I did some code for question 1 but still not finish yet. Please help me the last part and question 2 please. In R or R markdown is ok.
# Second Taylor approximation of f(x1, x2) = cos(x1*x2) around the point (1,0)
# The approximation is h(x1, x2) = 1pi^2/8*x1^2
library(rgl)
f < function(x1, x2){cos(x1*x2)}
x1 < seq(pi/4, pi/4, 0.1)
x2 < seq(pi/4, 3*pi/4, 0.1)
f1 < outer(x1, x2, FUN = f)
persp3d(x1, x2, f1, main = "Taylor approximation of cos(x1,x2)",
xlab="x1", ylab="x2", col = "red",axe =T, shade = 0.1, alpha = 0.5)
# The approximation
h < function(x1,x2){1(pi^2/8)*(x1^2)}
h1 < outer(x1, x2, FUN = h)
persp3d(x1, x2,h1,col = "blue", shade = 0.1 , add = TRUE, alpha = 0.5)
# The error of the approximation
error < abs(f1  h1)
persp3d(x1, x2, error, col = "green", alpha = 0.5)
filled.contour(x1, x2, error)
grid(nx = 20, ny = 20)

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