deletions | additions       diff --git a/To_derive_1_we_need__.tex b/To_derive_1_we_need__.tex  index dfb4e58..ac86a55 100644  --- a/To_derive_1_we_need__.tex  +++ b/To_derive_1_we_need__.tex 
...
To derive (1), we need to show that Now  $$ \vec{v}(t) \Delta t =\vec{x}(t) - \vec{x}(t-\Delta t) +  \frac{1}{2} \vec{a}(t) \Delta t^2 (\vec{x}(t+\Delta t) - \vec x(t-\Delta t))  $$ Now equivalently,  $$ \vec{v}(t) \Delta t = \vec{x}(t) - \vec{x}(t-\Delta t) + \frac{1}{2} (\vec{x}(t+\Delta t) - 2\vec x(t) + \vec x(t-\Delta t)) $$  But  $$\vec a(t)\Delta t^2=\vec{x}(t+\Delta t) - 2\vec x(t) + \vec x(t-\Delta t)$$  so  $$ \vec{v}(t) \Delta t = \vec{x}(t) - \vec{x}(t-\Delta t) + \frac{1}{2} (\vec{x}(t+\Delta t) - 2\vec \vec{a}(t) \Delta t^2 $$  Looking at (3) again,   $$\vec x(t+\Delta t)=2\vec x(t)-\vec x(t-\Delta t)+\vec a(t)\,\Delta t^2$$  $$\vec x(t+\Delta t)=\vec  x(t) + \vec x(t)-\vec  x(t-\Delta t)) $$ t)+\frac{1}{2}\vec a(t)\,\Delta t^2+\frac{1}{2}\vec a(t)\,\Delta t^2$$  $$ $$\vec x(t+\Delta t)=\vec x(t) +  \vec{v}(t) \Delta t = \frac{1}{2} (\vec{x}(t+\Delta t) - \vec x(t-\Delta t)) $$ +\frac{1}{2}\vec a(t)\,\Delta t^2$$