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Find the average value of the function f(x)=(2)/(x-11) from x=2 to x=8. Express your answer as a constant times ln 3. Answer: ◻ln 3

Find the average value of the function f(x)=2x−11 f(x)=\frac{2}{x-11} from x=2 x=2 to x=8 x=8 . Express your answer as a constant times ln⁡3 \ln 3 .\newlineAnswer: □ln⁡3\square \ln 3

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Q. Find the average value of the function f(x)=2x−11 f(x)=\frac{2}{x-11} from x=2 x=2 to x=8 x=8 . Express your answer as a constant times ln⁡3 \ln 3 .\newlineAnswer: □ln⁡3\square \ln 3
  1. Understand the concept: Understand the concept of the average value of a function.\newlineThe average value of a function f(x)f(x) on the interval [a,b][a, b] is given by the formula:\newlineAverage value = 1(b−a)∫abf(x) dx\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx\newlineHere, a=2a = 2, b=8b = 8, and f(x)=2(x−11)f(x) = \frac{2}{(x-11)}.
  2. Set up the integral: Set up the integral to find the average value.\newlineAverage value = (1/(8−2))×∫282(x−11) dx(1/(8-2)) \times \int_{2}^{8} \frac{2}{(x-11)} \,dx\newlineSimplify the coefficient (1/(8−2))(1/(8-2)):\newlineAverage value = (1/6)×∫282(x−11) dx(1/6) \times \int_{2}^{8} \frac{2}{(x-11)} \,dx
  3. Calculate the integral: Calculate the integral.\newlineThe integral of 2x−11\frac{2}{x-11} with respect to xx is 2⋅ln⁡∣x−11∣2 \cdot \ln|x-11|.\newlineSo, we need to evaluate this from x=2x=2 to x=8x=8.\newline∫282x−11 dx=[2⋅ln⁡∣x−11∣]28\int_{2}^{8} \frac{2}{x-11} \, dx = [2 \cdot \ln|x-11|]_{2}^{8}
  4. Evaluate at bounds: Evaluate the integral at the bounds and subtract.\newlinePlug in the upper bound x=8x=8:\newline2⋅ln⁡∣8−11∣=2⋅ln⁡∣−3∣2 \cdot \ln|8-11| = 2 \cdot \ln|-3|\newlineSince ln⁡∣−3∣=ln⁡(3)\ln|-3| = \ln(3) (because ln⁡\ln of an absolute value is the same as ln⁡\ln of the positive value), we have:\newline2⋅ln⁡(3)2 \cdot \ln(3)\newlineNow plug in the lower bound x=2x=2:\newline2⋅ln⁡∣2−11∣=2⋅ln⁡∣−9∣=2⋅ln⁡(9)2 \cdot \ln|2-11| = 2 \cdot \ln|-9| = 2 \cdot \ln(9)\newlineSince ln⁡(9)=2⋅ln⁡(3)\ln(9) = 2 \cdot \ln(3) (because 99 is 2⋅ln⁡∣8−11∣=2⋅ln⁡∣−3∣2 \cdot \ln|8-11| = 2 \cdot \ln|-3|00), we have:\newline2⋅ln⁡∣8−11∣=2⋅ln⁡∣−3∣2 \cdot \ln|8-11| = 2 \cdot \ln|-3|11\newlineNow subtract the two results:\newline2⋅ln⁡∣8−11∣=2⋅ln⁡∣−3∣2 \cdot \ln|8-11| = 2 \cdot \ln|-3|22
  5. Multiply by coefficient: Multiply by the coefficient to find the average value.\newlineAverage value = (16)×(−2×ln⁡(3))(\frac{1}{6}) \times (-2 \times \ln(3))\newlineSimplify the expression:\newlineAverage value = −13×ln⁡(3)-\frac{1}{3} \times \ln(3)

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