How do you find the average value of the function for #f(x)=1/x, 1

Question

How do you find the average value of the function for #f(x)=1/x, 1<=x<=4#?

Elijah Abram · Accepted Answer

#lnroot(3)4~~0.462# The #color(blue)"average value"# is. #1/(b-a)int_a^bf(x)dx# #=1/(4-1)int_1^4(1/x)dx# #=1/3[lnx]_1^4# #=1/3[ln4-ln1]# #=lnroot(3)(4)~~0.462" to 3 dec. places"#

Christopher Allers · Answer

undefined To find the average value of the function $ f(x) = \frac{1}{x} $ over the interval $ 1 \leq x \leq 4 $, you use the formula for the average value of a function over an interval:

$$ \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx $$

Where $ a $ and $ b $ are the endpoints of the interval.

In this case, $ a = 1 $ and $ b = 4 $. So, the average value is:

$$ \text{Average value} = \frac{1}{4-1} \int_{1}^{4} \frac{1}{x} \, dx $$

$$ = \frac{1}{3} \left[ \ln|x| \right]_{1}^{4} $$

$$ = \frac{1}{3} \left( \ln|4| - \ln|1| \right) $$

$$ = \frac{1}{3} \ln(4) $$

$$ = \frac{\ln(4)}{3} $$

Therefore, the average value of the function $ f(x) = \frac{1}{x} $ over the interval $ 1 \leq x \leq 4 $ is $ \frac{\ln(4)}{3} $.

How do you find the average value of the function for #f(x)=1/x, 1&lt;=x&lt;=4#?

How do you find the average value of the function for #f(x)=1/x, 1<=x<=4#?