Question

Using Taylor's expansion, expand f(x,y) = excosy at the neighborhood of (0,0) up to the third degree.

Answer 1

Using Taylor's expansion, the function f(x,y) = e^x*cos(y) can be expanded up to the third degree at the point (0,0) as 1 + x - (y^2/2) - (x^3/6), considering the partial derivatives of f with respect to x and y.

Step-by-step explanation:

The problem requires us to expand the function f(x,y) = excos(y) using Taylor's expansion around the point (0,0) up to the third degree. Taylor's expansion for a function of two variables can be represented as a sum of terms derived from the derivatives of the function evaluated at the expansion point. Here's the step-by-step process:

1. Compute the partial derivatives of f with respect to x and y up to the third order.
2. Evaluate these derivatives at the point (0,0).

The third degree Taylor expansion of f(x,y) at (0,0) is 1 + x - (y2/2) - (x3/6). This uses the fact that at (0,0), e0 = 1 and cos(0) = 1, and the derivatives of ex are powers of ex and the derivatives of cos(y) involve cos(y) and -sin(y).