Question

what is the runners average rate of change between the hours: 0.5 and 2? mph 1.5 and 2.5? mph

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).

Step2: Find the average rate of change between \(x_1 = 0.5\) and \(x_2=2\)

From the graph, when \(x = 0.5\), \(y = 2\) (assuming the point \((0.5,2)\)) and when \(x = 2\), \(y = 7\) (assuming the point \((2,7)\)).

$$ \frac{7 - 2}{2-0.5}=\frac{5}{1.5}=\frac{10}{3}\approx 3.33 $$

Step3: Find the average rate of change between \(x_1 = 1.5\) and \(x_2 = 2.5\)

From the graph, when \(x=1.5\), \(y = 6\) (assuming the point \((1.5,6)\)) and when \(x = 2.5\), \(y = 11\) (assuming the point \((2.5,11)\)).

$$ \frac{11 - 6}{2.5-1.5}=\frac{5}{1}=5 $$

Answer:

The runner's average rate of change between \(0.5\) and \(2\) is \(\frac{10}{3}\approx 3.33\) mph.
The runner's average rate of change between \(1.5\) and \(2.5\) is \(5\) mph.