Q. 54 - Calculus

Question

For each function f and interval [a, b] in Exercises 53–55, approximate the average value of f from x = a to x = b, using a sample size of at least eight.

$f\left(x\right)=\ln x, \left[1,5\right]$

Step-by-Step Solution

Answer

The average value of f from x = a to x = b, using a sample size of at least eight is  $2.5.$

Step 1. Given Information.

The given function and the interval is $f\left(x\right)=\ln x, \left[1,5\right].$

Step 2. Approximating the average value.

To approximate the average value of f from x = a to x = b, using a sample size of at least eight, we will use the formula: $\left(f\left(x_0\right)+......f\left(x_n\right)\right)△x, △x=\frac{b-a}{n}.$

Thus,

$$\begin{gathered} △x=\frac{5-1}{8} \\ △x=\frac{4}{8} \\ △x=\frac{1}{2} \end{gathered}$$

Now,

$$\begin{gathered} =\left(f\left(\frac{1}{2}\right)+f\left(1\right)+f\left(\frac{3}{2}\right)+f\left(2\right)+f\left(\frac{5}{2}\right)+f\left(3\right)+f\left(\frac{7}{2}\right)+f\left(4\right)\right) \\ \left(\frac{1}{2}\right) \\ =\left(\ln \left(\frac{1}{2}\right)+\ln \left(1\right)+\ln \left(\frac{3}{2}\right)+\ln \left(2\right)+\ln \left(\frac{5}{2}\right)+\ln \left(3\right)+\ln \left(\frac{7}{2}\right)+\ln \left(4\right)\right) \\ \left(\frac{1}{2}\right) \\ \approx 2.5 \end{gathered}$$

Therefore, the average value is $2.5.$