Modelos
Professor Fernando Bastos
Modelo Exponencial
\(f(x)=\beta_{0}e^{\beta_{1}x}\)
\(f(0)=\beta_{0}\quad \textrm{e}\quad f(x)=0\Rightarrow \beta_{0}=0\)
Derivadas
Derivada Primeira
\(f^{'}(x)=\beta_{0}\beta_{1}e^{\beta_{1}x}\)
Pontos Críticos
\(f^{'}(x)=0\Rightarrow \beta_{0}=0\quad \textrm{ou} \quad \beta_{1}=0\)
Logo, \(f\) não possuí pontos críticos.
Derivada segunda
\(f^{''}(x)=\beta_{0}\beta_{1}^{2}e^{\beta_{1}x}\)
\(f^{''}(x)=0\Rightarrow \beta_{0}=0\quad \textrm{ou} \quad \beta_{1}=0\)
Logo, \(f\) não possuí pontos de inflexão.
Assintotas
\({\displaystyle \lim_{x\rightarrow +\infty}\beta_{0}e^{\beta_{1}x}=+\infty}\)
\({\displaystyle \lim_{x\rightarrow -\infty}\beta_{0}e^{\beta_{1}x}=0}\)
Logo, a reta \(y=0\) é uma assintotal horizontal de \(f.\)
Reta Tangente
\(f^{'}(0)=\beta_{0}\beta_{1},\) logo, a reta tangente ao gráfico passando pelo ponto \(P(0,\beta_{0})\) é dada por:
\(y=\beta_{0}\beta_{1}x+\beta_{0}\)
Exemplo de Gráfico
f <- function(beta0,beta1,x){
beta0*exp(beta1*x)
}
beta0 <- 2
beta1 <- 0.3
curve(f(beta0,beta1,x),
from = -10,
to =10,
col = "black",
lwd = 2,
panel.first = grid(20, 20 ,lty = 1, lwd = 1),
xlim = c(-10,10),
ylim = c(-1,10),
main = bquote("Modelo Exponencial" ~y==.(beta0)*exp(.(beta1)*x)),
xlab = "Valores de x",
ylab = "Valores de y")
tangente <- function(beta0,beta1,x){
beta0+beta0*beta1*x
}
curve(tangente(beta0,beta1,x), from = -10,
to =10,
col = "deepskyblue",
lwd = 2,
add = TRUE)
#--------PONTO (0,0) NO PLANO CARTESIANO--------#
x <- c(0,0)
y <- c(0,0)
points(x,
y,
pch = 20,
col ="black")
points(c(0,0),
c(beta0,beta0),
pch = 20,
col ="red")
text(x + 0.5,
y - 0.5,
paste(x),
col = "blue")
text(c(0,0)+1.2,
c(beta0,beta0)-0.5,
"beta0",
col = "blue")
abline(h=0, lwd = 2, col = "blue")
abline(v=0, lwd = 2, col = "black")
points(c(-1/beta1,-1/beta1),
c(0,0),
pch = 20,
col ="red")
text(c(-1/beta1,-1/beta1)+0.5,
c(0,0)-0.5,
"P1",
col = "blue")
O ponto \(P1\) no gráfico acima é dado por \(P(-\frac{1}{\beta_{1}},0),\) ele representa a intercessão da reta tangente ao ponto \(P(0,\beta_{0})\) com a assintota \(y=0.\)
Modelo Gompertz
\(f(x)=\beta_{0}e^{-\beta_{1}e^{\beta_{2}x}}\)
\(f(0)=\beta_{0}e^{-\beta_{1}}\quad \textrm{e}\quad f(x)=0\Rightarrow \beta_{0}=0\)
Derivadas
Derivada Primeira
\(f^{'}(x)=-\beta_{0}\beta_{1}\beta_{2}e^{\beta_{2}x-\beta_{1}e^{\beta_{2}x}}\)
Pontos Críticos
\(f^{'}(x)=0\Rightarrow \beta_{0}=0\quad \textrm{ou} \quad \beta_{1}=0\quad \textrm{ou} \quad \beta_{2}=0\)
Logo, \(f\) não possuí pontos críticos.
Derivada segunda
\(f^{''}(x)=\Big(-\beta_{0}\beta_{1}\beta_{2}e^{\beta_{2}x-\beta_{1}e^{\beta_{2}x}}\Big)\Big(\beta_{2}-\beta_{1}\beta_{2}e^{\beta_{2}x}\Big)\)
\(f^{''}(x)=0\Rightarrow x=\frac{1}{\beta_{2}}\ln{\frac{1}{\beta_{1}}}\)
Como, \(f(\frac{1}{\beta_{2}}\ln{\frac{1}{\beta_{1}}})=\frac{\beta_{0}}{e}\) temos que o ponto \(P(\frac{1}{\beta_{2}}\ln{\frac{1}{\beta_{1}}},\frac{\beta_{0}}{e})\) é um possível ponto de inflexão de \(f\), o qual denotamos por PI no gráfico abaixo.
Assintotas
\({\displaystyle \lim_{x\rightarrow +\infty}\beta_{0}e^{-\beta_{1}e^{\beta_{2}x}}=0}\)
\({\displaystyle \lim_{x\rightarrow -\infty}\beta_{0}e^{-\beta_{1}e^{\beta_{2}x}}=\beta_{0}}\)
Logo, as retas \(y=0\) e \(y=\beta_{0}\) são assintotas horizontais de \(f.\)
Reta Tangente
\(f^{'}(\frac{1}{\beta_{2}}\ln{\frac{1}{\beta_{1}}})=-\frac{\beta_{0}\beta_{1}}{e},\) logo, a reta tangente ao gráfico passando pelo ponto PI é dada por:
\(y=\frac{\beta_{0}}{e}\Big(-\beta_{1}x+\frac{\beta_{1}}{\beta_{2}}\ln{\frac{1}{\beta_{1}}}+1\Big)\)
Exemplo de Gráfico
f <- function(beta0,beta1,beta2,x){
beta0*exp(-beta1*exp(beta2*x))
}
beta0 <- 8
beta1 <- 0.5
beta2 <- 0.5
curve(f(beta0, beta1, beta2,x),
from = -10,
to =10,
col = "black",
lwd = 2,
panel.first = grid(20, 20 ,lty = 1, lwd = 1),
xlim = c(-10,10),
ylim = c(-1,10),
main = bquote("Modelo Gompertz" ~y==.(beta0)*exp(-.(beta1)*exp(.(beta2)*x))),
xlab = "Valores de x",
ylab = "Valores de y")
tangente <- function(beta0,beta1,beta2,x){
-beta0*beta1*x/exp(1)+(beta0*beta1/exp(1))*(1/beta2)*log(1/beta1)+beta0/exp(1)
}
curve(tangente(beta0,beta1,beta2,x), from = -10,
to =10,
col = "deepskyblue",
lwd = 2,
add = TRUE)
#--------PONTO (0,0) NO PLANO CARTESIANO--------#
x <- c(0,0)
y <- c(0,0)
points(x,
y,
pch = 20,
col ="black")
text(x + 0.5,
y - 0.5,
paste(x),
col = "blue")
points(c((1/beta2)*log(1/beta1),(1/beta2)*log(1/beta1)),
c(beta0/exp(1),beta0/exp(1)),
pch = 20,
col ="red")
text(c((1/beta2)*log(1/beta1),(1/beta2)*log(1/beta1))+1,
c(beta0/exp(1),beta0/exp(1))+0.5,
"PI",
col = "blue")
abline(h=0, lwd = 1.5, col = "blue")
abline(h=beta0, lwd = 1.5, col = "blue")
abline(v=0, lwd = 2, col = "black")
points(c((1/beta2)*log(1/beta1)+1/beta1,(1/beta2)*log(1/beta1)+1/beta1),
c(0,0),
pch = 20,
col ="red")
text(c((1/beta2)*log(1/beta1)+1/beta1,(1/beta2)*log(1/beta1)+1/beta1)+0.5,
c(0,0)-0.5,
"P1",
col = "blue")
points(c((1/beta2)*log(1/beta1)+1/beta1-exp(1)/beta1,(1/beta2)*log(1/beta1)+1/beta1-exp(1)/beta1),
c(beta0,beta0),
pch = 20,
col ="red")
text(c((1/beta2)*log(1/beta1)+1/beta1-exp(1)/beta1,(1/beta2)*log(1/beta1)+1/beta1-exp(1)/beta1)+0.5,
c(beta0,beta0)+0.5,
"P2",
col = "blue")
O ponto \(P1\) no gráfico acima é dado por \(P\Big(\frac{1}{\beta_{2}}\ln{\frac{1}{\beta1}}+\frac{1}{\beta_{1}},0),\) ele representa a intercessão da reta tangente ao ponto \(PI=P\Big(\frac{1}{\beta_{2}}\ln{\frac{1}{\beta_{1}}},\frac{\beta_{0}}{e}\Big)\) com a assintota \(y=0.\)
O ponto \(P2\) no gráfico é dado por \(P\Big(\frac{1}{\beta_{2}}\ln{\frac{1}{\beta1}}+\frac{1}{\beta_{1}}-\frac{e}{\beta_{1}},\beta_{0}),\) ele representa a intercessão da reta tangente ao ponto \(PI=P\Big(\frac{1}{\beta_{2}}\ln{\frac{1}{\beta_{1}}},\frac{\beta_{0}}{e}\Big)\) com a assintota \(y=\beta_{0}.\)
Modelo Logístico
\(f(x)=\frac{\beta_{0}}{1+\beta_{1}e^{-\beta_{2}x}}\)
\(f(0)=\frac{\beta_{0}}{1+\beta_{1}}\quad \textrm{e}\quad f(x)=0\Rightarrow \beta_{0}=0\)
Derivadas
Derivada Primeira
\(f^{'}(x)=\frac{\beta_{0}\beta_{1}\beta_{2}e^{-\beta_{2}x}}{\Big(1+\beta_{1}e^{-\beta_{2}x}\Big)^{2}}\)
Pontos Críticos
\(f^{'}(x)=0\Rightarrow \beta_{0}=0\quad \textrm{ou} \quad \beta_{1}=0\quad \textrm{ou} \quad \beta_{2}=0\)
Logo, \(f\) não possuí pontos críticos.
Derivada segunda
\(f^{''}(x)=\frac{\beta_{0}\beta_{1}\beta_{2}^{2}e^{-\beta_{2}x}\Big(\beta_{1}e^{-\beta_{2}x}-1\Big)}{\Big(1+\beta_{1}e^{-\beta_{2}x}\Big)^{3}}\)
\(f^{''}(x)=0\Rightarrow x=\frac{1}{\beta_{2}}\ln{\beta_{1}}\)
Como, \(f(\frac{1}{\beta_{2}}\ln{\beta_{1}})=\frac{\beta_{0}}{2}\) temos que o ponto \(P(\frac{1}{\beta_{2}}\ln{\beta_{1}},\frac{\beta_{0}}{2})\) é um possível ponto de inflexão de \(f\), o qual denotamos por PI no gráfico abaixo.
Assintotas
\({\displaystyle \lim_{x\rightarrow +\infty}\frac{\beta_{0}}{1+\beta_{1}e^{-\beta_{2}x}}=\beta_{0}}\)
\({\displaystyle \lim_{x\rightarrow -\infty}\frac{\beta_{0}}{1+\beta_{1}e^{-\beta_{2}x}}=0}\)
Logo, as retas \(y=0\) e \(y=\beta_{0}\) são assintotas horizontais de \(f.\)
Reta Tangente
\(f^{'}(\frac{1}{\beta_{2}}\ln{\beta_{1}})=\frac{\beta_{0}\beta_{2}}{4},\) logo, a reta tangente ao gráfico passando pelo ponto PI é dada por:
\(y=\frac{\beta_{0}}{4}\Big(\beta_{2}x-\ln{\beta_{1}}+2\Big)\)
Exemplo de Gráfico
f <- function(beta0,beta1,beta2,x){
beta0/(1+beta1*exp(-beta2*x))
}
beta0 <- 8
beta1 <- 0.5
beta2 <- 0.5
curve(f(beta0, beta1, beta2,x),
from = -10,
to =10,
col = "black",
lwd = 2,
panel.first = grid(20, 20 ,lty = 1, lwd = 1),
xlim = c(-10,10),
ylim = c(-1,10),
main = bquote("Modelo Logistico" ~y==.(beta0)*exp(-.(beta1)*exp(.(beta2)*x))),
xlab = "Valores de x",
ylab = "Valores de y")
tangente <- function(beta0,beta1,beta2,x){
(beta0/4)*(beta2*x-log(beta1)+2)
}
curve(tangente(beta0,beta1,beta2,x), from = -10,
to =10,
col = "deepskyblue",
lwd = 2,
add = TRUE)
#--------PONTO (0,0) NO PLANO CARTESIANO--------#
x <- c(0,0)
y <- c(0,0)
points(x,
y,
pch = 20,
col ="black")
text(x + 0.5,
y - 0.5,
paste(x),
col = "blue")
points(c((1/beta2)*log(beta1),(1/beta2)*log(beta1)),
c(beta0/2,beta0/2),
pch = 20,
col ="red")
text(c((1/beta2)*log(beta1),(1/beta2)*log(beta1))-1.5,
c(beta0/2,beta0/2)+0.5,
"PI",
col = "blue")
abline(h=0, lwd = 1.5, col = "blue")
abline(h=beta0, lwd = 1.5, col = "blue")
abline(v=0, lwd = 2, col = "black")
points(c((1/beta2)*log(beta1)-2/beta2,(1/beta2)*log(beta1)-2/beta2),
c(0,0),
pch = 20,
col ="red")
text(c((1/beta2)*log(beta1)-2/beta2,(1/beta2)*log(beta1)-2/beta2)+0.5,
c(0,0)-0.5,
"P1",
col = "blue")
points(c((1/beta2)*log(beta1)+2/beta2,(1/beta2)*log(beta1)+2/beta2),
c(beta0,beta0),
pch = 20,
col ="red")
text(c((1/beta2)*log(beta1)+2/beta2,(1/beta2)*log(beta1)+2/beta2)-0.5,
c(beta0,beta0)+0.75,
"P2",
col = "blue")
O ponto \(P1\) no gráfico acima é dado por \(P\Big(\frac{1}{\beta_{2}}\ln{\beta1}-\frac{2}{\beta_{2}},0),\) ele representa a intercessão da reta tangente ao ponto \(PI=P\Big(\frac{1}{\beta_{2}}\ln{\beta_{1}},\frac{\beta_{0}}{2}\Big)\) com a assintota \(y=0.\)
O ponto \(P2\) no gráfico é dado por \(P\Big(\frac{1}{\beta_{2}}\ln{\beta_{1}}+\frac{2}{\beta_{2}},\beta_{0}),\) ele representa a intercessão da reta tangente ao ponto \(PI=P\Big(\frac{1}{\beta_{2}}\ln{\beta_{1}},\frac{\beta_{0}}{2}\Big)\) com a assintota \(y=\beta_{0}.\)
Modelo Richards
\(f(x)=\frac{\beta_{0}}{\Big(1+e^{\beta_{1}-\beta_{2}x}\Big)^{\frac{1}{\beta_{3}}}}\)
\(f(0)=\frac{\beta_{0}}{\Big(1+e^{\beta_{1}}\Big)^{1/\beta_{3}}}\quad \textrm{e}\quad f(x)=0\Rightarrow \beta_{0}=0\)
Derivadas
Derivada Primeira
\(f^{'}(x)=\frac{\beta_{0}\beta_{2}e^{\beta_{1}-\beta_{2}x}}{\beta_{3}\Big(1+e^{\beta_{1}-\beta_{2}x}\Big)^{\frac{1+\beta_{3}}{\beta_{3}}}}\)
Pontos Críticos
\(f^{'}(x)=0\Rightarrow \beta_{0}=0\quad \textrm{ou} \quad \beta_{2}=0\)
Logo, \(f\) não possuí pontos críticos.
Derivada segunda
\(f^{''}(x)=\dfrac{\beta_{0}\beta_{2}^{2}e^{\beta_{1}-\beta_2x}}{\beta_{3}^{2}(1+e^{\beta_{1}-\beta_{2}x})}\Bigg[\frac{e^{\beta_{1}-\beta_{2}x}-\beta_{3}}{\Big(1+e^{\beta_{1}-\beta_{2}x}\Big)^{\frac{\beta_{3}+1}{\beta_{3}}}}\Bigg]\)
\(f^{''}(x)=0\Rightarrow x=\frac{\beta_{1}-\ln{\beta_{3}}}{\beta_{2}}\)
Como, \(f(\frac{\beta_{1}-\ln{\beta_{3}}}{\beta_{2}})=\frac{\beta_{0}}{(1+\beta_{3})^{1/\beta_{3}}}\) temos que o ponto \(P(\frac{\beta_{1}-\ln{\beta_{3}}}{\beta_{2}},\frac{\beta_{0}}{(1+\beta_{3})^{1/\beta_{3}}})\) é um possível ponto de inflexão de \(f\), o qual denotamos por PI no gráfico abaixo.
Assintotas
\({\displaystyle \lim_{x\rightarrow +\infty}\frac{\beta_{0}}{\Big(1+e^{\beta_{1}-\beta_{2}x}\Big)^{\frac{1}{\beta_{3}}}}=\beta_{0}}\)
\({\displaystyle \lim_{x\rightarrow -\infty}\frac{\beta_{0}}{\Big(1+e^{\beta_{1}-\beta_{2}x}\Big)^{\frac{1}{\beta_{3}}}}=0}\)
Logo, as retas \(y=0\) e \(y=\beta_{0}\) são assintotas horizontais de \(f.\)
Reta Tangente
\(f^{'}(\frac{\beta_{1}-\ln{\beta_{3}}}{\beta_{2}})=\frac{\beta_{0}\beta_{2}}{\Big(1+\beta_{3}\Big)^{\frac{\beta_{3}+1}{\beta_{3}}}},\) logo, a reta tangente ao gráfico passando pelo ponto PI é dada por:
\(y=\frac{\beta_{0}\beta_{2}x-\beta_{0}\beta_{1}+\beta_{0}\ln{\beta_{3}+\beta_{0}+\beta_{0}\beta_{3}}}{(1+\beta_{3})^{1/\beta_{3}}(1+\beta_{3})}\)
Exemplo de Gráfico
f <- function(beta0,beta1,beta2,beta3,x){
beta0/(1+exp(beta1-beta2*x))^(1/beta3)
}
beta0 <- 8
beta1 <- -0.5
beta2 <- 0.5
beta3 <- 2
curve(f(beta0, beta1, beta2,beta3,x),
from = -10,
to =10,
col = "black",
lwd = 2,
panel.first = grid(20, 20 ,lty = 1, lwd = 1),
xlim = c(-10,10),
ylim = c(-1,10),
main = bquote("Modelo Richards" ~y==.(beta0)/(1+exp(.(beta1)-.(beta2)*x))^(1/.(beta3))),
xlab = "Valores de x",
ylab = "Valores de y")
tangente <- function(beta0,beta1,beta2,beta3,x){
(beta0*beta2*x-beta0*beta1+beta0*log(beta3)+beta0+beta0*beta3)/(((1+beta3)^(1/beta3))*(1+beta3))
}
curve(tangente(beta0,beta1,beta2,beta3,x), from = -10,
to =10,
col = "deepskyblue",
lwd = 2,
add = TRUE)
#--------PONTO (0,0) NO PLANO CARTESIANO--------#
x <- c(0,0)
y <- c(0,0)
points(x,
y,
pch = 20,
col ="black")
text(x + 0.5,
y - 0.5,
paste(x),
col = "blue")
points(c((1/beta2)*(beta1-log(beta3)),(1/beta2)*(beta1-log(beta3))),
c(beta0/((1+beta3)^(1/beta3)),beta0/((1+beta3)^(1/beta3))),
pch = 20,
col ="red")
text(c((1/beta2)*(beta1-log(beta3)),(1/beta2)*(beta1-log(beta3)))-1.5,
c(beta0/((1+beta3)^(1/beta3)),beta0/((1+beta3)^(1/beta3)))+0.5,
"PI",
col = "blue")
abline(h=0, lwd = 1.5, col = "blue")
abline(h=beta0, lwd = 1.5, col = "blue")
abline(v=0, lwd = 2, col = "black")
points(c((beta1-log(beta3)-1-beta3)/beta2,(beta1-log(beta3)-1-beta3)/beta2),
c(0,0),
pch = 20,
col ="red")
text(c((beta1-log(beta3)-1-beta3)/beta2,(beta1-log(beta3)-1-beta3)/beta2)+0.5,
c(0,0)-0.5,
"P1",
col = "blue")
points(c((((1+beta3)^((beta3+1)/beta3))+beta1-log(beta3)-1-beta3)/beta2,(((1+beta3)^((beta3+1)/beta3))+beta1-log(beta3)-1-beta3)/beta2),
c(beta0,beta0),
pch = 20,
col ="red")
text(c((((1+beta3)^((beta3+1)/beta3))+beta1-log(beta3)-1-beta3)/beta2,(((1+beta3)^((beta3+1)/beta3))+beta1-log(beta3)-1-beta3)/beta2)-0.5,
c(beta0,beta0)+0.75,
"P2",
col = "blue")
O ponto \(P1\) no gráfico acima é dado por \(P\Big(\frac{\beta_{1}-\ln{\beta_{3}-1-\beta_{3}}}{\beta2},0\Big),\) ele representa a intercessão da reta tangente ao ponto \(PI=P\Big(\frac{\beta_{1}-\ln{\beta_{3}}}{\beta_{2}},\frac{\beta_{0}}{(1+\beta_{3})^{1/\beta_{3}}}\Big)\) com a assintota \(y=0.\)
O ponto \(P2\) no gráfico é dado por \(P\Big(\frac{(1+\beta_{3})^{\frac{\beta_{3}+1}{\beta_{3}}}+\beta_{1}-\ln{\beta_{3}-1-\beta_{3}}}{\beta_{2}},\beta_{0}\Big),\) ele representa a intercessão da reta tangente ao ponto \(PI=P\Big(\frac{\beta_{1}-\ln{\beta_{3}}}{\beta_{2}},\frac{\beta_{0}}{(1+\beta_{3})^{1/\beta_{3}}}\Big)\) com a assintota \(y=\beta_{0}.\)
Modelo Weibull
\(f(x)=\beta_{0}-\beta_{1}e^{-\beta_{2}x^{\beta_{3}}}\)
\(f(0)=\beta_{0}-\beta_{1}\)
Caso 1
- Se \(\beta_{2}>0\) e \(\beta_{0}<\beta_{1}\) ou \(\beta_{2}<0\) e \(\beta_{0}>\beta_{1},\) temos que:
\(f(x)=0\Rightarrow x=e^{\frac{1}{\beta_{3}}\ln{\Big(\frac{1}{\beta_{2}}\ln{\frac{\beta_{1}}{\beta_{0}}}\Big)}}\)
Caso 2
- Se \(\beta_{2}>0\) e \(\beta_{0}>\beta_{1}\) ou \(\beta_{2}<0\) e \(\beta_{0}<\beta_{1},\) temos que:
Não existe \(x\) tal que \(f(x)=0.\)
Derivadas
Derivada Primeira
\(f^{'}(x)=\beta_{1}\beta_{2}\beta_{3}x^{\beta_{3}-1}e^{-\beta_{2}x^{\beta_{3}}}\)
Pontos Críticos
\(f^{'}(x)=0\Rightarrow x=0\)
Logo, \(x=0\) é um ponto crítico de \(f\).
Derivada segunda
\(f^{''}(x)=\beta_{1}\beta_{2}\beta_{3}e^{-\beta_{2}x^{\beta_{3}}}\Bigg[(\beta_{3}-1)x^{(\beta_{3}-2)}-\beta_{2}\beta_{3}x^{2(\beta_{3}-1)}\Bigg]\)
Caso 1
- \(x>0\)
\(f^{''}(x)=0\Rightarrow x=e^{\frac{1}{\beta_{3}}\ln{\Big(\frac{\beta_{3}-1}{\beta_{2}\beta_{3}}\Big)}}\)
Como, \(f\Bigg(e^{\frac{1}{\beta_{3}}\ln{\Big(\frac{\beta_{3}-1}{\beta_{2}\beta_{3}}\Big)}}\Bigg)=\beta_{0}-\beta_{1}\Big(\frac{\beta_{2}\beta_{3}}{\beta_{3}-1}\Big)^{\beta_{2}}\) temos que o ponto \(P\Big(e^{\frac{1}{\beta_{3}}\ln{\Big(\frac{\beta_{3}-1}{\beta_{2}\beta_{3}}\Big)}},\beta_{0}-\beta_{1}\Big(\frac{\beta_{2}\beta_{3}}{\beta_{3}-1}\Big)^{\beta_{2}}\Big)\) é um possível ponto de inflexão de \(f\), o qual denotamos por PI1 no gráfico abaixo.
Caso 2
- \(x<0\)
\(f^{''}(x)=0\Rightarrow x=-e^{\frac{1}{\beta_{3}}\ln{\Big(\frac{1-\beta_{3}}{\beta_{2}\beta_{3}}\Big)}}\)
Como, \(f\Bigg(-e^{\frac{1}{\beta_{3}}\ln{\Big(\frac{1-\beta_{3}}{\beta_{2}\beta_{3}}\Big)}}\Bigg)=\beta_{0}-\beta_{1}e^{\frac{1-\beta_{3}}{\beta_{3}}}\) temos que o ponto \(P\Big(-e^{\frac{1}{\beta_{3}}\ln{\Big(\frac{1-\beta_{3}}{\beta_{2}\beta_{3}}\Big)}},\beta_{0}-\beta_{1}e^{\frac{1-\beta_{3}}{\beta_{3}}}\Big)\) é outro possível ponto de inflexão de \(f\), o qual denotamos por PI2 no gráfico abaixo.
Assintotas
\({\displaystyle \lim_{x\rightarrow +\infty}(\beta_{0}-\beta_{1}e^{-\beta_{2}x^{\beta_{3}}})=\beta_{0}}\)
\({\displaystyle \lim_{x\rightarrow -\infty}(\beta_{0}-\beta_{1}e^{-\beta_{2}x^{\beta_{3}}})=-\infty}\)
Logo, a reta \(y=\beta_{0}\) é assintota horizontal de \(f.\)
Reta Tangente
\(f^{'}\Bigg(e^{\frac{1}{\beta_{3}}\ln{\Big(\frac{\beta_{3}-1}{\beta_{2}\beta_{3}}\Big)}}\Bigg)=\beta{1}\beta{2}\beta{3}\Bigg(\frac{\beta{3}-1}{\beta{2}\beta{3}}\Bigg)^{\frac{\beta{3}-1}{\beta{3}}}e^{\frac{1-\beta{3}}{\beta{3}}},\) logo, a reta tangente ao gráfico passando pelo ponto PI é dada por:
\(y=\frac{\beta_{0}\beta_{2}x-\beta_{0}\beta_{1}+\beta_{0}\ln{\beta_{3}+\beta_{0}+\beta_{0}\beta_{3}}}{(1+\beta_{3})^{1/\beta_{3}}(1+\beta_{3})}\)
Exemplo de Gráfico
f <- function(beta0,beta1,beta2,beta3,x){
beta0-beta1*exp(-beta2*(x^(beta3)))
}
# (beta2>0 e beta0<beta1) ou (beta2<0 e beta0>beta1)
beta0 <- 10
beta1 <- 30
beta2 <- 1
beta3 <- -2
## x>=0
curve(f(beta0, beta1, beta2,beta3,x),
from = -10,
to =10,
col = "black",
lwd = 2,
panel.first = grid(20, 20 ,lty = 1, lwd = 1),
xlim = c(-10,10),
ylim = c(-100,100),
main = bquote("Modelo Weibull" ~y==.(beta0)-.(beta1)*exp(-.(beta2)*x^(.(beta3)))),
xlab = "Valores de x",
ylab = "Valores de y")
#tangente <- function(beta0,beta1,beta2,beta3,x){
# (beta0*beta2*x-beta0*beta1+beta0*log(beta3)+beta0+beta0*beta3)/(((1+beta3)^(1/beta3))*(1+beta3))
#}
#curve(tangente(beta0,beta1,beta2,beta3,x), from = -10,
# to =10,
# col = "deepskyblue",
# lwd = 2,
# add = TRUE)
#--------PONTO (0,0) NO PLANO CARTESIANO--------#
x <- c(0,0)
y <- c(0,0)
points(x,
y,
pch = 20,
col ="black")
text(x + 0.5,
y - 0.5,
paste(x),
col = "blue")
points(c(((beta3-1)/(beta2*beta3))^(1/beta3),((beta3-1)/(beta2*beta3))^(1/beta3)),
c(beta0-beta1*exp((1-beta3)/beta3),beta0-beta1*exp((1-beta3)/beta3)),
pch = 20,
col ="red")
text(c(((beta3-1)/(beta2*beta3))^(1/beta3),((beta3-1)/(beta2*beta3))^(1/beta3))+1.5,
c(beta0-beta1*exp((1-beta3)/beta3),beta0-beta1*exp((1-beta3)/beta3))+0.5,
"PI1",
col = "blue")
points(c(-((beta3-1)/(beta2*beta3))^(1/beta3),-((beta3-1)/(beta2*beta3))^(1/beta3)),
c(beta0-beta1*exp((1-beta3)/beta3),beta0-beta1*exp((1-beta3)/beta3)),
pch = 20,
col ="red")
text(c(-((beta3-1)/(beta2*beta3))^(1/beta3),-((beta3-1)/(beta2*beta3))^(1/beta3))-1.5,
c(beta0-beta1*exp(((-1)^(beta3))*(1-beta3)/beta3),beta0-beta1*exp(((-1)^(beta3))*(1-beta3)/beta3))+0.5,
"PI2",
col = "blue")
#abline(h=0, lwd = 1.5, col = "blue")
abline(h=beta0, lwd = 1.5, col = "blue")
abline(v=0, lwd = 2, col = "black")
O ponto \(P1\) no gráfico acima é dado por \(P\Big(\frac{\beta_{1}-\ln{\beta_{3}-1-\beta_{3}}}{\beta2},0\Big),\) ele representa a intercessão da reta tangente ao ponto \(PI=P\Big(\frac{\beta_{1}-\ln{\beta_{3}}}{\beta_{2}},\frac{\beta_{0}}{(1+\beta_{3})^{1/\beta_{3}}}\Big)\) com a assintota \(y=0.\)
O ponto \(P2\) no gráfico é dado por \(P\Big(\frac{(1+\beta_{3})^{\frac{\beta_{3}+1}{\beta_{3}}}+\beta_{1}-\ln{\beta_{3}-1-\beta_{3}}}{\beta_{2}},\beta_{0}\Big),\) ele representa a intercessão da reta tangente ao ponto \(PI=P\Big(\frac{\beta_{1}-\ln{\beta_{3}}}{\beta_{2}},\frac{\beta_{0}}{(1+\beta_{3})^{1/\beta_{3}}}\Big)\) com a assintota \(y=\beta_{0}.\)
Modelo Autocatalitico
\(f(x)=\frac{\alpha}{1+\beta e^{-\gamma x}}\)
\(f(0)=\frac{\alpha}{1+\beta}\quad \textrm{e}\quad f(x)=0\Rightarrow \alpha=0\)
Derivadas
Derivada Primeira
\(f^{'}(x)=\gamma f(x)\Big[\frac{\alpha-f(x)}{\alpha}\Big]\)
Pontos Críticos
\(f^{'}(x)=0\Rightarrow f(x)=0\quad \textrm{ou} \quad f(x)=\alpha\Rightarrow \alpha=0 \quad \textrm{ou} \quad \beta=0\)
Logo, \(f\) não possuí pontos críticos.
Derivada segunda
\(f^{''}(x)=\gamma f^{'}(x)\Big[\frac{\alpha-2f(x)}{\alpha}\Big]\)
\(f^{''}(x)=0\Rightarrow f^{'}(x)=0\quad \textrm{ou} \quad f(x)=\frac{\alpha}{2}\Rightarrow x=-\frac{1}{\gamma}\ln{\frac{1}{\beta}}\)
Logo, \(-\frac{1}{\gamma}\ln{\frac{1}{\beta}}\) é um possível ponto de inflexão de \(f\).
Assintotas
\({\displaystyle \lim_{x\rightarrow +\infty}\frac{\alpha}{1+\beta e^{-\gamma x}}=\alpha}\)
\({\displaystyle \lim_{x\rightarrow -\infty}\frac{\alpha}{1+\beta e^{-\gamma x}}=0}\)
Logo, as retas \(y=0\) e \(y=\alpha\) são assintotas horizontais de \(f.\)
Reta Tangente
\(f^{'}(-\frac{1}{\gamma}\ln{\frac{1}{\beta}})=\frac{\gamma\alpha}{4},\) logo, a reta tangente ao gráfico passando pelo ponto \(P(-\frac{1}{\gamma}\ln{\frac{1}{\beta}},\frac{\gamma\alpha}{4})\) é dada por:
\(y=\frac{x\gamma\alpha}{4}+\frac{\alpha\ln{\frac{1}{\beta}}}{4}+\frac{\alpha}{2}\)
Exemplo de Gráfico
f <- function(alpha,beta,gamma,x){
alpha/(1+beta*exp(-gamma*x))
}
alpha <- 10
beta <- 0.5
gamma <- 0.5
curve(f(alpha,beta, gamma,x),
from = -100,
to =100,
col = "black",
lwd = 2,
panel.first = grid(20, 20 ,lty = 1, lwd = 1),
xlim = c(-10,10),
ylim = c(-1,10),
main = bquote("Modelo Exponencial" ~y==.(alpha)/(1+.(beta)*exp(-.(gamma)*x))),
xlab = "Valores de x",
ylab = "Valores de y")
tangente <- function(alpha,beta,gamma,x){
((gamma*alpha*x)/4)+((alpha*log(1/beta))/4)+(alpha/2)
}
curve(tangente(alpha,beta,gamma,x), from = -10,
to =10,
col = "deepskyblue",
lwd = 2,
add = TRUE)
#--------PONTO (0,0) NO PLANO CARTESIANO--------#
x <- c(0,0)
y <- c(0,0)
points(x,
y,
pch = 20,
col ="black")
text(x + 0.5,
y - 0.5,
paste(x),
col = "blue")
abline(h=0, lwd = 2, col = "blue")
abline(h=alpha, lwd = 2, col = "blue")
abline(v=0, lwd = 2, col = "black")
points(c((-1/gamma)*log(1/beta),(-1/gamma)*log(1/beta)),
c(alpha/2,alpha/2),
pch = 20,
col ="red")
text(c((-1/gamma)*log(1/beta),(-1/gamma)*log(1/beta))+1.2,
c(alpha/2,alpha/2)-0.5,
"PI",
col = "blue")
points(c((-1/gamma)*log(1/beta)-(2/gamma),(-1/gamma)*log(1/beta)-(2/gamma)),
c(0,0),
pch = 20,
col ="red")
text(c((-1/gamma)*log(1/beta)-(2/gamma),(-1/gamma)*log(1/beta)-(2/gamma))+0.5,
c(0,0)-0.5,
"P1",
col = "blue")
points(c((2/gamma)-((1/gamma)*log(1/beta)),(2/gamma)-((1/gamma)*log(1/beta))),
c(alpha,alpha),
pch = 20,
col ="red")
text(c((2/gamma)-((1/gamma)*log(1/beta)),(2/gamma)-((1/gamma)*log(1/beta)))+0.5,
c(alpha,alpha)-0.5,
"P2",
col = "blue")
O ponto \(P1\) no gráfico acima é dado por \(P(-\frac{1}{\gamma}\ln{\frac{1}{\beta}}-\frac{2}{\gamma},0),\) ele representa a intercessão da reta tangente ao ponto \(P(-\frac{1}{\gamma}\ln{\frac{1}{\beta}},\frac{\gamma\alpha}{4})\) com a assintota \(y=0.\)
O ponto \(P2\) no gráfico acima é dado por \(P(\frac{2}{\gamma}-\frac{1}{\gamma}\ln{\frac{1}{\beta}},\alpha),\) ele representa a intercessão da reta tangente ao ponto \(P(-\frac{1}{\gamma}\ln{\frac{1}{\beta}},\frac{\gamma\alpha}{4})\) com a assintota \(y=\alpha.\)