chapter 2 2.4 continuity fundamental theorem of calculus in this lecture you will learn the most...
Post on 04-Jan-2016
218 Views
Preview:
TRANSCRIPT
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).
animation
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
animation
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.
top related