Find the second derivative of F (x)

[~Sam~]

Homework Statement

Assuming sufficient differentiability, find second derivative of F(x) = integ[a,x] (t-x)² f(t) d(t)

Homework Equations

Probably Fund.Thm of Calculus and some properties

The Attempt at a Solution

I really have no idea..I tried evaluating but with t=x but I get zero..I have never done second derivative so I'm somewhat clueless. Anyone care to help and explain?

 

[Mark44]

The second derivative is the derivative of the first derivative. E.g., if f(x) = x^2, f'(x) = 2x, and f''(x) = d/dx(f'(x)) = d/dx(2x) = 2.

The Fundamental Theorem of Calculus can help you get F'(x), so take a careful look at this theorem and any examples that show how to use it. After that, just take the derivative of what you got for F'(x).

 

[Dick]

Use the Leibniz integral rule twice. Besides the term that you found equal to zero, there's another term that involves differentiating the integrand. That part is not necessarily zero.

 

[~Sam~]

I don't quite understand? I've never learned the leibniz rule. Can you exemplify it with this question? The f(t) part also confuses me.

 

[Dick]

Don't worry about the f(t) part. Your answer will be in terms of f(t). http://en.wikipedia.org/wiki/Leibniz_integral_rule Here's an easy example. Take F(x) to be the integral from a to x of x*t*dt. You only have to deal with the first and third terms in the rule. The first term you get just from putting t=x in the integrand, so x^2. That's the one you know. The last term says you should differentiate inside the integral. That gives you the integral from a to x of t*dt. Or x^2/2-t^2/2. Adding the two you get F'(x)=3*x^2/2-a^2/2. Compare that with what you get by actually integrating and then differentiating. Now try doing that with your example. What's F'(x)?