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What is the average value of (1)/(x) on the interval 4 <= x <= 8 ? Choose 1 answer: (A) (3)/(16) (B) (1)/(16) (C) (ln(32))/(4) (D) (ln(2))/(4)

What is the average value of 1x \frac{1}{x} on the interval 4≤x≤8 4 \leq x \leq 8 ?\newlineChoose 11 answer:\newline(A) 316 \frac{3}{16} \newline(B) 116 \frac{1}{16} \newline(C) ln⁡(32)4 \frac{\ln (32)}{4} \newline(D) ln⁡(2)4 \frac{\ln (2)}{4}

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Full solution

Q. What is the average value of 1x \frac{1}{x} on the interval 4≤x≤8 4 \leq x \leq 8 ?\newlineChoose 11 answer:\newline(A) 316 \frac{3}{16} \newline(B) 116 \frac{1}{16} \newline(C) ln⁡(32)4 \frac{\ln (32)}{4} \newline(D) ln⁡(2)4 \frac{\ln (2)}{4}
  1. Define Average Value Formula: To find the average value of a function f(x)f(x) on the interval [a,b][a, b], we use the formula:\newlineAverage value = 1(b−a)⋅∫abf(x) dx\frac{1}{(b-a)} \cdot \int_{a}^{b} f(x) \, dx\newlineHere, f(x)=1xf(x) = \frac{1}{x}, a=4a = 4, and b=8b = 8.
  2. Calculate Integral of 1/x1/x: First, we calculate the integral of 1/x1/x from 44 to 88. The integral of 1/x1/x dx is ln⁡∣x∣\ln|x|, so we evaluate ln⁡∣x∣\ln|x| from 44 to 88. ∫481xdx=ln⁡∣8∣−ln⁡∣4∣\int_{4}^{8} \frac{1}{x} dx = \ln|8| - \ln|4|
  3. Substitute Values into ln⁡∣x∣\ln|x|: Now we substitute the values of x=8x = 8 and x=4x = 4 into ln⁡∣x∣\ln|x|.ln⁡∣8∣−ln⁡∣4∣=ln⁡(8)−ln⁡(4)\ln|8| - \ln|4| = \ln(8) - \ln(4)
  4. Simplify Logarithmic Expression: We use the properties of logarithms to simplify the expression. ln⁡(8)−ln⁡(4)=ln⁡(84)=ln⁡(2)\ln(8) - \ln(4) = \ln\left(\frac{8}{4}\right) = \ln(2)
  5. Calculate Average Value: Now we calculate the average value using the formula.\newlineAverage value = (1/(8−4))×ln⁡(2)(1/(8-4)) \times \ln(2)\newlineAverage value = (1/4)×ln⁡(2)(1/4) \times \ln(2)
  6. Final Result: The average value of (1)/(x)(1)/(x) on the interval [4,8][4, 8] is (ln⁡(2))/(4)(\ln(2))/(4). This corresponds to answer choice (D)(D).

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