Question

The area under a velocity curve: Return to the very start of this chapter, and review the discussion of driving down a straight road with stoplights. Describe in your own words the relationship velocity, distance, and accumulation functions illustrated in that discussion. Then use what you know from the material you learned in this chapter to calculate the exact distance travelled in that situation.

Short answer

the graph in the example can be used to explain how velocity, distance, and accumulation functions relate to one another.

Step-by-step solution

Given information

Taking a straight road with stoplights as an example.

simplification

Driving down a straight road with stop lights requires a certain speed, which may be calculated as $v\left(t\right)=-0.22t^2+8.8t$.

The sum of the areas of the rectangles can be used to estimate the area under the curve of $v\left(t\right)$ is $\sum _{k=1}^{n}v\left(t_k\right)\Delta t$where $\Delta t=\frac{b-a}{n}$ and $t_k=a+k\Delta t$.

The exact distance travelled from one light to the following stop light is represented by the value found for the sum of the rectangle's areas. Considering that the velocity is a derivative of the distance travelled over time. i.e.$v=\frac{ds}{dt}$.

The distance travelled is determined by adding the areas of the functions in the aforementioned graph, where area is determined by $A\left(t\right)={\int }_{t=a}^{t=b}v\left(t\right).dt={\int }_0^s\frac{ds}{dt}dt={\int }_0^{s}ds$.

The actual distance travelled between the stop lights can be approximated by adding the sums of the areas of the rectangles, the area under the velocity graph, and the accumulation of the area of functions.

As a result, the graph in the example can be used to explain how velocity, distance, and accumulation functions relate to one another.