Questions: Find the average value of the function (f(x)=frac33 x-2) from (x=2) to (x=6). Express your answer as a constant times (ln 2).

Solution Steps

We start by calculating the definite integral of the function \( f(x) = \frac{3}{3x - 2} \) from \( x = 2 \) to \( x = 6 \):

\[ \int_{2}^{6} \frac{3}{3x - 2} \, dx = -\log(4) + \log(16) \]

Next, we compute the average value of the function over the interval \([2, 6]\) using the formula:

\[ \text{Average value} = \frac{1}{6 - 2} \left( -\log(4) + \log(16) \right) = \frac{1}{4} \left( -\log(4) + \log(16) \right) \]

This simplifies to:

\[ \text{Average value} = -0.25 \log(4) + 0.25 \log(16) \]

We can express the average value in terms of \( \ln 2 \):

\[ \log(4) = 2 \log(2) \quad \text{and} \quad \log(16) = 4 \log(2) \]

Thus, substituting these into our average value expression gives:

\[ \text{Average value} = -0.25(2 \log(2)) + 0.25(4 \log(2)) = -0.5 \log(2) + 1 \log(2) = 0.5 \log(2) \]

Final Answer