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Find the average value of the function
f(x)=(8)/(x-3) from
x=5 to
x=7. Write your answer as the logarithm of a single number in simplest form.
Answer:
ln(◻)

Find the average value of the function $f(x)=\frac{8}{x-3}$ from $x=5$ to $x=7$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$

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Find trigonometric ratios of special angles

Full solution

Q. Find the average value of the function

f(x)=\frac{8}{x-3}

from

x=5

to

x=7

. Write your answer as the logarithm of a single number in simplest form.

\newline

Answer:

\ln (\square)

Set Up Integral: To find the average value of a continuous function $f(x)$ on the interval $[a, b]$ , we use the formula: $\newline$ Average value = $(1/(b-a)) \times \int_{a}^{b} f(x) \, dx$ $\newline$ Here, $a = 5$ and $b = 7$ , so we need to integrate $f(x) = (8)/(x-3)$ from $x = 5$ to $x = 7$ and then multiply by $1/(7-5)$ .
Solve Integral: First, let's set up the integral: $\newline$ Average value = $(1/(7-5)) \times \int_{5}^{7} \frac{8}{x-3} \,dx$ $\newline$ This simplifies to: $\newline$ Average value = $(1/2) \times \int_{5}^{7} \frac{8}{x-3} \,dx$
Evaluate Integral: To integrate $\frac{8}{x-3}$ , we recognize this as a simple logarithmic integration problem. The integral of $\frac{1}{u} du$ is $\ln|u|$ , so the integral of $\frac{8}{x-3} dx$ is $8\ln|x-3|$ . Now we need to evaluate this from $x = 5$ to $x = 7$ .
Simplify Expression: Evaluating the integral, we get: $\newline$ $(\frac{1}{2}) \times [8\ln|7-3| - 8\ln|5-3|]$ $\newline$ This simplifies to: $\newline$ $(\frac{1}{2}) \times [8\ln(4) - 8\ln(2)]$
Apply Logarithm Properties: We can simplify the expression using logarithm properties. The difference of logarithms is the logarithm of the quotient: $\newline$ $(\frac{1}{2}) \times 8\ln(\frac{4}{2})$ $\newline$ Since $4/2$ is $2$ , this further simplifies to: $\newline$ $(\frac{1}{2}) \times 8\ln(2)$
Final Result: Multiplying through by $(\frac{1}{2}) \times 8$ gives us: $\newline$ $4\ln(2)$ $\newline$ This is the average value of the function $f(x)$ from $x = 5$ to $x = 7$ , expressed as the logarithm of a single number in simplest form.

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