Euler's Method Question 4

\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}

calculator allowed 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 $x=1$ ?

5.35

5.45

5.50

5.90

Approach

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