C\303\241lculo IILi\303\247\303\243o 29: Curvas de CrescimentoCrescimento IlimitadoConsideremos uma popula\303\247\303\243o que varia com o tempo, denominada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ0RidGNEY3Rj5GKw==. Seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R0Y9= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1JI21uR0YkNiRRIjBGJ0Y+Rj5GKw== e seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn a taxa de crescimento. A equa\303\247\303\243o diferencial que representa o crescimento ilimitado da popula\303\247\303\243o \303\251 dada por: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, com condi\303\247\303\243o inicial, 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restart:eq1:=Diff(P(t),t)=lambda*P(t);eq2:=eq1/P(t);eq3:=Int(lhs(eq2),t)=Int(rhs(eq2),t);eq4:=value(eq3);eq5:=map(exp,eq4);Embora a resposta esteja correta, o Maple n\303\243o insere as constantes de integra\303\247\303\243o. Neste caso, \303\251 preciso inser\303\255-las manualmente, i.e.,eq4a:=lhs(eq4)=rhs(eq4)+K;eq5a:=map(exp,eq4a);eq6:=expand(eq5a);eq7:=value(subs(t=0,eq6));eq8:=subs(exp(K)=P[0],eq6);\303\211 poss\303\255vel automatizar todo o processo com o Maple:ic:=P(0)=P[0]:dsolve({eq1,ic},P(t));Ex. 1 Utilizando o modelo de crescimento ilimitado, verificar quanto tempo leva para uma popula\303\247\303\243o de 1000 indiv\303\255duos, com taxa de crescimento de 1% ao ano, aumentar para 1500 indiv\303\255duos.restart:eq1:=Diff(P(t),t)=.01*P(t);ic:=P(0)=1000:eq2:=dsolve({eq1,ic},P(t));plot(rhs(eq2),t=0..50);answer:=fsolve(1500=rhs(eq2),t);Crescimento LimitadoSuponhamos que a popula\303\247\303\243o seja limitada por algum valor m\303\241ximo, a capacidade de suporte, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRJG1heEYnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn. Neste caso, a taxa de crescimento n\303\243o depende da popula\303\247\303\243o, mas da oportunidade para crescimento, a capacidade dispon\303\255vel, 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, e de um par\303\242metro do crescimento, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn.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restart:eq1:=Diff(P(t),t)=lambda*(P_max-P(t));ic:=P(0)=P[0];eq2:=eq1/(P_max-P(t));eq3:=Int(lhs(eq2),t)=Int(rhs(eq2),t)+K;eq4:=value(eq3);eq5:=map(exp,eq4);eq6:=P(t)=solve(eq5,P(t));eq7:=expand(simplify(eq6));eq8:=expand(subs(t=0,eq7));const:=solve(eq8,exp(K));eq9:=subs(exp(K)=const,eq7);Por fim, utilizamos a fun\303\247\303\243o interna do Maple, dsolve.dsolve({eq1,ic},P(t));Ex. 2Suponhamos que um lago seja capaz de suportar uma popula\303\247\303\243o de 1000 peixes do tipo tuvira. Se 10 tuviras s\303\243o inseridas no lago, apresentando um par\303\242metro de crescimento de 200%, quanto tempo levar\303\241 para que se chegue a uma popula\303\247\303\243o com 900 tuviras.restart:eq1:=Diff(P(t),t)=2*(1000-P(t));ic:=P(0)=10;eq2:=dsolve({eq1,ic},P(t));plot(rhs(eq2),t=0..4);A curva aproxima-se de 1000 assintoticamente. No entanto, buscamos o ponto onde 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.fsolve(rhs(eq2)=999.5,t);Crescimento Log\303\255sticoEm muitas circunst\303\242ncias, uma popula\303\247\303\243o pode, inicialmente, apresentar um modelo de crescimento ilimitado. Isto acontece enquanto os recursos excedem as necessidades da popula\303\247\303\243o. Com o passar do tempo, a limita\303\247\303\243o relativa dos recursos e a competi\303\247\303\243o resultante entre os indiv\303\255duos resulta em um modelo de crescimento limitado. Esta situa\303\247\303\243o \303\251 representada pela equa\303\247\303\243o log\303\255stica, 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 .restart:eq1:=Diff(P(t),t)=lambda*P(t)-K*P(t)^2;ic:=P(0)=P[0];eq2:=dsolve({eq1,ic},P(t));Para uma interpreta\303\247\303\243o do par\303\242metro K, deve-se encontrar a ass\303\255ntota horizontal da curva que representa a popula\303\247\303\243o m\303\241xima (capacidade de suporte).eq3:=Pop[max]=limit(rhs(eq2),t=infinity) assuming lambda>0;K:=solve(eq3,K);eq4:=eq1; Ex. 3Em um meio com capacidade de suporte igual a 1000, tra\303\247ar o gr\303\241fico da curva de crescimento log\303\255stico de uma popula\303\247\303\243o inicial de 10 indiv\303\255duos, com taxa de crescimento de 10% ao ano.restart:eq1:=Diff(P(t),t)=lambda*P(t)-lambda/Pop[max]*P(t)^2:ic:=P(0)=10: lambda:=.01: Pop[max]:=1000:eq1;pop:=dsolve({eq1,ic},P(t));plot(rhs(pop),t=0..1000);Em geral, tem-se os dados da popula\303\247\303\243o, cujos par\303\242metros da curva de crescimento apropriada devem ser encontrados.Ex. 4Dada a tabela de dados a seguir, selecionar um modelo de popula\303\247\303\243o, encontrar os par\303\242metros do modelo e tra\303\247ar o gr\303\241fico da curva obtida pelo modelo e os pontos representados pelos dados originais no mesmo conjunto de eixos.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restart: with(plots):pts:=[[2,6],[4,9],[6,13],[8,20],[10,30]];pointplot(pts);Pelo gr\303\241fico, verifica-se que, ao menos neste intervalo, a popula\303\247\303\243o pode ser modelada por uma curva de crescimento ilimitado, 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. \303\211 preciso determinar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R0Y9. eq1:=P(t)=exp(lambda*t)*P[0];eq2:=map(log,eq1);eq3:=simplify(eq2,symbolic);Esta \303\251 uma equa\303\247\303\243o linear em t. Pode-se, no entanto, utilizar a fun\303\247\303\243o de ajuste, contida no pacote stats, ap\303\263s a convers\303\243o dos valores para as vari\303\241veis da popula\303\247\303\243o em logar\303\255tmos.with(stats):pop:=[seq(pts[k,2],k=1..nops(pts))];lpop:=evalf(map(log,pop));tvals:=[seq(pts[k,1],k=1..nops(pts))];eq4:=fit[leastsquare[[t,p],p=lambda*t+P,{lambda,P}]]([tvals,lpop]);eq5:=P(t)=exp(rhs(eq4));eq6:=expand(eq5);P1:=plot(rhs(eq6),t=0..10):P2:=pointplot(pts):display(P1,P2);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R0Y9 \303\251 igual a 4, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn \303\251 igual a 0.2. Ex. 5Repetir o Prob# 4 com os dados a seguir. 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:with(plots):pts:=[[0,200],[4,760],[8,930],[12,980],[16,995],[20,998]];pointplot(pts,symbol=circle,color=red);Com base no gr\303\241fico, pode-se adotar um modelo de crescimento limitado. A popula\303\247\303\243o m\303\241xima \303\251 estimada pelo gr\303\241fico em 1000.eq1:=diff(P(t),t)=lambda*(1000-P(t));dsolve(eq1,P(t));A fim de simplificar a tarefa de ajuste da curva, deve-se fazer uma mudan\303\247a de coordenadas. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiUtRiw2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEidEYnRjZGOUZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGXG8tRiM2Ji1JI21uR0YkNiRRJTEwMDBGJ0ZALUY9Ni1RKCZtaW51cztGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGaG8tRiM2JS1GLDYlUSJQRidGNkY5RjxGVkYrRitGKw==.eq2:=subs(P(t)=1000-Q(t),eq1);eq3:=dsolve(eq2,Q(t));value(subs(t=0,eq3));Da tabela de valores, verifica-se que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiUtRiw2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtSSNtbkdGJDYkUSIwRidGQEZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGXW8tRiM2Ji1GZm42JFElMTAwMEYnRkAtRj02LVEoJm1pbnVzO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZoby1GIzYlLUYsNiVRIlBGJ0Y2RjlGPEZWRitGK0Yr = 800. Pode-se ent\303\243o calcular Q(t) para cada instante de tempo e proceder como no caso do crescimento ilimitado.Q(t):=[seq(1000-pts[k,2],k=1..6)];Qvals:=evalf(map(log,Q(t)));Tvals:=[seq(4*k,k=0..5)];with(stats):eq4:=fit[leastsquare[[t,Q], Q=a+lambda*t,{a,lambda}]]([Tvals,Qvals]);A substitui\303\247\303\243o deve ser desfeita, de forma o obter-se as coordenadas originais.eq5:=1000-P=evalf(exp(rhs(eq4)));P:=solve(eq5,P);P:=expand(P);P1:=plot(P,t=0..20,color=blue):P2:=pointplot(pts,symbol=circle,color=red):display(P1,P2);Suponhamos que n\303\243o se tenha uma estimativa da popula\303\247\303\243o m\303\241xima boa o suficiente, ou mesmo da popula\303\247\303\243o no instante zero.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restart:with(plots):with(stats):pts := [[1, 410], [2, 560], [3, 675], [4, 760], [5, 820], [6, 870], [7, 900], [8, 925], [9, 945], [10, 960]]:pointplot(pts);Neste caso, \303\251 necess\303\241rio estimar valores para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1JJm1mcmFjR0YkNigtRiM2Iy1JI21vR0YkNi9RMCZEaWZmZXJlbnRpYWxEO0YnLyUrZm9yZWdyb3VuZEdRLlsxNDQsMTQ0LDE0NF1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvSSttc2VtYW50aWNzR0YkUSZpbmVydEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlQtRiM2JUYrLUYjNiRGNi1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvRj5RJ2l0YWxpY0YnRisvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmJvLyUpYmV2ZWxsZWRHRkUtRjc2LVExJkludmlzaWJsZVRpbWVzO0YnRj1GQ0ZGRkhGSkZMRk5GUEZSRlUtRiM2JS1GLDYlUSJQRidGaG5GW28tRjc2LVEwJkFwcGx5RnVuY3Rpb247RidGPUZDRkZGSEZKRkxGTkZQRlJGVS1JKG1mZW5jZWRHRiQ2JC1GIzYjRmVuRj1GK0Yr e ajustar os dados fornecidos \303\240 equa\303\247\303\243o diferencial original. Como uma estimativa para a derivada em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRImpGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== , utiliza-se a inclina\303\247\303\243o da secante que conecta 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 a 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, i.e.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. Pop:=[seq(pts[k,2],k=1..10)];Neste caso, os valores sucessivos de t diferem por uma unidade, de forma que o denominador \303\251 igual a um.for k to 9 do dPt[k]:=Pop[k+1]-Pop[k] od:dP:=[seq(dPt[k],k=1..9)];A equa\303\247\303\243o diferencial apropriada \303\251 dada por 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.Popvals:=[Pop[j] $ j=1..nops(Pop)-1];fiteq2:=fit[leastsquare[[p,dp], dp=A+B*p,{A,B}]]([Popvals,dP]);eq2:=evalf(eq2);eq3:=Diff(p(t),t)=256.8917482-.2566679874*p(t);ic:=p(5)=820: # Select a point in the middle of the range as a boundary condition. eq4:=dsolve({eq3,ic},p(t));eq4:=evalf(eq4);P1:=plot(rhs(eq4),t=0..10):P2:=pointplot(pts):display(P1,P2):A resposta ainda pode ser melhorada. Estes resultados foram obtidos pela aproxima\303\247\303\243o da derivada em um ponto, atrav\303\251s da diferen\303\247a adiantada: 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 = 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. Pode-se melhorar os resultados atrav\303\251s da diferen\303\247a central: 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=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. Calcula-se assim os valores intermedi\303\241rios de f(x) tomando-se a m\303\251dia de valores sucessivos de f(x). Com este procedimento, pode-se determinar a inclina\303\247\303\243o da secante que une dois pontos \303\240 derivada do ponto m\303\251dio entre estes dois pontos. Os dados podem ser modificados, de forma a tomar valores intermedi\303\241rios entre os valores originais.pmod:=[seq((pts[k,2]+pts[k+1,2])/2.0,k=1..9)];Os valores obtidos pelo operador de diferen\303\247as centradas com os dados modificados s\303\243o os mesmos obtidos pelo operador de diferen\303\247as adiantadas com os dados originais. O fato de estarem associados aos dados modificados faz toda a diferen\303\247a. As diferen\303\247as adiantadas s\303\243o especificadas como dP=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m:=[seq((pts[k+1,2]-pts[k,2])/1.0,k=1..9)];eq5:=fit[leastsquare[[p,dp],dp=a*p+b,{a,b}]]([pmod,m]);eq6:=diff(p(t),t)=-.2943477008*p(t)+294.6269537;ic:=p(4.5)=790:eq7:=dsolve({eq6,ic},p(t));eq7:=evalf(eq7);P1:=plot(rhs(eq7),t=0..10):P2:=pointplot(pts):display(P1,P2);Verifica-se uma melhora significativa.Ex. 6Utilizando dos dados da tabela a seguir, encontrar a curva de crescimento da popula\303\247\303\243o.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restart: with(plots): with(stats):pts := [[4, 32], [8, 100], [12, 270], [16, 551], [20, 803], [24, 931], [28, 978], [32, 993], [36, 998], [40, 999]];pointplot(pts);Tem-se claramente uma curva de crescimento log\303\255stico. Dada a experi\303\252ncia com trabalhos anteriores, utilizaremos agora a diferen\303\247a central.pmod:=[seq((pts[t+1,2]+pts[t,2])/2.0,t=1..9)];m:=[seq((pts[t+1,2]-pts[t,2])/4.0,t=1..9)];eq1:=fit[leastsquare[[dp,p],dp=a*p+b*p^2,{a,b}]]([m,pmod]);eq2:=diff(p(t),t)=.2873151540*p(t)-.2877335513e-3*p(t)^2;ic:=p(20)=803:eq3:=dsolve({eq2,ic},p(t));eq3:=evalf(eq3);P1:=plot(rhs(eq3),t=0..40):P2:=pointplot(pts):display(P1,P2);Aplica\303\247\303\265esProb #1Quanto tempo leva para uma popula\303\247\303\243o que diminui em 2% ao ano cair para 1/2 de seu tamanho inicial?Prob #2Em que instante a taxa de crescimento de uma popula\303\247\303\243o descrita pela curva de crescimento log\303\255stico \303\251 m\303\241xima?Prob #3Ajustar a curva de crescimento ao seguinte conjunto de 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Prob # 4Ajustar a curva de crescimento limitado aos dados a seguir. Encontrar a popula\303\247\303\243o inicial e a capacidade de suporte do meio.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Prob # 5Ajustar a curva de crescimento log\303\255stico aos dados a seguir, e encontrar as popula\303\247\303\265es m\303\241xima e inicial.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