Question

expand xy^2+cos(xy) up to fourth degree terms using maclaurin's expansion

Answer 1

The definition of the taylor expansion in 2 dimensions is given as:
\[\sum_{m=0}^{\infty} \sum_{n = 0}^{\infty} \frac{(x-x_0)^m(y-y_0)^n}{n! m!} \left. \frac{\partial^{n+m} f}{\partial x^m \partial y^n} \right\|_{(x,y) = (x_0,y_0)}\]
Using (x_0,y_0) = (0,0) gives the McLaurin series. Therefore you simply need calculate up to n=m=4. (Remember, all possible combinations so: (0,0),(0,1),(0,2),(0,3),(0,4),(1,0),(1,1),...

Answer 2

It's kinda a feather but the xy^2 term will eventually go to zero.