qualitatively correct as the figure shows. Using other step sizes As suggested in the introduction, the Euler method is more accurate if the step size is smaller. The table below shows the result with different step sizes. The top row corresponds to the example in the previous section, and the second row is illustrated in the figure. {| class="wikitable" |- ! step size !! result of Euler's method !! error |- | 1 || 16.00 || 38.60 |- | 0.25 || 35.53 || 19.07 |- | 0.1 || 45.26 || 9.34 |- | 0.05 || 49.56 || 5.04 |- | 0.025 || 51.98 || 2.62 |- | 0.0125 || 53.26 || 1.34 |} The error recorded in the last column of the table is the difference between the exact solution at and the Euler approximation. In the bottom of the table, the step size is half the step size in the previous row, and the error is also approximately half the error in the previous row. This suggests that the error is roughly proportional to the step size, at least for fairly small values of the step size. This is true in general, also for other equations; see the section Global truncation error for more details. Other methods, such as the midpoint method also illustrated in the figures, behave more favourably: the global error of the midpoint method is roughly proportional to the square of the step size. For this reason, the Euler method is said to be a first-order method, while the midpoint method is second order. We can extrapolate from the above table that the step size needed to get an answer that is correct to three decimal places is approximately 0.00001, meaning that we need 400,000 steps. This large number of steps entails a high computational cost. For this reason, higher-order methods are employed such as Runge–Kutta methods or linear multistep methods, especially if a high accuracy is desired. Derivation The Euler method can be derived in a number of ways. Firstly, there is the geometrical description above. Another possibility is to consider the Taylor expansion of the function around : The differential equation states that . If this is substituted in the Taylor expansion and the quadratic and higher-order terms are ignored, the Euler method arises. The Taylor expansion is used below to analyze the error committed by the Euler method, and it can be extended