Let f(x) = integral cos^2(t)dt from 0 to x for all x. Show that f has an inverse . Let c = f(pi/4), find (f^-1)'(c).

I think I can do the second part, but how do I show if there is an inverse? I know the function has to be increasing or decreasing but I'm not sure what to do with cos^2t

Question

Let f(x) = integral cos^2(t)dt from 0 to x for all x. Show that f has an inverse . Let c = f(pi/4), find (f^-1)'(c).

I think I can do the second part, but how do I show if there is an inverse? I know the function has to be increasing or decreasing but I'm not sure what to do with cos^2t

auctoratrox · Answer

say y = cos^2x and switch the variables and solve for y again. I got that the inverse of cos^2x is cos^-1(x^(1/2))

anonymous · Answer

oh so we ignore the integral..?

anonymous · Answer

Actually can anyone help me with the second part too? I can't figure out what value of t makes the function pi/4..

auctoratrox · Answer

do you mean c times the derivative of the inverse function?

anonymous · Answer

http://img811.imageshack.us/img811/639/3179eb0220514ce4bc93818.png This is the problem. I'm studying for final and I"ve never seen inverse function with integrals..

anonymous · Answer

$$
\int_a^bf^{-1}=bf^{-1}(b)-af^{-1}(a)-\int_{f^{-1}(a)}^{f^{-1}(b)}f
$$

anonymous · Answer

That's if $f$ is increasing. Probably different if $f$ is decreasing. I haven't actually spent the time to work through the proofs yet.