IDNLearn.com: Where questions are met with accurate and insightful answers. Our platform is designed to provide quick and accurate answers to any questions you may have.
Sagot :
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]
[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]
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Find reliable answers at IDNLearn.com. Thanks for stopping by, and come back for more trustworthy solutions.
