Question

the graph shows the momentum y, in newton - seconds, of an object x seconds after the object started moving, for ( 0leq xleq8 ). what is the average rate of change, in newton - seconds per second, in the momentum of the object from ( x = 2 ) to ( x = 6 )?

Explanation:

Step1: Identify the formula for average rate of change

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

Step2: Determine the values of \(f(a)\) and \(f(b)\)

From the graph, when \(x = 2\) (i.e., \(a = 2\)), \(y=f(2)=6\) (newton - seconds). When \(x = 6\) (i.e., \(b = 6\)), \(y=f(6)=8\) (newton - seconds).

Step3: Substitute into the formula

Substitute \(a = 2\), \(b = 6\), \(f(a)=6\), and \(f(b)=8\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{8 - 6}{6-2}\).

Step4: Simplify the expression

\(\frac{8 - 6}{6 - 2}=\frac{2}{4}=\frac{1}{2}\)

Answer:

\(\frac{1}{2}\)