#a.)
def f(x): return (r*x*e^-(x/N)^2)
r= 2 ; N= 1000
#We verify our calculations by substituting the two equilibria for Pn in the original equation.
f(0)
#so this one must be correctly a fixed point.
n(f(sqrt(N^2*log(r))))
#This one as well must be a fixed point.
for x in range(0,101): print x,; print "implies",; print n(f(x))
var ('x')
plot(f(x),0,sqrt(N^2*log(r)))
0 implies 0.000000000000000 1 implies 1.99999800000100 2 implies 3.99998400003200 3 implies 5.99994600024300 4 implies 7.99987200102399 5 implies 9.99975000312497 6 implies 11.9995680077759 7 implies 13.9993140168067 8 implies 15.9989760327673 9 implies 17.9985420590474 10 implies 19.9980000999967 11 implies 21.9973381610445 12 implies 23.9965442488201 13 implies 25.9956063712721 14 implies 27.9945125377889 15 implies 29.9932507593181 16 implies 31.9918090484865 17 implies 33.9901754197202 18 implies 35.9883378893639 19 implies 37.9862844758011 20 implies 39.9840031995734 21 implies 41.9814820835007 22 implies 43.9787091528006 23 implies 45.9756724352082 24 implies 47.9723599610954 25 implies 49.9687597635908 26 implies 51.9648598786992 27 implies 53.9606483454209 28 implies 55.9561132058712 29 implies 57.9512425054003 30 implies 59.9460242927116 31 implies 61.9404466199823 32 implies 63.9344975429817 33 implies 65.9281651211907 34 implies 67.9214374179213 35 implies 69.9143025004351 36 implies 71.9067484400631 37 implies 73.8987633123239 38 implies 75.8903351970432 39 implies 77.8814521784727 40 implies 79.8721023454085 41 implies 81.8622737913102 42 implies 83.8519546144194 43 implies 85.8411329178787 44 implies 87.8297968098496 45 implies 89.8179344036316 46 implies 91.8055338177803 47 implies 93.7925831762258 48 implies 95.7790706083912 49 implies 97.7649842493103 50 implies 99.7503122397460 51 implies 101.735042726309 52 implies 103.719163861573 53 implies 105.702663804197 54 implies 107.685530719041 55 implies 109.667752777280 56 implies 111.649318156529 57 implies 113.630215040953 58 implies 115.610431621392 59 implies 117.589956095470 60 implies 119.568776667719 61 implies 121.546881549694 62 implies 123.524258960088 63 implies 125.500897124851 64 implies 127.476784277308 65 implies 129.451908658274 66 implies 131.426258516168 67 implies 133.399822107135 68 implies 135.372587695160 69 implies 137.344543552183 70 implies 139.315677958216 71 implies 141.285979201461 72 implies 143.255435578423 73 implies 145.224035394028 74 implies 147.191766961736 75 implies 149.158618603662 76 implies 151.124578650686 77 implies 153.089635442569 78 implies 155.053777328073 79 implies 157.016992665071 80 implies 158.979269820664 81 implies 160.940597171295 82 implies 162.900963102866 83 implies 164.860356010850 84 implies 166.818764300406 85 implies 168.776176386496 86 implies 170.732580693994 87 implies 172.687965657807 88 implies 174.642319722981 89 implies 176.595631344821 90 implies 178.547888989003 91 implies 180.499081131686 92 implies 182.449196259626 93 implies 184.398222870290 94 implies 186.346149471970 95 implies 188.292964583891 96 implies 190.238656736332 97 implies 192.183214470729 98 implies 194.126626339796 99 implies 196.068880907632 100 implies 198.009966749834 0 implies 0.000000000000000 1 implies 1.99999800000100 2 implies 3.99998400003200 3 implies 5.99994600024300 4 implies 7.99987200102399 5 implies 9.99975000312497 6 implies 11.9995680077759 7 implies 13.9993140168067 8 implies 15.9989760327673 9 implies 17.9985420590474 10 implies 19.9980000999967 11 implies 21.9973381610445 12 implies 23.9965442488201 13 implies 25.9956063712721 14 implies 27.9945125377889 15 implies 29.9932507593181 16 implies 31.9918090484865 17 implies 33.9901754197202 18 implies 35.9883378893639 19 implies 37.9862844758011 20 implies 39.9840031995734 21 implies 41.9814820835007 22 implies 43.9787091528006 23 implies 45.9756724352082 24 implies 47.9723599610954 25 implies 49.9687597635908 26 implies 51.9648598786992 27 implies 53.9606483454209 28 implies 55.9561132058712 29 implies 57.9512425054003 30 implies 59.9460242927116 31 implies 61.9404466199823 32 implies 63.9344975429817 33 implies 65.9281651211907 34 implies 67.9214374179213 35 implies 69.9143025004351 36 implies 71.9067484400631 37 implies 73.8987633123239 38 implies 75.8903351970432 39 implies 77.8814521784727 40 implies 79.8721023454085 41 implies 81.8622737913102 42 implies 83.8519546144194 43 implies 85.8411329178787 44 implies 87.8297968098496 45 implies 89.8179344036316 46 implies 91.8055338177803 47 implies 93.7925831762258 48 implies 95.7790706083912 49 implies 97.7649842493103 50 implies 99.7503122397460 51 implies 101.735042726309 52 implies 103.719163861573 53 implies 105.702663804197 54 implies 107.685530719041 55 implies 109.667752777280 56 implies 111.649318156529 57 implies 113.630215040953 58 implies 115.610431621392 59 implies 117.589956095470 60 implies 119.568776667719 61 implies 121.546881549694 62 implies 123.524258960088 63 implies 125.500897124851 64 implies 127.476784277308 65 implies 129.451908658274 66 implies 131.426258516168 67 implies 133.399822107135 68 implies 135.372587695160 69 implies 137.344543552183 70 implies 139.315677958216 71 implies 141.285979201461 72 implies 143.255435578423 73 implies 145.224035394028 74 implies 147.191766961736 75 implies 149.158618603662 76 implies 151.124578650686 77 implies 153.089635442569 78 implies 155.053777328073 79 implies 157.016992665071 80 implies 158.979269820664 81 implies 160.940597171295 82 implies 162.900963102866 83 implies 164.860356010850 84 implies 166.818764300406 85 implies 168.776176386496 86 implies 170.732580693994 87 implies 172.687965657807 88 implies 174.642319722981 89 implies 176.595631344821 90 implies 178.547888989003 91 implies 180.499081131686 92 implies 182.449196259626 93 implies 184.398222870290 94 implies 186.346149471970 95 implies 188.292964583891 96 implies 190.238656736332 97 implies 192.183214470729 98 implies 194.126626339796 99 implies 196.068880907632 100 implies 198.009966749834 |
x=1
1 Traceback (click to the left of this block for traceback) ... NameError: name 'f' is not defined 1
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_2.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("eD0xCndoaWxlIHggPD0gMjAwMDoKICAgcHJpbnQgeAogICB4ID0gbihmKHgpKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmp76uhPQ/___code___.py", line 4, in <module>
exec compile(u'while x <= _sage_const_2000 :\n print x\n x = n(f(x))
File "", line 3, in <module>
NameError: name 'f' is not defined
|
while x <= 2000:
print x
x = n(f(x))
#The function does not oscillate between any points
|
|
#b.)
def f(x): return (r*x*e^-(x/N)^2)
r= 3 ; N= 1000
f(0)
n(f(sqrt(N^2*log(r))))
for x in range(0,101): print x,; print "implies",; print n(f(x))
var ('x')
plot(f(x),0,sqrt(N^2*log(r)))
0 implies 0.000000000000000 1 implies 2.99999700000150 2 implies 5.99997600004800 3 implies 8.99991900036450 4 implies 11.9998080015360 5 implies 14.9996250046875 6 implies 17.9993520116639 7 implies 20.9989710252101 8 implies 23.9984640491510 9 implies 26.9978130885711 10 implies 29.9970001499950 11 implies 32.9960072415668 12 implies 35.9948163732301 13 implies 38.9934095569081 14 implies 41.9917688066833 15 implies 44.9898761389771 16 implies 47.9877135727298 17 implies 50.9852631295803 18 implies 53.9825068340459 19 implies 56.9794267137016 20 implies 59.9760047993601 21 implies 62.9722231252510 22 implies 65.9680637292010 23 implies 68.9635086528123 24 implies 71.9585399416431 25 implies 74.9531396453862 26 implies 77.9472898180488 27 implies 80.9409725181313 28 implies 83.9341698088069 29 implies 86.9268637581004 30 implies 89.9190364390675 31 implies 92.9106699299735 32 implies 95.9017463144725 33 implies 98.8922476817861 34 implies 101.882156126882 35 implies 104.871453750653 36 implies 107.860122660095 37 implies 110.848144968486 38 implies 113.835502795565 39 implies 116.822178267709 40 implies 119.808153518113 41 implies 122.793410686965 42 implies 125.777931921629 43 implies 128.761699376818 44 implies 131.744695214774 45 implies 134.726901605447 46 implies 137.708300726670 47 implies 140.688874764339 48 implies 143.668605912587 49 implies 146.647476373965 50 implies 149.625468359619 51 implies 152.602564089463 52 implies 155.578745792359 53 implies 158.553995706296 54 implies 161.528296078561 55 implies 164.501629165920 56 implies 167.473977234793 57 implies 170.445322561430 58 implies 173.415647432088 59 implies 176.384934143205 60 implies 179.353165001579 61 implies 182.320322324541 62 implies 185.286388440132 63 implies 188.251345687277 64 implies 191.215176415963 65 implies 194.177862987410 66 implies 197.139387774251 67 implies 200.099733160702 68 implies 203.058881542740 69 implies 206.016815328274 70 implies 208.973516937324 71 implies 211.928968802192 72 implies 214.883153367635 73 implies 217.836053091041 74 implies 220.787650442604 75 implies 223.737927905493 76 implies 226.686867976028 77 implies 229.634453163854 78 implies 232.580665992110 79 implies 235.525488997607 80 implies 238.468904730996 81 implies 241.410895756942 82 implies 244.351444654299 83 implies 247.290534016274 84 implies 250.228146450609 85 implies 253.164264579743 86 implies 256.098871040991 87 implies 259.031948486710 88 implies 261.963479584471 89 implies 264.893447017231 90 implies 267.821833483504 91 implies 270.748621697528 92 implies 273.673794389438 93 implies 276.597334305435 94 implies 279.519224207954 95 implies 282.439446875837 96 implies 285.357985104498 97 implies 288.274821706094 98 implies 291.189939509694 99 implies 294.103321361447 100 implies 297.014950124750 0 implies 0.000000000000000 1 implies 2.99999700000150 2 implies 5.99997600004800 3 implies 8.99991900036450 4 implies 11.9998080015360 5 implies 14.9996250046875 6 implies 17.9993520116639 7 implies 20.9989710252101 8 implies 23.9984640491510 9 implies 26.9978130885711 10 implies 29.9970001499950 11 implies 32.9960072415668 12 implies 35.9948163732301 13 implies 38.9934095569081 14 implies 41.9917688066833 15 implies 44.9898761389771 16 implies 47.9877135727298 17 implies 50.9852631295803 18 implies 53.9825068340459 19 implies 56.9794267137016 20 implies 59.9760047993601 21 implies 62.9722231252510 22 implies 65.9680637292010 23 implies 68.9635086528123 24 implies 71.9585399416431 25 implies 74.9531396453862 26 implies 77.9472898180488 27 implies 80.9409725181313 28 implies 83.9341698088069 29 implies 86.9268637581004 30 implies 89.9190364390675 31 implies 92.9106699299735 32 implies 95.9017463144725 33 implies 98.8922476817861 34 implies 101.882156126882 35 implies 104.871453750653 36 implies 107.860122660095 37 implies 110.848144968486 38 implies 113.835502795565 39 implies 116.822178267709 40 implies 119.808153518113 41 implies 122.793410686965 42 implies 125.777931921629 43 implies 128.761699376818 44 implies 131.744695214774 45 implies 134.726901605447 46 implies 137.708300726670 47 implies 140.688874764339 48 implies 143.668605912587 49 implies 146.647476373965 50 implies 149.625468359619 51 implies 152.602564089463 52 implies 155.578745792359 53 implies 158.553995706296 54 implies 161.528296078561 55 implies 164.501629165920 56 implies 167.473977234793 57 implies 170.445322561430 58 implies 173.415647432088 59 implies 176.384934143205 60 implies 179.353165001579 61 implies 182.320322324541 62 implies 185.286388440132 63 implies 188.251345687277 64 implies 191.215176415963 65 implies 194.177862987410 66 implies 197.139387774251 67 implies 200.099733160702 68 implies 203.058881542740 69 implies 206.016815328274 70 implies 208.973516937324 71 implies 211.928968802192 72 implies 214.883153367635 73 implies 217.836053091041 74 implies 220.787650442604 75 implies 223.737927905493 76 implies 226.686867976028 77 implies 229.634453163854 78 implies 232.580665992110 79 implies 235.525488997607 80 implies 238.468904730996 81 implies 241.410895756942 82 implies 244.351444654299 83 implies 247.290534016274 84 implies 250.228146450609 85 implies 253.164264579743 86 implies 256.098871040991 87 implies 259.031948486710 88 implies 261.963479584471 89 implies 264.893447017231 90 implies 267.821833483504 91 implies 270.748621697528 92 implies 273.673794389438 93 implies 276.597334305435 94 implies 279.519224207954 95 implies 282.439446875837 96 implies 285.357985104498 97 implies 288.274821706094 98 implies 291.189939509694 99 implies 294.103321361447 100 implies 297.014950124750 |
x=1
|
|
while x <= 2000:
print x
x = n(f(x))
#The function oscillates between 1283.71275330666 and 741.151903683755
#With Po= 1. This loop allows us to see that the system oscillates between (approx.) 741 and 1283, starting at P10.
WARNING: Output truncated! 1 2.99999700000150 8.99991000061200 26.9975431560116 80.9336180942653 241.215645282380 682.743082336629 1285.10126597814 739.311853084456 1284.01876398598 740.746306470987 1283.78162828514 741.060610800271 1283.72832592365 741.131262201464 1283.71627786270 741.147231880287 1283.71355120440 741.150846068819 1283.71293394614 741.151664245763 1283.71279420289 741.151849475724 1283.71276256547 741.151891411190 1283.71275540283 741.151900905275 1283.71275378123 741.151903054714 1283.71275341410 741.151903541342 1283.71275333098 741.151903651514 1283.71275331217 741.151903676456 1283.71275330791 741.151903682103 1283.71275330694 741.151903683382 1283.71275330672 741.151903683671 1283.71275330667 741.151903683736 1283.71275330666 741.151903683751 1283.71275330666 741.151903683754 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 ... 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 WARNING: Output truncated! 1 2.99999700000150 8.99991000061200 26.9975431560116 80.9336180942653 241.215645282380 682.743082336629 1285.10126597814 739.311853084456 1284.01876398598 740.746306470987 1283.78162828514 741.060610800271 1283.72832592365 741.131262201464 1283.71627786270 741.147231880287 1283.71355120440 741.150846068819 1283.71293394614 741.151664245763 1283.71279420289 741.151849475724 1283.71276256547 741.151891411190 1283.71275540283 741.151900905275 1283.71275378123 741.151903054714 1283.71275341410 741.151903541342 1283.71275333098 741.151903651514 1283.71275331217 741.151903676456 1283.71275330791 741.151903682103 1283.71275330694 741.151903683382 1283.71275330672 741.151903683671 1283.71275330667 741.151903683736 1283.71275330666 741.151903683751 1283.71275330666 741.151903683754 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 ... 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 1283.71275330666 741.151903683755 |
#c.)
def f(x): return (r*x*e^-(x/N)^2)
r= 3.7 ; N= 1000
f(0)
n(f(sqrt(N^2*log(r))))
for x in range(0,101): print x,; print "implies",; print n(f(x))
var ('x')
plot(f(x),0,sqrt(N^2*log(r)))
#Calculations similar to those of part (b) show that for r=3.7,the equilibria are 0 and 1143.82377123846. Now we calculate f'(x) at those points, and get: f'(0)= 3.7 and f'(1143.82377123846)= -1.616665639. So both equilibria are unstable.
0 implies 0.000000000000000 1 implies 3.69999630000185 2 implies 7.39997040005920 3 implies 11.0999001004496 4 implies 14.7997632018944 5 implies 18.4995375057812 6 implies 22.1992008143854 7 implies 25.8987309310924 8 implies 29.5981056606195 9 implies 33.2973028092377 10 implies 36.9963001849938 11 implies 40.6950755979323 12 implies 44.3936068603171 13 implies 48.0918717868534 14 implies 51.7898481949094 15 implies 55.4875139047384 16 implies 59.1848467397001 17 implies 62.8818245264824 18 implies 66.5784250953233 19 implies 70.2746262802320 20 implies 73.9704059192107 21 implies 77.6657418544763 22 implies 81.3606119326812 23 implies 85.0549940051352 24 implies 88.7488659280265 25 implies 92.4422055626430 26 implies 96.1349907755935 27 implies 99.8271994390286 28 implies 103.518809430862 29 implies 107.209798634990 30 implies 110.900144941517 31 implies 114.589826246967 32 implies 118.278820454516 33 implies 121.967105474203 34 implies 125.654659223154 35 implies 129.341459625805 36 implies 133.027484614117 37 implies 136.712712127799 38 implies 140.397120114530 39 implies 144.080686530175 40 implies 147.763389339006 41 implies 151.445206513924 42 implies 155.126116036676 43 implies 158.806095898076 44 implies 162.485124098222 45 implies 166.163178646718 46 implies 169.840237562894 47 implies 173.516278876018 48 implies 177.191280625524 49 implies 180.865220861224 50 implies 184.538077643530 51 implies 188.209829043671 52 implies 191.880453143910 53 implies 195.549928037765 54 implies 199.218231830225 55 implies 202.885342637968 56 implies 206.551238589578 57 implies 210.215897825764 58 implies 213.879298499575 59 implies 217.541418776619 60 implies 221.202236835281 61 implies 224.861730866934 62 implies 228.519879076162 63 implies 232.176659680975 64 implies 235.832050913021 65 implies 239.486031017806 66 implies 243.138578254910 67 implies 246.789670898200 68 implies 250.439287236046 69 implies 254.087405571538 70 implies 257.734004222700 71 implies 261.379061522703 72 implies 265.022555820083 73 implies 268.664465478951 74 implies 272.304768879212 75 implies 275.943444416775 76 implies 279.580470503768 77 implies 283.215825568753 78 implies 286.849488056936 79 implies 290.481436430382 80 implies 294.111649168228 81 implies 297.740104766896 82 implies 301.366781740302 83 implies 304.991658620072 84 implies 308.614713955751 85 implies 312.235926315017 86 implies 315.855274283889 87 implies 319.472736466942 88 implies 323.088291487514 89 implies 326.701917987919 90 implies 330.313594629655 91 implies 333.923300093618 92 implies 337.531013080307 93 implies 341.136712310037 94 implies 344.740376523144 95 implies 348.341984480199 96 implies 351.941514962214 97 implies 355.538946770849 98 implies 359.134258728623 99 implies 362.727429679118 100 implies 366.318438487192 0 implies 0.000000000000000 1 implies 3.69999630000185 2 implies 7.39997040005920 3 implies 11.0999001004496 4 implies 14.7997632018944 5 implies 18.4995375057812 6 implies 22.1992008143854 7 implies 25.8987309310924 8 implies 29.5981056606195 9 implies 33.2973028092377 10 implies 36.9963001849938 11 implies 40.6950755979323 12 implies 44.3936068603171 13 implies 48.0918717868534 14 implies 51.7898481949094 15 implies 55.4875139047384 16 implies 59.1848467397001 17 implies 62.8818245264824 18 implies 66.5784250953233 19 implies 70.2746262802320 20 implies 73.9704059192107 21 implies 77.6657418544763 22 implies 81.3606119326812 23 implies 85.0549940051352 24 implies 88.7488659280265 25 implies 92.4422055626430 26 implies 96.1349907755935 27 implies 99.8271994390286 28 implies 103.518809430862 29 implies 107.209798634990 30 implies 110.900144941517 31 implies 114.589826246967 32 implies 118.278820454516 33 implies 121.967105474203 34 implies 125.654659223154 35 implies 129.341459625805 36 implies 133.027484614117 37 implies 136.712712127799 38 implies 140.397120114530 39 implies 144.080686530175 40 implies 147.763389339006 41 implies 151.445206513924 42 implies 155.126116036676 43 implies 158.806095898076 44 implies 162.485124098222 45 implies 166.163178646718 46 implies 169.840237562894 47 implies 173.516278876018 48 implies 177.191280625524 49 implies 180.865220861224 50 implies 184.538077643530 51 implies 188.209829043671 52 implies 191.880453143910 53 implies 195.549928037765 54 implies 199.218231830225 55 implies 202.885342637968 56 implies 206.551238589578 57 implies 210.215897825764 58 implies 213.879298499575 59 implies 217.541418776619 60 implies 221.202236835281 61 implies 224.861730866934 62 implies 228.519879076162 63 implies 232.176659680975 64 implies 235.832050913021 65 implies 239.486031017806 66 implies 243.138578254910 67 implies 246.789670898200 68 implies 250.439287236046 69 implies 254.087405571538 70 implies 257.734004222700 71 implies 261.379061522703 72 implies 265.022555820083 73 implies 268.664465478951 74 implies 272.304768879212 75 implies 275.943444416775 76 implies 279.580470503768 77 implies 283.215825568753 78 implies 286.849488056936 79 implies 290.481436430382 80 implies 294.111649168228 81 implies 297.740104766896 82 implies 301.366781740302 83 implies 304.991658620072 84 implies 308.614713955751 85 implies 312.235926315017 86 implies 315.855274283889 87 implies 319.472736466942 88 implies 323.088291487514 89 implies 326.701917987919 90 implies 330.313594629655 91 implies 333.923300093618 92 implies 337.531013080307 93 implies 341.136712310037 94 implies 344.740376523144 95 implies 348.341984480199 96 implies 351.941514962214 97 implies 355.538946770849 98 implies 359.134258728623 99 implies 362.727429679118 100 implies 366.318438487192 |
x=1
while x <= 2000:
print x
x = n(f(x))
#The function does not oscillate between any points
Traceback (click to the left of this block for traceback) ... SyntaxError: invalid syntax Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_2.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("eD0xCmRlZiAKd2hpbGUgeCA8PSAyMDAwOgogICBwcmludCB4CiAgIHggPSBuKGYoeCkpCiAgICM="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpoNyHnq/___code___.py", line 4
def
^
SyntaxError: invalid syntax
|
#d.)
def f(x): return (r*x*e^-(x/N)^2)
r= 3.8 ; N= 1000
f(0)
n(f(sqrt(N^2*log(r))))
for x in range(0,101): print x,; print "implies",; print n(f(x))
var ('x')
plot(f(x),0,sqrt(N^2*log(r)))
#Calculations similar to those of part (b) show that for r=3.8,the equilibria are 0 and 1155.422462. Now we calculate f'(x) at those points, and get: f'(0)= 3.8 and f'(1155.422462)=-1.67002128 . So both equilibria are unstable.
0 implies 0.000000000000000 1 implies 3.79999620000190 2 implies 7.59996960006080 3 implies 11.3998974004617 4 implies 15.1997568019456 5 implies 18.9995250059374 6 implies 22.7991792147742 7 implies 26.5986966319328 8 implies 30.3980544622579 9 implies 34.1972299121901 10 implies 37.9962001899937 11 implies 41.7949425059846 12 implies 45.5934340727581 13 implies 49.3916521054170 14 implies 53.1895738217988 15 implies 56.9871764427043 16 implies 60.7844371921244 17 implies 64.5813332974684 18 implies 68.3778419897915 19 implies 72.1739405040220 20 implies 75.9696060791894 21 implies 79.7648159586513 22 implies 83.5595473903212 23 implies 87.3537776268956 24 implies 91.1474839260813 25 implies 94.9406435508225 26 implies 98.7332337695284 27 implies 102.525231856300 28 implies 106.316615091155 29 implies 110.107360760260 30 implies 113.897446156152 31 implies 117.686848577966 32 implies 121.475545331665 33 implies 125.263513730262 34 implies 129.050731094050 35 implies 132.837174750827 36 implies 136.622822036120 37 implies 140.407650293415 38 implies 144.191636874382 39 implies 147.974759139098 40 implies 151.756994456276 41 implies 155.538320203489 42 implies 159.318713767397 43 implies 163.098152543969 44 implies 166.876613938714 45 implies 170.654075366900 46 implies 174.430514253782 47 implies 178.205908034829 48 implies 181.980234155943 49 implies 185.753470073690 50 implies 189.525593255517 51 implies 193.296581179986 52 implies 197.066411336989 53 implies 200.835061227975 54 implies 204.602508366177 55 implies 208.368730276832 56 implies 212.133704497405 57 implies 215.897408577812 58 implies 219.659820080644 59 implies 223.420916581393 60 implies 227.180675668666 61 implies 230.939074944418 62 implies 234.696092024167 63 implies 238.451704537217 64 implies 242.205890126886 65 implies 245.958626450720 66 implies 249.709891180718 67 implies 253.459662003556 68 implies 257.207916620804 69 implies 260.954632749147 70 implies 264.699788120611 71 implies 268.443360482776 72 implies 272.185327599004 73 implies 275.925667248652 74 implies 279.664357227299 75 implies 283.401375346958 76 implies 287.136699436303 77 implies 290.870307340881 78 implies 294.602176923339 79 implies 298.332286063635 80 implies 302.060612659261 81 implies 305.787134625461 82 implies 309.511829895445 83 implies 313.234676420614 84 implies 316.955652170771 85 implies 320.674735134342 86 implies 324.391903318589 87 implies 328.107134749832 88 implies 331.820407473663 89 implies 335.531699555160 90 implies 339.240989079105 91 implies 342.948254150203 92 implies 346.653472893289 93 implies 350.356623453551 94 implies 354.057683996742 95 implies 357.756632709393 96 implies 361.453447799030 97 implies 365.148107494385 98 implies 368.840590045612 99 implies 372.530873724500 100 implies 376.218936824684 0 implies 0.000000000000000 1 implies 3.79999620000190 2 implies 7.59996960006080 3 implies 11.3998974004617 4 implies 15.1997568019456 5 implies 18.9995250059374 6 implies 22.7991792147742 7 implies 26.5986966319328 8 implies 30.3980544622579 9 implies 34.1972299121901 10 implies 37.9962001899937 11 implies 41.7949425059846 12 implies 45.5934340727581 13 implies 49.3916521054170 14 implies 53.1895738217988 15 implies 56.9871764427043 16 implies 60.7844371921244 17 implies 64.5813332974684 18 implies 68.3778419897915 19 implies 72.1739405040220 20 implies 75.9696060791894 21 implies 79.7648159586513 22 implies 83.5595473903212 23 implies 87.3537776268956 24 implies 91.1474839260813 25 implies 94.9406435508225 26 implies 98.7332337695284 27 implies 102.525231856300 28 implies 106.316615091155 29 implies 110.107360760260 30 implies 113.897446156152 31 implies 117.686848577966 32 implies 121.475545331665 33 implies 125.263513730262 34 implies 129.050731094050 35 implies 132.837174750827 36 implies 136.622822036120 37 implies 140.407650293415 38 implies 144.191636874382 39 implies 147.974759139098 40 implies 151.756994456276 41 implies 155.538320203489 42 implies 159.318713767397 43 implies 163.098152543969 44 implies 166.876613938714 45 implies 170.654075366900 46 implies 174.430514253782 47 implies 178.205908034829 48 implies 181.980234155943 49 implies 185.753470073690 50 implies 189.525593255517 51 implies 193.296581179986 52 implies 197.066411336989 53 implies 200.835061227975 54 implies 204.602508366177 55 implies 208.368730276832 56 implies 212.133704497405 57 implies 215.897408577812 58 implies 219.659820080644 59 implies 223.420916581393 60 implies 227.180675668666 61 implies 230.939074944418 62 implies 234.696092024167 63 implies 238.451704537217 64 implies 242.205890126886 65 implies 245.958626450720 66 implies 249.709891180718 67 implies 253.459662003556 68 implies 257.207916620804 69 implies 260.954632749147 70 implies 264.699788120611 71 implies 268.443360482776 72 implies 272.185327599004 73 implies 275.925667248652 74 implies 279.664357227299 75 implies 283.401375346958 76 implies 287.136699436303 77 implies 290.870307340881 78 implies 294.602176923339 79 implies 298.332286063635 80 implies 302.060612659261 81 implies 305.787134625461 82 implies 309.511829895445 83 implies 313.234676420614 84 implies 316.955652170771 85 implies 320.674735134342 86 implies 324.391903318589 87 implies 328.107134749832 88 implies 331.820407473663 89 implies 335.531699555160 90 implies 339.240989079105 91 implies 342.948254150203 92 implies 346.653472893289 93 implies 350.356623453551 94 implies 354.057683996742 95 implies 357.756632709393 96 implies 361.453447799030 97 implies 365.148107494385 98 implies 368.840590045612 99 implies 372.530873724500 100 implies 376.218936824684 |
x=1
while x <= 2000:
print x
x = n(f(x))
#the function does not oscilate around any points.
WARNING: Output truncated! 1 3.79999620000190 14.4397770485382 54.8597129488310 207.840452836032 756.402880229300 1622.02377776529 443.842524601279 1385.02671228922 772.920949082756 1616.09989190115 450.786239733717 1397.98345840354 752.521379230355 1623.17931923073 442.496259827208 1382.47429323098 776.965567025715 1614.40477684373 452.786128611905 1401.65040246403 746.789159929313 1624.71860488926 440.707079178696 1379.06191091444 782.385890463897 1611.98476178141 455.651227343884 1406.85316355743 738.688216859829 1626.55037724904 438.584131408836 1374.98304847046 788.884060191766 1608.86199250564 459.365654999337 1413.50934539649 728.380215465444 1628.29138555173 436.572626324334 1371.08839654675 795.107900254359 1605.64740642265 463.209666826552 1420.29186327345 717.942697123740 1629.37063741085 435.328753097858 1368.66548201232 798.989103848317 1603.53330178238 465.748972755357 1424.71313692700 711.175552933212 1629.69752461559 434.952467000382 1367.93033480724 800.168105832420 1602.87457723436 ... 1426.27231126946 708.796105800046 1629.74208681980 434.901187240942 1367.83007130648 800.328954557264 1602.78411443201 466.650982216379 1426.27231126945 708.796105800048 1629.74208681980 434.901187240942 1367.83007130648 800.328954557265 1602.78411443201 466.650982216380 1426.27231126946 708.796105800046 1629.74208681980 434.901187240942 1367.83007130648 800.328954557264 1602.78411443201 466.650982216379 1426.27231126945 708.796105800048 1629.74208681980 434.901187240942 1367.83007130648 800.328954557265 1602.78411443201 466.650982216380 1426.27231126946 708.796105800046 1629.74208681980 434.901187240942 1367.83007130648 800.328954557264 1602.78411443201 466.650982216379 1426.27231126945 708.796105800048 1629.74208681980 434.901187240942 1367.83007130648 800.328954557265 1602.78411443201 466.650982216380 1426.27231126946 708.796105800046 1629.74208681980 434.901187240942 1367.83007130648 800.328954557264 1602.78411443201 466.650982216379 1426.27231126945 708.796105800048 1629.74208681980 434.901187240942 WARNING: Output truncated! 1 3.79999620000190 14.4397770485382 54.8597129488310 207.840452836032 756.402880229300 1622.02377776529 443.842524601279 1385.02671228922 772.920949082756 1616.09989190115 450.786239733717 1397.98345840354 752.521379230355 1623.17931923073 442.496259827208 1382.47429323098 776.965567025715 1614.40477684373 452.786128611905 1401.65040246403 746.789159929313 1624.71860488926 440.707079178696 1379.06191091444 782.385890463897 1611.98476178141 455.651227343884 1406.85316355743 738.688216859829 1626.55037724904 438.584131408836 1374.98304847046 788.884060191766 1608.86199250564 459.365654999337 1413.50934539649 728.380215465444 1628.29138555173 436.572626324334 1371.08839654675 795.107900254359 1605.64740642265 463.209666826552 1420.29186327345 717.942697123740 1629.37063741085 435.328753097858 1368.66548201232 798.989103848317 1603.53330178238 465.748972755357 1424.71313692700 711.175552933212 1629.69752461559 434.952467000382 1367.93033480724 800.168105832420 1602.87457723436 ... 1426.27231126946 708.796105800046 1629.74208681980 434.901187240942 1367.83007130648 800.328954557264 1602.78411443201 466.650982216379 1426.27231126945 708.796105800048 1629.74208681980 434.901187240942 1367.83007130648 800.328954557265 1602.78411443201 466.650982216380 1426.27231126946 708.796105800046 1629.74208681980 434.901187240942 1367.83007130648 800.328954557264 1602.78411443201 466.650982216379 1426.27231126945 708.796105800048 1629.74208681980 434.901187240942 1367.83007130648 800.328954557265 1602.78411443201 466.650982216380 1426.27231126946 708.796105800046 1629.74208681980 434.901187240942 1367.83007130648 800.328954557264 1602.78411443201 466.650982216379 1426.27231126945 708.796105800048 1629.74208681980 434.901187240942 1367.83007130648 800.328954557265 1602.78411443201 466.650982216380 1426.27231126946 708.796105800046 1629.74208681980 434.901187240942 1367.83007130648 800.328954557264 1602.78411443201 466.650982216379 1426.27231126945 708.796105800048 1629.74208681980 434.901187240942 |
#e.)
def f(x): return (r*x*e^-(x/N)^2)
r= 4.2 ; N= 1000
f(0)
n(f(sqrt(N^2*log(r))))
for x in range(0,2001): print x,; print "implies",; print n(f(x))
var ('x')
plot(f(x),0,sqrt(N^2*log(r)))
WARNING: Output truncated! full_output.txt 1000 implies 1545.09365292006 1001 implies 1543.54701674838 1002 implies 1541.99730583670 1003 implies 1540.44453562649 1004 implies 1538.88872155279 1005 implies 1537.32987904400 1006 implies 1535.76802352182 1007 implies 1534.20317040102 1008 implies 1532.63533508936 1009 implies 1531.06453298744 1010 implies 1529.49077948854 1011 implies 1527.91408997851 1012 implies 1526.33447983561 1013 implies 1524.75196443037 1014 implies 1523.16655912547 1015 implies 1521.57827927560 1016 implies 1519.98714022729 1017 implies 1518.39315731882 1018 implies 1516.79634588006 1019 implies 1515.19672123232 1020 implies 1513.59429868825 1021 implies 1511.98909355167 1022 implies 1510.38112111747 1023 implies 1508.77039667144 1024 implies 1507.15693549014 1025 implies 1505.54075284080 1026 implies 1503.92186398116 1027 implies 1502.30028415933 1028 implies 1500.67602861367 1029 implies 1499.04911257267 1030 implies 1497.41955125480 1031 implies 1495.78735986838 1032 implies 1494.15255361143 1033 implies 1492.51514767161 1034 implies 1490.87515722599 1035 implies 1489.23259744101 1036 implies 1487.58748347229 1037 implies 1485.93983046454 1038 implies 1484.28965355138 1039 implies 1482.63696785529 1040 implies 1480.98178848740 1041 implies 1479.32413054743 1042 implies 1477.66400912352 1043 implies 1476.00143929211 1044 implies 1474.33643611782 1045 implies 1472.66901465334 1046 implies 1470.99918993928 1047 implies 1469.32697700406 1048 implies 1467.65239086376 1049 implies 1465.97544652203 1050 implies 1464.29615896995 1051 implies 1462.61454318592 1052 implies 1460.93061413549 1053 implies 1459.24438677131 1054 implies 1457.55587603295 1055 implies 1455.86509684680 1056 implies 1454.17206412594 1057 implies 1452.47679277005 1058 implies 1450.77929766524 ... 1943 implies 187.133819769679 1944 implies 186.503780621937 1945 implies 185.875441743903 1946 implies 185.248800298779 1947 implies 184.623853447227 1948 implies 184.000598347411 1949 implies 183.379032155029 1950 implies 182.759152023355 1951 implies 182.140955103275 1952 implies 181.524438543326 1953 implies 180.909599489734 1954 implies 180.296435086449 1955 implies 179.684942475183 1956 implies 179.075118795452 1957 implies 178.466961184606 1958 implies 177.860466777870 1959 implies 177.255632708380 1960 implies 176.652456107222 1961 implies 176.050934103463 1962 implies 175.451063824195 1963 implies 174.852842394564 1964 implies 174.256266937813 1965 implies 173.661334575314 1966 implies 173.068042426603 1967 implies 172.476387609422 1968 implies 171.886367239747 1969 implies 171.297978431830 1970 implies 170.711218298232 1971 implies 170.126083949857 1972 implies 169.542572495991 1973 implies 168.960681044334 1974 implies 168.380406701036 1975 implies 167.801746570733 1976 implies 167.224697756581 1977 implies 166.649257360291 1978 implies 166.075422482164 1979 implies 165.503190221122 1980 implies 164.932557674749 1981 implies 164.363521939321 1982 implies 163.796080109839 1983 implies 163.230229280066 1984 implies 162.665966542561 1985 implies 162.103288988708 1986 implies 161.542193708757 1987 implies 160.982677791852 1988 implies 160.424738326065 1989 implies 159.868372398433 1990 implies 159.313577094986 1991 implies 158.760349500785 1992 implies 158.208686699952 1993 implies 157.658585775703 1994 implies 157.110043810381 1995 implies 156.563057885489 1996 implies 156.017625081724 1997 implies 155.473742479004 1998 implies 154.931407156507 1999 implies 154.390616192699 2000 implies 153.851366665367 WARNING: Output truncated! full_output.txt 1000 implies 1545.09365292006 1001 implies 1543.54701674838 1002 implies 1541.99730583670 1003 implies 1540.44453562649 1004 implies 1538.88872155279 1005 implies 1537.32987904400 1006 implies 1535.76802352182 1007 implies 1534.20317040102 1008 implies 1532.63533508936 1009 implies 1531.06453298744 1010 implies 1529.49077948854 1011 implies 1527.91408997851 1012 implies 1526.33447983561 1013 implies 1524.75196443037 1014 implies 1523.16655912547 1015 implies 1521.57827927560 1016 implies 1519.98714022729 1017 implies 1518.39315731882 1018 implies 1516.79634588006 1019 implies 1515.19672123232 1020 implies 1513.59429868825 1021 implies 1511.98909355167 1022 implies 1510.38112111747 1023 implies 1508.77039667144 1024 implies 1507.15693549014 1025 implies 1505.54075284080 1026 implies 1503.92186398116 1027 implies 1502.30028415933 1028 implies 1500.67602861367 1029 implies 1499.04911257267 1030 implies 1497.41955125480 1031 implies 1495.78735986838 1032 implies 1494.15255361143 1033 implies 1492.51514767161 1034 implies 1490.87515722599 1035 implies 1489.23259744101 1036 implies 1487.58748347229 1037 implies 1485.93983046454 1038 implies 1484.28965355138 1039 implies 1482.63696785529 1040 implies 1480.98178848740 1041 implies 1479.32413054743 1042 implies 1477.66400912352 1043 implies 1476.00143929211 1044 implies 1474.33643611782 1045 implies 1472.66901465334 1046 implies 1470.99918993928 1047 implies 1469.32697700406 1048 implies 1467.65239086376 1049 implies 1465.97544652203 1050 implies 1464.29615896995 1051 implies 1462.61454318592 1052 implies 1460.93061413549 1053 implies 1459.24438677131 1054 implies 1457.55587603295 1055 implies 1455.86509684680 1056 implies 1454.17206412594 1057 implies 1452.47679277005 1058 implies 1450.77929766524 ... 1943 implies 187.133819769679 1944 implies 186.503780621937 1945 implies 185.875441743903 1946 implies 185.248800298779 1947 implies 184.623853447227 1948 implies 184.000598347411 1949 implies 183.379032155029 1950 implies 182.759152023355 1951 implies 182.140955103275 1952 implies 181.524438543326 1953 implies 180.909599489734 1954 implies 180.296435086449 1955 implies 179.684942475183 1956 implies 179.075118795452 1957 implies 178.466961184606 1958 implies 177.860466777870 1959 implies 177.255632708380 1960 implies 176.652456107222 1961 implies 176.050934103463 1962 implies 175.451063824195 1963 implies 174.852842394564 1964 implies 174.256266937813 1965 implies 173.661334575314 1966 implies 173.068042426603 1967 implies 172.476387609422 1968 implies 171.886367239747 1969 implies 171.297978431830 1970 implies 170.711218298232 1971 implies 170.126083949857 1972 implies 169.542572495991 1973 implies 168.960681044334 1974 implies 168.380406701036 1975 implies 167.801746570733 1976 implies 167.224697756581 1977 implies 166.649257360291 1978 implies 166.075422482164 1979 implies 165.503190221122 1980 implies 164.932557674749 1981 implies 164.363521939321 1982 implies 163.796080109839 1983 implies 163.230229280066 1984 implies 162.665966542561 1985 implies 162.103288988708 1986 implies 161.542193708757 1987 implies 160.982677791852 1988 implies 160.424738326065 1989 implies 159.868372398433 1990 implies 159.313577094986 1991 implies 158.760349500785 1992 implies 158.208686699952 1993 implies 157.658585775703 1994 implies 157.110043810381 1995 implies 156.563057885489 1996 implies 156.017625081724 1997 implies 155.473742479004 1998 implies 154.931407156507 1999 implies 154.390616192699 2000 implies 153.851366665367 |
#Calculations similar to those of part (b) show that for r=4.2,the equilibria are 0 and 1197.950135. Now we calculate f'(x) at those points, and get: f'(0)= 4.2 and f'(1197.950135)= -1.87016905. So both equilibria are unstable.
|
|
x=1
while x <= 2000:
print x
x = n(f(x))
#the function does not oscilate around any points.
WARNING: Output truncated! 1 4.19999580000210 17.6396711940868 74.0635699571127 309.365334494813 1180.74456516661 1230.07116328117 1137.78282988091 1309.46183510295 990.060197254931 1560.29639296784 574.322318420563 1734.42143328887 359.709221120891 1327.41641576366 957.226434311062 1608.15373367222 508.654062915176 1649.32882996165 456.196049561200 1556.02899677139 580.419214754779 1740.53651554933 353.387733838021 1309.98044046564 989.107732104282 1561.73658343102 572.273511968788 1732.29881055984 361.922446932878 1333.45236550446 946.258640291519 1623.26534850486 488.964153172995 1616.93516447862 497.150581829907 1630.78828288517 479.350895523642 1599.96995827805 519.528043020443 1665.85857085065 436.197740979416 1514.60768549964 641.596637194634 1785.40060157810 309.459526464851 1181.03522200103 1229.52964072234 1138.79772443448 1307.60516561929 993.472003011864 1555.11345902771 581.732337532020 1741.81413179796 352.077213865024 1306.32962380742 995.817600845464 1551.52863533775 586.891144755507 ... 559.753430955500 1718.58577322417 376.457241382883 1372.19776678651 876.841947735959 1707.11395985834 388.932549595374 1404.19846492794 821.008225076196 1757.35153922134 336.420962476827 1261.76738245818 1078.46746604769 1415.57168961638 801.536143823846 1770.74280556130 323.341436396255 1223.22205415026 1150.61978063366 1285.90066904158 1033.55402978755 1491.60686243347 677.089010588840 1798.01352863808 297.873713176361 1144.84598003286 1296.51814334589 1013.90331371739 1523.31997184446 628.432241675377 1778.25114151465 316.174114954699 1201.60268179170 1191.11727628214 1210.72083508671 1174.04552517232 1242.53932525964 1114.42399951471 1351.85990766868 913.054843163475 1666.05082029163 435.968728055419 1514.11487999065 642.345913756684 1785.76686362187 309.118423641989 1179.98237034643 1231.49099821626 1135.12193348248 1314.32443629034 981.138740707756 1573.66909372769 555.470378102196 1713.60131528479 381.842234570719 1386.15437882487 852.308986533478 1731.25704302064 363.012262388507 1336.41139491822 WARNING: Output truncated! 1 4.19999580000210 17.6396711940868 74.0635699571127 309.365334494813 1180.74456516661 1230.07116328117 1137.78282988091 1309.46183510295 990.060197254931 1560.29639296784 574.322318420563 1734.42143328887 359.709221120891 1327.41641576366 957.226434311062 1608.15373367222 508.654062915176 1649.32882996165 456.196049561200 1556.02899677139 580.419214754779 1740.53651554933 353.387733838021 1309.98044046564 989.107732104282 1561.73658343102 572.273511968788 1732.29881055984 361.922446932878 1333.45236550446 946.258640291519 1623.26534850486 488.964153172995 1616.93516447862 497.150581829907 1630.78828288517 479.350895523642 1599.96995827805 519.528043020443 1665.85857085065 436.197740979416 1514.60768549964 641.596637194634 1785.40060157810 309.459526464851 1181.03522200103 1229.52964072234 1138.79772443448 1307.60516561929 993.472003011864 1555.11345902771 581.732337532020 1741.81413179796 352.077213865024 1306.32962380742 995.817600845464 1551.52863533775 586.891144755507 ... 559.753430955500 1718.58577322417 376.457241382883 1372.19776678651 876.841947735959 1707.11395985834 388.932549595374 1404.19846492794 821.008225076196 1757.35153922134 336.420962476827 1261.76738245818 1078.46746604769 1415.57168961638 801.536143823846 1770.74280556130 323.341436396255 1223.22205415026 1150.61978063366 1285.90066904158 1033.55402978755 1491.60686243347 677.089010588840 1798.01352863808 297.873713176361 1144.84598003286 1296.51814334589 1013.90331371739 1523.31997184446 628.432241675377 1778.25114151465 316.174114954699 1201.60268179170 1191.11727628214 1210.72083508671 1174.04552517232 1242.53932525964 1114.42399951471 1351.85990766868 913.054843163475 1666.05082029163 435.968728055419 1514.11487999065 642.345913756684 1785.76686362187 309.118423641989 1179.98237034643 1231.49099821626 1135.12193348248 1314.32443629034 981.138740707756 1573.66909372769 555.470378102196 1713.60131528479 381.842234570719 1386.15437882487 852.308986533478 1731.25704302064 363.012262388507 1336.41139491822 |
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