: \[\begin{aligned} \Naturals ~~&=~~ \text{the set of all \textbf{natural} numbers, i.e. the set of nonnegative integers: } 0, 1, 2, 3, 4, \ldots\ \[6pt] \Integers ~~&=~~ \text{the set of all integers: } 0, \pm 1, \pm 2, \pm 3, \pm 4, \ldots\ \[6pt] \Rationals ~~&=~~ \text{the set of all \textbf{rational} numbers $\frac{m}{n}$, where $m$ and $n$ are integers, with $n \ne 0$}\ 6pt] \Reals ~~&=~~ \text{the set of all real numbers}\end{aligned} \nonumber

: \[\begin{aligned} \text{instantaneous velocity at $t$} ~~&=~~ \text{limit of average velocity over $\ival{t}{t+\Delta t}$ as $\Delta t$ approaches to 0}\ \[6pt] &=~~ \lim_{\Delta t \to 0} ~\frac{\Delta s}{\Delta t}\ \[8pt] &=~~ \lim_{\Delta t \to 0} ~(-32t ~-~ 16\Delta t)\ \[6pt] &=~~ -32t - 16(0)\ 6pt] &=~~ -32t\end{aligned} \nonumber

s(t)=−16t2+100s(t) = -16t^2 + 100: \[\begin{aligned} \dfrac{\Delta s}{\Delta t} ~~&=~~ \dfrac{s(t + \Delta t) ~-~ s(t)} {\Delta t}\ \[8pt] &=~~ \dfrac{-16(t+\Delta t)^2 ~+~ 100 ~-~ (-16t^2 ~+~ 100)}{\Delta t}\ \[8pt] &=~~ \dfrac{-16t^2 ~-~ 32t\Delta t ~-~ 16(\Delta t)^2 ~+~ 100 ~+~ 16t^2 ~-~ 100}{\Delta t}\ \[8pt] &=~~ \dfrac{-32t\Delta t ~-~ 16(\Delta t)^2}{\Delta t} ~~=~~ \dfrac{\cancel{\Delta t} \,(-32t ~-~