deletions | additions       diff --git a/Then_vec_v_t_Delta__.tex b/Then_vec_v_t_Delta__.tex  index 8a20b58..c579c46 100644  --- a/Then_vec_v_t_Delta__.tex  +++ b/Then_vec_v_t_Delta__.tex 
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And to show (2) follows from (3) we just need to show that  $$\vec{v}(t + \Delta t) - \vec{v}(t) = \frac{\vec{a}(t) + \vec{a}(t + \Delta t)}{2} \Delta t$$Or  $$ \frac{\vec{x}(t+2\Delta t) - \vec{x}(t) - \vec{x}(t+\Delta t) + \vec{x}(t-\Delta t)}{2 \Delta t} = \frac{\vec{a}(t) + \vec{a}(t + \Delta t)}{2} \Delta t$$  $$ \vec{x}(t+2\Delta t) - \vec{x}(t) - \vec{x}(t+\Delta t) + \vec{x}(t-\Delta t) = (\vec{a}(t) + \vec{a}(t + \Delta t)) \Delta t^2$$