Let \(y=f(x)\) be the solution to the differential equation \(\dfrac{dy}{dx}=x-y-1 \) with the initial condition \(f(1)=-2\). What is the approximation for \(f(1.4)\) if Euler's method is used, starting at \(x=1\) with two steps of equal size?

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Starting at \((1,-2)\), we have a slope of:

$$ \frac{dy}{dx}\Big|_{x=1, y=-2}=1-(-2)-1 = 2 $$ $$ f(1.2)=f(1)+0.2(2) = -2+0.4 = -1.6$$

At \((1.2,-1.6)\)

$$ \frac{dy}{dx} \Big|_{x=1.2, y=-1.6} = 1.2-(-1.6)-1 = 1.8 $$ $$ f(1.4)=f(1.2)+0.2(1.8) = -1.6+0.36 = \boxed{-1.24}$$