1. CHAPTER 2 2.4 Continuity Fundamental Theorem of Calculus In this lecture you will learn the most important relation between derivatives and areas (definite integrals). animation

2.  b a f (x) dx = –  a b f (x) dx  a a f (x) dx = 0 Comparison Properties of the Integral 1.If f (x) >= 0 for a = 0. 2.If f (x) >= g (x) for a <= x <= b, then  a b f (x) dx >=  a b g (x) dx. 1.If m <= f (x) <= M for a <= x <= b, then m(b-a) <=  a b f (x) dx <= M(b-a).

3. Example Estimate the value of the integral  -1 1 e x 2 dx.

4. ``Area so far’’ function. Let g(x) be the area between the lines: t=a, and t=x, and under the graph of the function f(t) above the T-axis. animation g’(x) = f(x) where g(x) =  a x f(t) dt.

5. Example Find the derivative with respect to x of  -2 x t 2 dt.

6. Example Find the derivative with respect to x of  -3 2 x sin t dt.

7. Example Find the derivative with respect to x of  -x 2 cos t dt.