Limits and velocity

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\todo }[0]{} \newcommand {\mooculus }[0]{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\tkzTabSlope }[1]{\foreach \x /\y /\z in {#1}{\node [left = 3pt] at (Z\x 1) {\scriptsize $\y $}; \node [right = 3pt] at (Z\x 1) {\scriptsize $\z $}; }} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\sagemath }[0]{\textsf {SageMath}} \newcommand {\d }[0]{\mathop {}\!d} \newcommand {\l }[0]{\ell } \newcommand {\ddx }[0]{\frac {d}{\d x}} \newcommand {\ddt }[0]{\frac {d}{\d t}} \newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\unit }[0]{\mathop {}\!\mathrm } \newcommand {\eval }[1]{\bigg [ #1 \bigg ]} \newcommand {\seq }[1]{\left ( #1 \right )} \newcommand {\epsilon }[0]{\varepsilon } \newcommand {\phi }[0]{\varphi } \newcommand {\iff }[0]{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\proj }{\vec {proj}} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\arrowvec }[0]{\overrightarrow } \newcommand {\vec }[0]{\mathbf } \newcommand {\veci }[0]{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }[0]{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }[0]{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }[0]{\boldsymbol {\l }} \newcommand {\uvec }[1]{\mathbf {\hat {#1}}} \newcommand {\utan }[0]{\mathbf {\hat {t}}} \newcommand {\unormal }[0]{\mathbf {\hat {n}}} \newcommand {\ubinormal }[0]{\mathbf {\hat {b}}} \newcommand {\dotp }[0]{\bullet } \newcommand {\cross }[0]{\boldsymbol \times } \newcommand {\grad }[0]{\boldsymbol \nabla } \newcommand {\divergence }[0]{\grad \dotp } \newcommand {\curl }[0]{\grad \cross } \newcommand {\lto }[0]{\mathop {\longrightarrow \,}\limits } \newcommand {\bar }[0]{\overline } \newcommand {\surfaceColor }[0]{violet} \newcommand {\surfaceColorTwo }[0]{redyellow} \newcommand {\sliceColor }[0]{greenyellow} \newcommand {\vector }[1]{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }[0]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\descriptionlabel }[1]{\hspace {\labelsep }\textbf {#1:}}$

Two young mathematicians discuss limits and instantaneous velocity.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Hey Riley, I’ve been thinking about limits.
Riley: That is awesome.
Devyn: I know! You know limits remind me of something…How a GPS or a phone computes velocity!
Riley: Huh. A GPS can calculate our location. Then, to compute velocity from position, it must look at $\frac {\text {change in position}}{\text {change in time}}$
Devyn: And then we study this as the change in time gets closer and closer to zero.
Riley: Just like with limits at zero, we can study something by looking near a point, but not exactly at a point.
Devyn: O.M.G. Life’s a rich tapestry.
Riley: Poet, you know it.

Suppose you take a road trip from Columbus Ohio to Urbana-Champaign Illinois. Moreover, suppose your position is modeled by $s(t) = 36t^2 -4.8t^3 \qquad \text {(miles West of Columbus)}$ where $t$ is measured in hours and runs from $0$ to $5$ hours.

What is the average velocity for the entire trip?

Remember, $\text {change in distance} = \text {rate}\cdot \text {change in time}.$

So, $\frac {\Delta \text {distance}}{\Delta \text {time}} = \text {rate}.$

So, $\frac {\Delta \text {distance}}{\Delta \text {time}} = \frac {300}{5}.$

The average velocity is $\answer {60}$ miles per hour.

Use a calculator to estimate the instantaneous velocity at $t=2$ .

Remember, $\text {change in distance} = \text {rate}\cdot \text {change in time}.$

So, $\frac {\Delta \text {distance}}{\Delta \text {time}} = \text {rate}.$

Compute $\frac {36 t^2 -4.8 t^3 -\left (36\cdot 2^2 -4.8\cdot 2^3\right ) }{ t-2}$ for $t$ closer, and closer to $2$ .

The instantaneous velocity, (rounded to the nearest tenth) is $\answer {86.4}$ miles per hour.

Considering the work above, when we want to compute instantaneous velocity, we need to compute $\frac {\text {change in position}}{\text {change in time}}$ when (choose all that apply):

The “change in time” is zero. The “change in time” gets closer and closer to zero. The “change in time” approaches zero. The “change in time” is near zero. The “change in time” goes to zero.

Computing average velocities for smaller, and smaller, values of $t-2$ as we did above is tedious. Nevertheless, this is exactly how a GPS determines velocity from position! To avoid these tedious calculations, we would really like to have a formula.