Expert: Paul Klarreich - 4/7/2006

Question
Hi Paul,
My AP exam is 3 weeks away and I'm studying hard!
I'm hoping you can help me with the following question:
If F' is a continuous function for all real x, then limit as h approaches zero of [1/h]*[intgeral of F'(x) from a to a+h is F'(a).
Why is this true?

Answer
Hi, Jeff,

Subject:  limit of integral
Question:  Hi Paul,
My AP exam is 3 weeks away and I'm studying hard!
I'm hoping you can help me with the following question:
If F' is a continuous function for all real x, then limit as h approaches zero of

[1/h]*[intgeral of F'(x) from a to a+h is F'(a).
Why is this true?
-----------------------------------------------
Let F'(x) = f(x) (just a change of notation here)

You want the limit of this as h -> 0:

1  {a+h
--- |    f(x) dx
h  }a

This is the fundamental theorem of calculus.  It says:

{b
|  f(x) dx = F(b) - F(a)
}a

where F is any antiderivative of f.  (which is what your problem, and my notation, says.)

Applying this:

1  {a+h            1
--- |    f(x) dx = ---[F(a + h) - F(a)] =
h  }a              h

F(a + h) - F(a)
---------------
     h

But the limit of that as  h-> 0 is simply the derivative of the function at a, I.E.

F'(a)

I think that's all you need.