To find the average value of the function [tex]\( f(x) = \frac{6x}{5} - 1 \)[/tex] over the interval [tex]\(\left[-\frac{13}{6}, \frac{23}{6}\right] \)[/tex], we can use the formula for the average value of a function over an interval [tex]\([a, b]\)[/tex]:
[tex]\[
\text{Average Value} = \frac{1}{b-a} \int_a^b f(x) \, dx
\][/tex]
Here, our interval [tex]\([a, b]\)[/tex] is [tex]\(\left[-\frac{13}{6}, \frac{23}{6}\right]\)[/tex]. So, [tex]\( a = -\frac{13}{6} \)[/tex] and [tex]\( b = \frac{23}{6} \)[/tex].
Now, let's compute the integral [tex]\(\int_a^b f(x) \, dx\)[/tex]:
[tex]\[
\int_{-\frac{13}{6}}^{\frac{23}{6}} \left( \frac{6x}{5} - 1 \right) dx
\][/tex]
We can split this integral into two parts:
[tex]\[
\int_{-\frac{13}{6}}^{\frac{23}{6}} \left( \frac{6x}{5} \right) dx - \int_{-\frac{13}{6}}^{\frac{23}{6}} 1 \, dx
\][/tex]
First, evaluate [tex]\( \int_{-\frac{13}{6}}^{\frac{23}{6}} \frac{6x}{5} \, dx \)[/tex]:
[tex]\[
\int_{-\frac{13}{6}}^{\frac{23}{6}} \frac{6x}{5} \, dx = \frac{6}{5} \int_{-\frac{13}{6}}^{\frac{23}{6}} x \, dx
\][/tex]
The integral of [tex]\( x \)[/tex] is [tex]\(\frac{x^2}{2}\)[/tex], so:
[tex]\[
\frac{6}{5} \left[ \frac{x^2}{2} \right]_{-\frac{13}{6}}^{\frac{23}{6}} = \frac{6}{5} \left( \frac{\left(\frac{23}{6}\right)^2}{2} - \frac{\left(\frac{13}{6}\right)^2}{2} \right)
\][/tex]
Calculating each term separately:
[tex]\[
\left(\frac{23}{6}\right)^2 = \frac{529}{36}, \quad \left(\frac{13}{6}\right)^2 = \frac{169}{36}
\][/tex]
Thus,
[tex]\[
\frac{6}{5} \left( \frac{529}{36 \cdot 2} - \frac{169}{36 \cdot 2} \right) = \frac{6}{5} \left( \frac{529 - 169}{72} \right) = \frac{6}{5} \left( \frac{360}{72} \right) = \frac{6}{5} \left( 5 \right) = 6
\][/tex]
Next, evaluate [tex]\( \int_{-\frac{13}{6}}^{\frac{23}{6}} 1 \, dx \)[/tex]:
[tex]\[
\int_{-\frac{13}{6}}^{\frac{23}{6}} 1 \, dx = \left[ x \right]_{-\frac{13}{6}}^{\frac{23}{6}} = \frac{23}{6} - \left(-\frac{13}{6}\right) = \frac{23}{6} + \frac{13}{6} = \frac{36}{6} = 6
\][/tex]
Putting everything together:
[tex]\[
\int_{-\frac{13}{6}}^{\frac{23}{6}} \left( \frac{6x}{5} - 1 \right) dx = 6 - 6 = 0
\][/tex]
Finally, divide by the length of the interval, [tex]\( b - a \)[/tex]:
[tex]\[
b - a = \frac{23}{6} - \left(-\frac{13}{6}\right) = \frac{36}{6} = 6
\][/tex]
[tex]\[
\text{Average Value} = \frac{1}{b-a} \int_a^b f(x) \, dx = \frac{1}{6} \times 0 = 0
\][/tex]
Thus, the average value of the function [tex]\( f(x) = \frac{6x}{5} - 1 \)[/tex] over the interval [tex]\(\left[-\frac{13}{6}, \frac{23}{6}\right]\)[/tex] is:
[tex]\[
\boxed{0}
\][/tex]
