Find the average value of the function f(x)=(4)/(6-x) from x=7 to x=9. Write your answer as the logarithm of a single number in simplest form. Answer: ln(◻)

Question

Find the average value of the function  f(x)=(4)/(6-x) from  x=7 to  x=9. Write your answer as the logarithm of a single number in simplest form. Answer:  ln(◻)

Bytelearn · Accepted Answer

Set Up Integral: The average value of a continuous function f(x) on the interval [a, b] is given by the formula: Average value = (1/(b-a)) * ∫[a to b] f(x) dx Here, a = 7, b = 9, and f(x) = (4)/(6-x). First, we need to set up the integral to find the average value.Calculate Integral: Now we calculate the integral of f(x) from x=7 to x=9. ∫[7 to 9] (4)/(6-x) dx To integrate this function, we can use a substitution method. Let u = 6-x, then du = -dx.Substitution Method: Substituting u into the integral, we get: ∫[7 to 9] (4)/(6-x) dx = -∫[-1 to -3] (4)/u du Notice that the limits of integration have also changed according to the substitution u = 6-x. When x = 7, u = -1, and when x = 9, u = -3.Evaluate Integral: Now we integrate (4)/u with respect to u: ∫(4)/u du = 4 * ln|u| Evaluating this from -1 to -3 gives us: 4 * ln|-3| - 4 * ln|-1|Simplify Expression: Since ln|-1| = ln(1) = 0, the expression simplifies to: 4 * ln|-3| - 4 * 0 = 4 * ln(3)Find Average Value: Now we need to divide this result by (b-a) to find the average value: Average value = (1/(9-7)) * 4 * ln(3) = (1/2) * 4 * ln(3) = 2 * ln(3)Final Result: The average value of the function f(x) from x=7 to x=9 is therefore 2 * ln(3). This is the logarithm of a single number, as required.

Find the average value of the function $f(x)=\frac{4}{6-x}$ from $x=7$ to $x=9$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$

Full solution

More problems from Find trigonometric ratios of special angles

Related topics

Find the average value of the function f(x)=46−x f(x)=\frac{4}{6-x} f(x)=6−x4​ from x=7 x=7 x=7 to x=9 x=9 x=9. Write your answer as the logarithm of a single number in simplest form.\newlineAnswer: ln⁡(□) \ln (\square) ln(□)

Find the average value of the function $f(x)=\frac{4}{6-x}$ from $x=7$ to $x=9$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$