Find the average value of the function f(x)=(6)/(3-4x) from x=1 to x=7. Express your answer as a constant times ln 5. Answer: ◻ln 5

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Find the average value of the function  f(x)=(6)/(3-4x) from  x=1 to  x=7. Express your answer as a constant times  ln 5. Answer:  ◻ln 5

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Understand the concept: Understand the concept of average value of a function. The average value of a function f(x) on the interval [a, b] is given by the formula: Average value = (1/(b-a)) * ∫ from a to b of f(x) dx Here, a = 1 and b = 7.Set up the integral: Set up the integral for the average value. Average value = (1/(7-1)) * ∫ from 1 to 7 of (6)/(3-4x) dx Simplify the coefficient: (1/(7-1)) = 1/6 So, Average value = (1/6) * ∫ from 1 to 7 of (6)/(3-4x) dxSimplify the coefficient: Simplify the integral. The integral becomes (1/6) * ∫ from 1 to 7 of (6)/(3-4x) dx = (1/6) * 6 * ∫ from 1 to 7 of (1)/(3-4x) dx The 6's cancel out, leaving us with ∫ from 1 to 7 of (1)/(3-4x) dxSimplify the integral: Perform a substitution to solve the integral. Let u = 3 - 4x, then du = -4 dx, or dx = -1/4 du. When x = 1, u = 3 - 4(1) = -1. When x = 7, u = 3 - 4(7) = -25. Now, the integral is ∫ from -1 to -25 of (-1/4) * (1/u) du.Perform a substitution: Evaluate the integral with the new limits of integration. The integral becomes (-1/4) * ∫ from -1 to -25 of (1/u) du = (-1/4) * [ln|u|] from -1 to -25.Evaluate the integral: Calculate the value of the integral. Plugging in the limits, we get (-1/4) * (ln|-25| - ln|-1|) = (-1/4) * (ln(25) - ln(1)). Since ln(1) = 0, this simplifies to (-1/4) * ln(25).Calculate the value: Simplify the expression. ln(25) can be written as 2 * ln(5), so the expression becomes (-1/4) * 2 * ln(5) = (-1/2) * ln(5).Simplify the expression: Multiply by the coefficient from the average value formula. The average value is (1/6) times the integral, so we multiply (-1/2) * ln(5) by 1/6. Average value = (1/6) * (-1/2) * ln(5) = (-1/12) * ln(5).Multiply by the coefficient: Check for any mathematical errors. Reviewing the steps, there are no apparent mathematical errors.

Find the average value of the function $f(x)=\frac{6}{3-4 x}$ from $x=1$ to $x=7$ . Express your answer as a constant times $\ln 5$ . $\newline$ Answer: $\square\ln 5$

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Find the average value of the function f(x)=63−4x f(x)=\frac{6}{3-4 x} f(x)=3−4x6​ from x=1 x=1 x=1 to x=7 x=7 x=7. Express your answer as a constant times ln⁡5 \ln 5 ln5.\newlineAnswer: □ln⁡5 \square\ln 5 □ln5

Find the average value of the function $f(x)=\frac{6}{3-4 x}$ from $x=1$ to $x=7$ . Express your answer as a constant times $\ln 5$ . $\newline$ Answer: $\square\ln 5$