Find the average value of f(x) = 1 ⁄(1 + e ^x) ^3 on the interval [0, ln 2].

Question

Find the average value of f(x) = 1 ⁄(1 + e^ x) ^3 on the interval [0, ln 2].

Answer 1

Recall the formula for average value, where a and b are the endpoints of the interval [a , b]:

[f(b) - f(a)] / (b - a)

Here, a = 0 and b = ln(2).

( [1 / (1 + e ^(ln(2)) ^3] - [1 ⁄ (1 + e ^(0)) ^3] ) / (ln(2) - 0)

( [1 / (1 + 2)^3 ] - [1 / (1 + 1)^3 ] ) / ln(2)

( (1 / 27) - (1 / 8) ) / ln(2)

The exact answer is -19 / (216 ln(2)), and an approximate answer is -0.127.