deletions | additions       diff --git a/To_derive_1_we_need__.tex b/To_derive_1_we_need__.tex  index bcd9b05..18c1a0d 100644  --- a/To_derive_1_we_need__.tex  +++ b/To_derive_1_we_need__.tex 
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To derive (1), we need to show that  $$ \vec{v}(t) \Delta t = \vec{x}(t) - \vec{x}(t-\Delta t) + \frac{1}{2} \vec{a}(t) \Delta t^2 $$ $$\vec a(t)\Delta t^2=\vec{x}(t+\Delta t) - 2\vec x(t) + \vec x(t-\Delta t)$$  so  $$ \vec{v}(t) \Delta t = \vec{x}(t) - \vec{x}(t-\Delta t) + \frac{1}{2} (\vec{x}(t+\Delta t) - 2\vec x(t) + \vec x(t-\Delta t)) $$  $$ \vec{v}(t) \Delta t = \frac{1}{2} (\vec{x}(t+\Delta t) - \vec x(t-\Delta t)) $$