Question

the accompanying graph shows the total distance s traveled by a bicyclist after t hours. using the graph, answer parts (a) through (c)

Explanation:

Step1: Recall average - speed formula

The average speed $v_{avg}$ over an interval $[t_1,t_2]$ is given by $v_{avg}=\frac{s(t_2)-s(t_1)}{t_2 - t_1}$, where $s(t)$ is the position - function (distance traveled as a function of time).

Step2: For the interval $[2.5,3.5]$

From the graph, assume $s(2.5)=18$ miles and $s(3.5)=28$ miles. Then $v_{avg}=\frac{s(3.5)-s(2.5)}{3.5 - 2.5}=\frac{28 - 18}{1}=10$ mph.

Step3: Recall instantaneous - speed formula

The instantaneous speed at $t = a$ is the slope of the tangent line to the $s - t$ graph at $t=a$.

Step4: For $t=\frac{1}{2}$ hour

Estimate the slope of the tangent line at $t = \frac{1}{2}$ hour. By drawing a tangent line at $t=\frac{1}{2}$ and using two points on the tangent line to calculate the slope. Let's assume two points on the tangent line: $(0,0)$ and $(1,12.2)$. The slope $m=\frac{12.2-0}{1 - 0}=12.2$ mph.

Step5: For $t = 2$ hours

At $t = 2$ hours, the graph has a horizontal tangent line. The slope of a horizontal line is $0$. So the instantaneous speed at $t = 2$ hours is $0$ mph.

Answer:

For the average - speed over $[2.5,3.5]$: C. 10 mph
For the instantaneous speed at $t=\frac{1}{2}$ hour: C. 12.2 mph
For the instantaneous speed at $t = 2$ hours: A. 0 mph