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)

We need to find the average value of the function \( f(x) = \frac{1}{x} \) over the interval \([4,8]\). The formula for the average value of a function \( f(x) \) on the interval \([a, b]\) is given by: \[ \text{Average} = \frac{1}{b-a} \int_a^b f(x) \, dx \] For this problem, \( a = 4 \) and \( b = 8 \), so: \[ \text{Average} = \frac{1}{8-4} \int_4^8 \frac{1}{x} \, dx = \frac{1}{4} \int_4^8 \frac{1}{x} \, dx \] The integral of \( \frac{1}{x} \) is: \[ \int \frac{1}{x} \, dx = \ln |x| + C \] Evaluating the definite integral: \[ \int_4^8 \frac{1}{x} \, dx = \left[ \ln |x| \right]_4^8 = \ln(8) - \ln(4) \] Using the properties of logarithms: \[ \ln(8) - \ln(4) = \ln\left(\frac{8}{4}\right) = \ln(2) \] Thus, the average value is: \[ \text{Average} = \frac{1}{4} \ln(2) \] Comparing this with the given options, we see that option (D) is: \[ \frac{\ln(2)}{4} \] Therefore, the correct answer is (D).