What is the average value of \( \frac{1}{x} \) on the interval \( 4 \leq x \leq 8 \) ? Choose 1 answer: (A) \( \frac{3}{16} \) (B) \( \frac{1}{16} \) (C) \( \frac{\ln (32)}{4} \) (D) \( \frac{\ln (2)}{4} \) (a)

To find the average value of the function \( \frac{1}{x} \) on the interval \( [4, 8] \), we need to use the formula for the average value of a continuous function \( f \) over the interval \( [a, b] \): \[ \text{Average Value} = \frac{1}{b-a} \int_a^b f(x) \, dx \] In this case, \( f(x) = \frac{1}{x} \), \( a = 4 \), and \( b = 8 \). First, calculate the integral: \[ \int_4^8 \frac{1}{x} \, dx = [\ln |x|]_4^8 = \ln(8) - \ln(4) = \ln\left(\frac{8}{4}\right) = \ln(2) \] Now, calculate the average value: \[ \text{Average Value} = \frac{1}{8-4} \cdot \ln(2) = \frac{1}{4} \ln(2) \] Thus, the average value of \( \frac{1}{x} \) on the interval \( [4, 8] \) is \( \frac{\ln(2)}{4} \). The correct answer is (D) \( \frac{\ln (2)}{4} \).