Adicionando derivadas parciais
f <- function(x, y) 300 - 15 * x^2 - 10 * y^2 + 5 * x * y
x_seq <- seq(-5, 5, length.out = 200)
y_seq <- seq(-6, 6, length.out = 200)
dados <- expand_grid(x = x_seq, y = y_seq) %>%
mutate(z = f(x, y)) %>%
mutate(z = if_else(z>0, z, NA_real_)) #somente valores positivos de z
# Matriz z: linhas = y, colunas = x
z_mat <- dados %>%
pivot_wider(names_from = x, values_from = z) %>%
arrange(y) %>%
select(-y) %>%
as.matrix()
# Curvas na superfície
curva_x1 <- tibble(x = 1, y = y_seq) %>% mutate(z = f(x, y)) %>% filter(z>0) # x = 1
curva_y1 <- tibble(x = x_seq, y = 1) %>% mutate(z = f(x, y)) %>% filter(z>0) # y = 1
# Extensão dos planos: onde f(1, y) > 0 e f(x, 1) > 0
y_lim <- range(curva_x1$y)
x_lim <- range(curva_y1$x)
# Limites verticais dos planos
zmin <- 0 # min(dados$z)
zmax <- 300 # max(dados$z)
# Cores sólidas para os planos
cor_azul <- list(c(0, 1), c("royalblue", "royalblue"))
cor_vermelha <- list(c(0, 1), c("firebrick", "firebrick"))
plot_ly() %>%
# Superfície
add_surface(x = ~x_seq, y = ~y_seq, z = ~z_mat,
colorscale = "Viridis", opacity = 0.9,
name = "f(x,y)", showscale = FALSE) %>%
# Plano x = 1
add_surface(x = matrix(1, 2, 2),
y = matrix(c(y_lim[1], y_lim[2], y_lim[1], y_lim[2]), 2, 2),
z = matrix(c(zmin, zmin, zmax, zmax), 2, 2),
colorscale = cor_azul, opacity = 0.25,
showscale = FALSE, name = "plano x = 1") %>%
# Plano y = 1
add_surface(x = matrix(c(x_lim[1], x_lim[2], x_lim[1], x_lim[2]), 2, 2),
y = matrix(1, 2, 2),
z = matrix(c(zmin, zmin, zmax, zmax), 2, 2),
colorscale = cor_vermelha, opacity = 0.25,
showscale = FALSE, name = "plano y = 1") %>%
# Curva x = 1 sobre a superfície
add_trace(data = curva_x1, x = ~x, y = ~y, z = ~z,
type = "scatter3d", mode = "lines",
line = list(color = "blue", width = 7),
name = "curva x = 1") %>%
# Curva y = 1 sobre a superfície
add_trace(data = curva_y1, x = ~x, y = ~y, z = ~z,
type = "scatter3d", mode = "lines",
line = list(color = "red", width = 7),
name = "curva y = 1") %>%
layout(
title = "f(x,y) = 300 - 15x² - 10y² + 5xy",
scene = list(
xaxis = list(title = "x"),
yaxis = list(title = "y"),
zaxis = list(title = "f(x,y)")
)
)
