Consider the curve y=f(x)=(2^x)-1

Find a value for a such that the average value of the function f(x) on the interval [0,a] is equal to 1.

Question

Consider the curve y=f(x)=(2^x)-1

Find a value for a such that the average value of the function f(x) on the interval [0,a] is equal to 1.

anonymous · Answer

(1/a-0) ∫ [0,a] (2^x-1)dx = 1 
I do not know how to solve for a

perl · Answer

did you apply the value a

anonymous · Answer

I found the antiderivative and I applied the first fundamental theorem of calculus. I'm getting stuck with simplification.

1/a[(2^a/ln(2))-a)-(2^0/ln(2))]

perl · Answer

ok

anonymous · Answer

I've gotten as far as:
2^a-aln(2)-1/aln(2)=1

perl · Answer

that looks right so far

anonymous · Answer

2^a-1=-2aln(2)

perl · Answer

we can really only approximate it

perl · Answer

a=1.812647074

anonymous · Answer

Yeah, that's what I got. So there's easy no way to simplify?

perl · Answer

no, not really.

perl · Answer

unless you want to leave your answer in terms of the Lambert function

anonymous · Answer

No, I think this is good. Thanks for the help.