deletions | additions       diff --git a/vec_x_t_Delta_t__1.tex b/vec_x_t_Delta_t__1.tex  index 9b0b437..9930b14 100644  --- a/vec_x_t_Delta_t__1.tex  +++ b/vec_x_t_Delta_t__1.tex 
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$$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t) \Delta t + \frac{1}{2} \vec{a}(t) \Delta t^2 + \frac{1}{6} \vec{b}(t) \Delta t^3 + O(\Delta t^4)$$  $$\vec{v}(t + \Delta t) = \vec{v}(t) + \frac{\vec{a}(t) + \vec{a}(t + \Delta t)}{2} \Delta t$$  $$\vec{v}(t + \Delta t) = \vec{v}(t) + \vec{a}(t) \Delta t + \frac{1}{2} \vec{b}(t) \Delta t^2 + O(\Delta t^3)$$ $$\vec{v}(t) = \vec{v}(t-\Delta t) + \frac{\vec{a}(t-\Delta t) + \vec{a}(t)}{2} \Delta t$$  $$\vec{v}(t) = \vec{v}(t-\Delta t) + \vec{a}(t-\Delta t) \Delta t + \frac{1}{2} \vec{b}(t-\Delta t) \Delta t^2 + O(\Delta t^3)$$