Confusing integration problem

[Nyithra]

Let F(x) = ₁∫^√x e^{t^2} dt. Find F(1), F'(1), and F''(1)

Now...I figure I am just approaching this problem the wrong way, but with x = 1 the integral would be from 1 to 1 right? So all of the answers would be zero?

That answer doesn't feel right since this is on a take home quiz. I'm not looking for this to be solved for me, I just want a push in the right direction.

 

[tkhunny]

This is a definiion problem, just to see if you are paying attention.

Remember that F'(1) does NOT mean what the notation actually indicates if hte standard interpretation of functino notation is to be enforced. This notation means a little variation. F'(1) means 1) Find F'(x), and 2) Substitute x = 1. It does NOT mean the other diretion, as it always would be zero.

You need a chain rule. \(\displaystyle F'(x) = \frac{e^{x}}{2\cdot\sqrt{x}}\)

 

[Nyithra]

Thanks for responding!

Your answer would make sense to me if the same variable on the integral was being used in the function, but t is being used in the function, so how would that work?

I am also looking for more help with finding F(1), if it doesn't just equal zero that is.

 

[tkhunny]

You are exactly correct on F(1). Limits are the same? Zero (0). Good call.

It does not matter that 't' is there. Change 't' to 's' and observe that there is no consequence. It's called a "Dummy Variable". It is a function of 'x', not matter the name of the Dummy Variable.

 

[Nyithra]

Alright, thank you so much for your help, this problem is very clear now.