The graph of the velocity, \(v\), of an object moving along a straight line is given above. The velocity is measured in m/min and time in minutes.

What is the average velocity on the time interval \([0,6]\)?

The average velocity is

\[ \overline {v}=\answer {1.5} m/min \]

What value of \(c\) verifies the Mean Value Theorem for integrals? In other words, at what time \(c\) does the velocity equal the average value of the velocity on \([0,6]\)?

We have to solve the equation
\[ v(t)=\overline {v}. \]
Since, \(v(t)=2\) for \(0\le t \le 3\), it follows that the solution is in the interval \([3,6]\). But, from the graph it follows that \(v(t)=-\frac {2}{3}(t-6)\), for \(3\le t \le 6\).

So, we have to solve the equation

\[ -\frac {2}{3}(t-6)=\overline {v}. \]
\[ v(c)=\overline {v} \]
at
\[ c = \answer {3.75} min \]