Given the function f (x, y) = e−x · sin(x + y) a) Calculate its gradient at the point (0, π)

To calculate the gradient of a function f(x, y) = e^{-x} \cdot \sin(x + y) at the point (0, \pi), we need to find the partial derivatives with respect to x and y at that point.

Given function: f(x, y) = e^{-x} \cdot \sin(x + y)

To find the gradient, we need to calculate the partial derivatives:
\frac{\partial f}{\partial x} and \frac{\partial f}{\partial y}

\frac{\partial f}{\partial x} = -e^{-x} \cdot \sin(x + y) + e^{-x} \cdot \cos(x + y)
\frac{\partial f}{\partial y} = e^{-x} \cdot \cos(x + y)

Now, evaluate these partial derivatives at the point (0, \pi):
\frac{\partial f}{\partial x} \Bigg|_{(0, \pi)} = -e^0 \cdot \sin(0 + \pi) + e^0 \cdot \cos(0 + \pi) = -\sin(\pi) + \cos(\pi) = -1
\frac{\partial f}{\partial y} \Bigg|_{(0, \pi)} = e^0 \cdot \cos(0 + \pi) = \cos(\pi) = -1

Therefore, the gradient of the function f(x, y) = e^{-x} \cdot \sin(x + y) at the point (0, \pi) is \nabla f(0, \pi) = (-1, -1) .

\boxed{\nabla f(0, \pi) = (-1, -1)}