Let \(y=f(x)\) be the solution to the differential equation \(\dfrac{dy}{dx}=1+2y\) with the initial condition \(f(0)=1\). What is the approximation for \(f(1)\) if Euler's method is used, starting at \(x=0\) with a step size of 0.5?

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We start at \((0,1)\). At this point, we have a slope of:

$$ \frac{dy}{dx}\Big|_{y=1} = 1+2(1) = 3 $$ $$ f(0.5)=f(0)+3(0.5) = 1+1.5 = 2.5$$

At \((0.5,2.5)\)

$$ \frac{dy}{dx}\Big|_{y=2.5}=1+2(2.5)=6 $$ $$ f(1)=f(0.5)+6(0.5) = \boxed{5.5} $$