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7-8.

For each function below, find its average value on the given integral and state the time $t$ at which it equals its average value.

1. $g(t) = 3t + 6$ for [$0, 8$]

1. $g(t) = 3e^t$ for [$0, 1$]

Read the Math Note about how to compute the Mean Value of $g(x)$, given $g(x)$.

To find the the time the function is at its average value, let $g(t) =$ the average value and solve for $t$.

Average Value $= 3e − 3$. Now find the time, $t$, that $g(t) =$ its average value.

The Mean Value Theorem states that a differentiable function will reach its average (mean) value at least once on any closed interval.
Check your values of $t$ in parts (a) and (b). Are they within the the given closed intervals?