Question

Consider again the stoplight example from the reading. In making an approximation for distance travelled, why do we assume that velocity is constant on small subintervals? What are some different ways that we could choose which velocity to use on each subinterval? Illustrate a couple of these ways with graphs that involve rectangles.

Short answer

$The\mathrm{area}\mathrm{under}\mathrm{the}\mathrm{graph}\mathrm{of}\mathrm{the}\mathrm{funclion}{f}^{\mathrm{\prime }}\text{is related lo the funclion}f.$

Step-by-step solution

Step1. Given Information

$\text{Consider the discussion at the end of the stoplight example.}$

$\text{The objective is to determine if the area under the graph of the function}f\text{is related to the}$

$\text{derivative}{f}^{\mathrm{\prime }}\text{or vice-versa.}$

Step2. Representation

$\text{Let,}v\left(t\right)\text{represents velocity and}s\left(t\right)\text{represents distance.}$

$\text{In the example,}$

$v\left(t\right)=s^{\mathrm{\prime }}\left(t\right)$

localid="1650180807321" $\text{Therefore, theareaunder thegraphofthe funclion}{f}^{\mathrm{\prime }}\text{is related lo the funclion}f\text{.}$