Can you carry on?
Hi I feel very lost.
Question is if f is a differentiable function such that f(x) is never zero and
definite integral (I will use "S") b=x; a=0 f(t)dt = (f(x))^2 for all x, find f.
So my first thought is to let y = f(x) so S b=x; a=0 f(t) dt = y^2
then S b=x; a=0 2y = y^2 but I still don't have a clue what f is and if I jumped too far with the 2y
Please help. I can't find an example anywhere in my textbook on how to find f.
This calculus beginner
(The point here is that you don't have to be at all sure what f(x) is to determine f'(x) - and if that's a point you already got then you weren't asking for further help so forgive the following attempt.)
Anyway, just in case a picture helps...
The fundamental theroem tells us...
... where the vertical line differentiates downwards (anti-differentiates up) with respect to x.
And the problem says that , so we're entitled to say...
But is a composite function, requiring the chain-rule, which we can depict so...
... where straight continuous lines differentiate downwards (anti-diff up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (which is the inner function of the composite and hence subject to the chain rule).
So we get to...
... the bottom row to be solved for f ' (x) as you discussed.
Don't integrate - balloontegrate!
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