Final answer:
The degree 2 Taylor polynomial for the function f(x) = eˣsinx is x + x².
Step-by-step explanation:
To find the degree 2 Taylor polynomial for the function f(x) = eˣsinx, we need to find the second-degree Taylor expansion of the function around a specific point. The Taylor polynomial of degree 2 is given by:
T₂(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2, where f(a), f'(a), and f''(a) are the values of the function, its first derivative, and its second derivative at the point a. Let's calculate the derivatives and expand the polynomial:
f(x) = eˣsinx
f'(x) = eˣsinx + eˣcosx
f''(x) = 2eˣcosx
T₂(x) = f(0) + f'(0)(x - 0) + f''(0)(x - 0)²/2
T₂(x) = 0 + (1)(x) + (2)(x²)/2
T₂(x) = x + x²