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Of course still better results can be expected by taking intermediate values. This is what happens when using the [[Runge-Kutta]] method, especially the one of grade 4 or 5 are most useful.
A completely different method is the use of [[Taylor series]]. In that case we write: <math>r = r_0 + r'_0 t + r''_0 \frac{t^2}{2!} +
but rather than developing up to some higher derivative in r only, one can develop in r and v (that is r') by writing <math>r = f r_0 + g r'_0</math>and then write out the factors ''f'' and ''g'' in a series.
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