" }}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 55 "The normal simplifying command also facto
rs the result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "D(f)(x); \+
normal(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "By setting only th
e numerator to zero we can find all roots, since fsolve on a polynomia
l will find all roots:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f
solve((x-3)*(2*x^3-x^2-59*x+48)): evalf(%,3);" }}}{EXCHG {PARA 0 "" 0
"" {TEXT -1 96 "\"map\" is a way to apply a function to each input in \+
a list of numbers, to get the corresponding " }{XPPEDIT 18 0 "y;" "6#%
\"yG" }{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;" "6#\"#5" }{TEXT -1 2 ", " }{XPPEDIT
18 0 "10^0;" "6#*$\"#5\"\"!" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "10^(
-2);" "6#)\"#5,$\"\"#!\"\"" }{TEXT -1 5 ", so " }{TEXT 256 46 "no one \+
window can capture all features at once" }{TEXT -1 6 ". The " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 54 "-values are on about the sa
me scales: 1, not like the " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1
8 "-values." }}{PARA 0 "" 0 "" {TEXT -1 32 "Also notice that the facto
rs of " }{XPPEDIT 18 0 "x-3;" "6#,&%\"xG\"\"\"\"\"$!\"\"" }{TEXT -1
23 " (critical number) and " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1
95 " (v-asym) obviously change sign in the derivative as it crosses th
rough zero but the factor of " }{XPPEDIT 18 0 "x-1;" "6#,&%\"xG\"\"\"F
%!\"\"" }{TEXT -1 217 " (v-asym) does not; the remaining 3 critical nu
mbers are zeros of a cubic, so since they are all distinct, the cubic \+
must change sign as one crosses these zeros, so all critical numbers c
orrespond 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 59 "fsolve(360-954*x+759*x^2-133*x^3+3*x^5-15*x^4): 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 -1 222 "We expected 3 points of inflectio
n from the sign chart for f alone with its asymptote information. To c
heck that the signs agree as we cross the 3 roots of the 6th degree fa
ctor, we plot to see if the sign always changes:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 54 "plot(360-954*x+759*x^2-133*x^3+3*x^5-15*x^4,
x=-10..0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(360-954*
x+759*x^2-133*x^3+3*x^5-15*x^4,x=2..4);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 52 "plot(360-954*x+759*x^2-133*x^3+3*x^5-15*x^4,x=4..8);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "So yes, they all change sign,
so all correspond to points of inflection." }}}}}{EXCHG {PARA 0 "" 0
"" {TEXT -1 101 "Now we scan from left to right zooming to see the beh
avior in each interval at the appropriate scale:" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 45 "plot(f(x),x=-500..-4); #shows left hor asympt
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(x),x=-10..-3);"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(f(x),x=-1.5..2.5,y=-
500..300,numpoints=1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
65 "plot(f(x),x=2.5..10,numpoints=100); #shows right hor asympt start
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(f(x),x=10..200,num
points=100); # shows right hor asympt " }}}}}{MARK "0 0 0" 0 }
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