Final answer:
To find the quadratic Taylor polynomial for f(x)=e^xsin(x) about a=0, we calculate its first and second derivatives, evaluate them at a=0, and construct the polynomial, resulting in P_2(x) = x + x^2.
Step-by-step explanation:
The task is to find the quadratic Taylor polynomial for the function f(x) = exsin(x) around the point a = 0. The Taylor polynomial of degree two for a function f about a point a is given by:
P2(x) = f(a) + f'(a)(x-a) + (f''(a)/2)(x-a)2
First, we must calculate the first and second derivatives of f(x):
- f'(x) = exsin(x) + excos(x)
- f''(x) = 2excos(x)
Next, we evaluate these derivatives at a = 0:
- f'(0) = e0sin(0) + e0cos(0) = 1
- f''(0) = 2e0cos(0) = 2
Finally, we can write the quadratic Taylor polynomial:
P2(x) = f(0) + f'(0)x + (f''(0)/2)x2
Which simplifies to:
P2(x) = 1x + (2/2)x2
Or,
P2(x) = x + x2