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)

Define Average Value Formula: To find the average value of a function f(x) on the interval [a, b], we use the formula: Average value = (1/(b-a)) * ∫ from a to b of f(x) dx Here, f(x) = 1/x, a = 4, and b = 8.Calculate Integral of 1/x: First, we calculate the integral of 1/x from 4 to 8. The integral of 1/x dx is ln|x|, so we evaluate ln|x| from 4 to 8. ∫ from 4 to 8 of (1/x) dx = ln|8| - ln|4|Substitute Values into ln|x|: Now we substitute the values of x = 8 and x = 4 into ln|x|. ln|8| - ln|4| = ln(8) - ln(4)Simplify Logarithmic Expression: We use the properties of logarithms to simplify the expression. ln(8) - ln(4) = ln(8/4) = ln(2)Calculate Average Value: Now we calculate the average value using the formula. Average value = (1/(8-4)) * ln(2) Average value = (1/4) * ln(2)Final Result: The average value of (1)/(x) on the interval [4, 8] is (ln(2))/(4). This corresponds to answer choice (D).

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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 $\frac{1}{x}$ on the interval $4 \leq x \leq 8$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{3}{16}$ $\newline$ (B) $\frac{1}{16}$ $\newline$ (C) $\frac{\ln (32)}{4}$ $\newline$ (D) $\frac{\ln (2)}{4}$

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Math Problems

Algebra 2

Roots of rational numbers

Full solution

Q. What is the average value of

\frac{1}{x}

on the interval

4 \leq x \leq 8

\newline

Choose

1

answer:

\newline

(A)

\frac{3}{16}

\newline

(B)

\frac{1}{16}

\newline

(C)

\frac{\ln (32)}{4}

\newline

(D)

\frac{\ln (2)}{4}

Define Average Value Formula: To find the average value of a function $f(x)$ on the interval $[a, b]$ , we use the formula: $\newline$ Average value = $\frac{1}{(b-a)} \cdot \int_{a}^{b} f(x) \, dx$ $\newline$ Here, $f(x) = \frac{1}{x}$ , $a = 4$ , and $b = 8$ .
Calculate Integral of $1/x$ : First, we calculate the integral of $1/x$ from $4$ to $8$ . The integral of $1/x$ dx is $\ln|x|$ , so we evaluate $\ln|x|$ from $4$ to $8$ . $\int_{4}^{8} \frac{1}{x} dx = \ln|8| - \ln|4|$
Substitute Values into $\ln|x|$ : Now we substitute the values of $x = 8$ and $x = 4$ into $\ln|x|$ . $\ln|8| - \ln|4| = \ln(8) - \ln(4)$
Simplify Logarithmic Expression: We use the properties of logarithms to simplify the expression. $\ln(8) - \ln(4) = \ln\left(\frac{8}{4}\right) = \ln(2)$
Calculate Average Value: Now we calculate the average value using the formula. $\newline$ Average value = $(1/(8-4)) \times \ln(2)$ $\newline$ Average value = $(1/4) \times \ln(2)$
Final Result: The average value of $(1)/(x)$ on the interval $[4, 8]$ is $(\ln(2))/(4)$ . This corresponds to answer choice $(D)$ .