Find the average value of the function f(x)=(4)/(2x-11) from x=6 to x=10. Write your answer as the logarithm of a single number in simplest form. Answer: ln(◻)

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Find the average value of the function  f(x)=(4)/(2x-11) from  x=6 to  x=10. Write your answer as the logarithm of a single number in simplest form. Answer:  ln(◻)

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Calculate Interval Length: To find the average value of the function f(x) = 4/(2x - 11) over the interval [6, 10], we need to integrate the function over this interval and then divide by the length of the interval. The average value formula is given by: Average value = (1/(b - a)) * ∫ from a to b of f(x) dx Here, a = 6 and b = 10.Set Up Integral: First, we calculate the length of the interval [6, 10], which is b - a. Length of interval = 10 - 6 = 4Perform Substitution: Next, we set up the integral of f(x) from 6 to 10. ∫ from 6 to 10 of (4/(2x - 11)) dx We will need to use a substitution to solve this integral. Let u = 2x - 11, then du = 2 dx.Change Limits: We need to change the limits of integration to match our substitution. When x = 6, u = 2(6) - 11 = 1. When x = 10, u = 2(10) - 11 = 9. Now we can rewrite the integral in terms of u. ∫ from 1 to 9 of (4/u) * (1/2) du The (1/2) comes from the fact that du = 2 dx, so dx = (1/2) du.Simplify Integral: Simplify the integral. ∫ from 1 to 9 of (4/u) * (1/2) du = (1/2) * ∫ from 1 to 9 of (4/u) du = (1/2) * ∫ from 1 to 9 of (4u^(-1)) du = (1/2) * 4 * ∫ from 1 to 9 of u^(-1) du = 2 * ∫ from 1 to 9 of u^(-1) duIntegrate u^(-1): Now we integrate u^(-1) with respect to u from 1 to 9. ∫ of u^(-1) du is the natural logarithm function, ln(u). So, 2 * ∫ from 1 to 9 of u^(-1) du = 2 * [ln(u)] from 1 to 9 = 2 * (ln(9) - ln(1))Simplify Expression: Since ln(1) = 0, we simplify the expression. 2 * (ln(9) - ln(1)) = 2 * ln(9)Find Average Value: Now we divide by the length of the interval to find the average value. Average value = (1/4) * 2 * ln(9) = (1/2) * ln(9)Simplify ln(9): We can simplify ln(9) since 9 is a perfect square. ln(9) = ln(3^2) = 2 * ln(3) So, the average value becomes: Average value = (1/2) * 2 * ln(3) = ln(3)

Find the average value of the function $f(x)=\frac{4}{2 x-11}$ from $x=6$ to $x=10$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$

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Find the average value of the function $f(x)=\frac{4}{2 x-11}$ from $x=6$ to $x=10$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$