$$ \begin{array}{|c||c|c|c|c|c|} \hline x & 0.0 & 0.5 & 1.0 & 1.5& 2.0 \\ \hline f'(x) & 1.0 & 0.7 & 0.5 & 0.4 & 0.3 \\ \hline \end{array} $$

Let \(y=f(x)\) be the solution to the differential equation \(\dfrac{dy}{dx}=f'(x)\) with initial condition \(f(1)=5\). Selected values of \(f'(x)\) are given in the table above. What is the approximation for \(f(2)\) if Euler's method is used with a step size of \(0.5\), starting at \(x=1\) ?

Summary Submit

Starting at \(x=1\):

$$ f(1.5)=f(1)+ f'(1)(0.5) = 5 + 0.5(0.5) = 5.25 $$ $$ f(2)=f(1.5)+f'(1.5)(0.5)= 5.25 + 0.4(0.5) = 5.45 $$