Final answer:
The first-degree Taylor series expansion of eˣ˥ about the point (0,0) is 1 + xy, with the first-degree term being xy.
Step-by-step explanation:
The Taylor series expansion of eˣ˥ about the point (0,0) up to the first-degree terms can be found using the general formula for a Taylor series expansion. Since we're looking for the expansion around the origin, the series will be based on the derivatives at that point. The function eˣ˥ has a value of 1 at the origin, and its first partial derivatives with respect to x (holding y constant) is yeˣ˥, and with respect to y (holding x constant) is xeˣ˥. At the origin, both of these derivatives are 0. Therefore, the first-degree terms in the Taylor series expansion of eˣ˥ would be x plus y, without any contribution from the exponential terms (since x=0, y=0 make these terms vanish). The first-degree term or linear approximation of the function around the point (0,0) is thus 1 + xy, which is the answer we're looking for.