Final answer:
The first 3 nonzero terms of the Taylor series for f(x) = sin(πx/2) centered at 0 are: (π/2)x, -(π²/2)x², and -(π³/8)x³.
Step-by-step explanation:
To find the first 3 nonzero terms of the Taylor series for f(x) = sin(πx/2) centered at 0, we can start by finding the derivatives of the function. The first derivative is f'(x) = (π/2)cos(πx/2), the second derivative is f''(x) = -(π²/4)sin(πx/2), and the third derivative is f'''(x) = -(π³/8)cos(πx/2).
Next, we evaluate these derivatives at x = 0 to get the coefficients of the Taylor series. The first nonzero term is (π/2) * 1! * x = (π/2)x, the second term is -(π²/4) * 2! * x² = -(π²/2)x², and the third term is -(π³/8) * 3! * x³ = -(π³/8)x³.
So, the first 3 nonzero terms of the Taylor series for f(x) = sin(πx/2) centered at 0 are: (π/2)x, -(π²/2)x², and -(π³/8)x³.