Find the average value of the function f(x)=(3)/(9-4x) from x=3 to x=6. Express your answer as a constant times ln 5. Answer: ◻ln 5

Question

Find the average value of the function  f(x)=(3)/(9-4x) from  x=3 to  x=6. Express your answer as a constant times  ln 5. Answer:  ◻ln 5

Bytelearn · Accepted Answer

Understand average value 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 = 3 and b = 6, so we need to integrate f(x) = 3/(9-4x) from x = 3 to x = 6 and then multiply by the reciprocal of the interval's length.Set up integral for average value: Set up the integral for the average value. Average value = (1/(6-3)) * ∫ from 3 to 6 of (3/(9-4x)) dx This simplifies to: Average value = (1/3) * ∫ from 3 to 6 of (3/(9-4x)) dxPerform substitution to simplify integral: Perform a substitution to simplify the integral. Let u = 9 - 4x, then du = -4 dx or dx = -du/4. When x = 3, u = 9 - 4(3) = 9 - 12 = -3. When x = 6, u = 9 - 4(6) = 9 - 24 = -15. Now we can rewrite the integral in terms of u: Average value = (1/3) * ∫ from -3 to -15 of (-3/4) * (1/u) duEvaluate the integral: Evaluate the integral. The integral of 1/u with respect to u is ln|u|, so we have: Average value = (1/3) * (-3/4) * [ln|u|] from -3 to -15 This simplifies to: Average value = (-1/4) * [ln|-15| - ln|-3|] Since ln|a| - ln|b| = ln|a/b|, we can combine the logarithms: Average value = (-1/4) * ln(15/3) Average value = (-1/4) * ln(5)Check for errors and write final answer: Check for math errors and write the final answer. We have correctly applied the rules of logarithms and the integral was evaluated correctly. The negative sign comes from the fact that du was -4 dx, which we accounted for in the integral. Final answer: (-1/4)ln(5)

Find the average value of the function $f(x)=\frac{3}{9-4 x}$ from $x=3$ to $x=6$ . Express your answer as a constant times $\ln 5$ . $\newline$ Answer: $\square\ln 5$

Full solution

More problems from Negative exponents

Related topics

Find the average value of the function f(x)=39−4x f(x)=\frac{3}{9-4 x} f(x)=9−4x3​ from x=3 x=3 x=3 to x=6 x=6 x=6. 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 $f(x)=\frac{3}{9-4 x}$ from $x=3$ to $x=6$ . Express your answer as a constant times $\ln 5$ . $\newline$ Answer: $\square\ln 5$