Homework SourseIt would be nice if average values AV of integrable function
It would be nice if average values (AV) of integrable functions obeyed the following rules on an interval [a,b].
a) AV(f + g) = AV(f) + AV(g)
b) AV(kf) = k*AV(f) for any number k
c) AV(f) ? AV(g) if f(x) ? g(x) on [a,b]
Do these rules ever hold? Give reasons for your answers.
a) AV(f + g) = AV(f) + AV(g)
b) AV(kf) = k*AV(f) for any number k
c) AV(f) ? AV(g) if f(x) ? g(x) on [a,b]
Do these rules ever hold? Give reasons for your answers.
Solution
a.) yes
b.) yes
c.) yes
av( f+g)
= [(f+g)dx]/(b-a)
= [f dx + gdx]/(b-a)
= [f dx]/(b-a) + [gdx]/(b-a)
= av(f) + av(g)
av( kf)
= [kfdx]/(b-a)]
= k[fdx]/(b-a)
= k av(f)
f g..........for each x on [a,b]
implies
f dx g dx
so
[f dx]/(b-a) [g dx]/(b-a)........provided b-a > 0 ie b > a
the interval [a, b] is always written so that b > a
![It would be nice if average values (AV) of integrable functions obeyed the following rules on an interval [a,b]. a) AV(f + g) = AV(f) + AV(g) b) AV(kf) = k*AV(f](/WebImages/43/it-would-be-nice-if-average-values-av-of-integrable-function-1133976-1761606489-0.webp "It would be nice if average values (AV) of integrable functions obeyed the following rules on an interval [a,b]. a) AV(f + g) = AV(f) + AV(g) b) AV(kf) = k*AV(f")
