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).

In this lecture you will learn the most important relation between derivatives and areas (definite integrals).

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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 <= x <= b, then a

b

f (x) dx >= 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).

Example Estimate the value of the integral -1

1

e x2 dx .

``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.

0.2 0.4 0.6 0.8 1

-0.2

-0.1

0.1

0.2

Area :0.0440554

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g’(x) = f(x)

where

g(x) = a

x

f(t) dt.

Example Find the derivative with respect to x of -2

x t

2 dt.

Example Find the derivative with respect to x of -3

2 x sin t dt.

Example Find the derivative with respect to x of -x

2 cos t dt.