A particle moves along the \(x\)-axis. The velocity of the particle at time \(t\) is given by \(v(t)\), and the acceleration of the particle at time \(t\) is given by \(a(t)\). Which of the following gives the average velocity of the particle from \(t=0\) to time \(t=5\) ?

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The average value of a function \(y\) is given by:

$$ \frac{1}{b-a}\int\limits_a^b f(x)   dx $$

For the velocity:

$$ \frac{1}{5-0} \int \limits_0^5 v(t)   dt $$ $$ = \boxed{\frac{1}{5}\int\limits_0^5 v(t)   dt } $$

\(\frac{1}{5} \int\limits_0^5 |v(t)|   dt \) refers to the average speed of the particle. Speed uses the magnitude of the velocity and disregards the direction.