Taylor Approximations 2 Variables

## Description

This interact displays the nth order Taylor approximation, for n from 1 to 10 of the function

(1)
\begin{align} f(x, y) = \sin (x^2 + y^2)\cos (y)\exp\left(-\frac{x^2 + y^2}{2}\right) \end{align}

## Sage Cell

#### Code

var('x y')
var('xx yy')
G = sin(xx^2 + yy^2) * cos(yy) * exp(-0.5*(xx^2+yy^2))
def F(x,y):
return G.subs(xx=x).subs(yy=y)
plotF = plot3d(F, (0.4, 2), (0.4, 2), adaptive=True, color='blue')
@interact
def _(x0=(0.5,1.5), y0=(0.5, 1.5),
order=[1..10]):
F0 = float(G.subs(xx=x0).subs(yy=y0))
P = (x0, y0, F0)
dot = point3d(P, size=15, color='red')
plot = dot + plotF
approx = F0
for n in range(1, order+1):
for i in range(n+1):
if i == 0:
deriv = G.diff(yy, n)
elif i == n:
deriv = G.diff(xx, n)
else:
deriv = G.diff(xx, i).diff(yy, n-i)
deriv = float(deriv.subs(xx=x0).subs(yy=y0))
coeff = binomial(n, i)/factorial(n)
approx += coeff * deriv * (x-x0)^i * (y-y0)^(n-i)
plot += plot3d(approx, (x, 0.4, 1.6),
(y, 0.4, 1.6), color='red', opacity=0.7)
pretty_print(html('$F(x,y) = e^{-(x^2+y^2)/2} \\cos(y) \\sin(x^2+y^2)$'))
show(plot)


none

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