Answer:
f(x, y) = Sin(x*y)
We want the second order taylor expansion around x = 0, y = 0.
This will be:
[tex]f(x,y) = f(0,0) + \frac{df(0,0)}{dx} x + \frac{df(0,0)}{dy} y + \frac{1}{2} \frac{d^2f(0,0)}{dx^2} x^2 +\frac{1}{2} \frac{d^2f(0,0)}{dy^2}y^2 + \frac{d^2f(0,0)}{dydx} x*y[/tex]
So let's find all the terms:
Remember that:
[tex]\frac{dsin(ax)}{dx} = a*cos(ax)[/tex]
[tex]\frac{dcos(ax)}{dx} = -a*cos(ax)[/tex]
f(0,0) = sin(0*0) = 1.
[tex]\frac{df(0,0)}{dx}*x = y*cos(0*0)*x = x*y[/tex]
[tex]\frac{df(0,0)}{dy} *y = x*cos(00)*y = x*y[/tex]
[tex]\frac{1}{2} \frac{d^2f(0,0)}{dx^2}*x^2 = -\frac{1}{2} *y^2*sin(0*0)*x^2 = 0[/tex]
[tex]\frac{1}{2} \frac{d^2f(0,0)}{dy^2}*y^2 = -\frac{1}{2} *x^2*sin(0*0)*y^2 = 0[/tex]
[tex]\frac{d^2f(0,0)}{dxdy} x*y = (cos(0*0) -x*y*sin(0*0))*x*y = x*y[/tex]
Then we have that the taylor expansion of second order around x = 0 and y = 0 is:
sin(x,y) = x*y + x*y + x*y = 3*x*y
