Final answer:
To create the Taylor approximations for the function f(x, y) at the origin, we calculate the function's partial derivatives at the origin and construct the Taylor series expansion up to the desired order, either quadratic or cubic.
Step-by-step explanation:
The question asks about finding the quadratic and cubic approximations of the function f(x, y) = 5 sin x cos y near the origin using Taylor's formula. To find these approximations, we need to calculate the partial derivatives of the function at the origin and use them to construct the Taylor series expansion up to the quadratic and cubic terms.
For the quadratic approximation, we will use terms up to second order derivatives. For the cubic approximation, we will include terms up to third order derivatives. Typically, we evaluate the function and its partial derivatives at the origin, and since sine and cosine functions are bounded, we would expect certain terms to vanish at the origin, simplifying our Taylor series expansion.
To calculate these approximations accurately, one needs to understand the process of obtaining partial derivatives and constructing a Taylor series expansion around a given point for a multivariable function.