6. Multiple Choice 1 point Let \( y=f(x) \) be the solution to the differential equation \( \frac{d y}{d x}=x+y \) with initial condition \( f(1)=2 \). What is the approximation for \( f(2) \) if Euler's method is used, starting at \( x=1 \) with a step size of 0.5 ?

To use Euler's method to approximate \( f(2) \) with a step size of 0.5, we begin at the initial condition \( f(1) = 2 \). 1. First step: At \( x=1 \), \( y=f(1)=2 \). \[ \frac{dy}{dx} = 1 + 2 = 3 \] Update \( y \): \[ f(1.5) \approx f(1) + \frac{dy}{dx} \cdot \text{step size} = 2 + 3 \cdot 0.5 = 2 + 1.5 = 3.5 \] 2. Second step: Now at \( x=1.5 \) and \( y=f(1.5)=3.5 \). \[ \frac{dy}{dx} = 1.5 + 3.5 = 5 \] Update \( y \): \[ f(2) \approx f(1.5) + \frac{dy}{dx} \cdot \text{step size} = 3.5 + 5 \cdot 0.5 = 3.5 + 2.5 = 6 \] Therefore, the approximation for \( f(2) \) using Euler's method is \( \boxed{6} \).