Fundamental theorem of calculus Deriving the Theorem Let Apply the definition of the derivative Rule for Integrals Deriving the Theorem This is average value of f from x to x h ID: 529984
Download Presentation The PPT/PDF document "Section 5.4a" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Section 5.4a
Fundamental theorem of calculusSlide2
Deriving the Theorem
Let
Apply the definition of the derivative:
Rule for Integrals!Slide3
Deriving the Theorem
This is average value of
f
from
x to x
+ h. Assuming that f iscontinuous, it takes on its averagevalue at least once in the interval…
For some
c
between
x and
x + hBack to the proof…Slide4
Deriving the Theorem
What happens to
c
as
h
goes to zero???
As
x
+
h
gets closer to x, it forces c to approach x…
Since
f
is continuous, this means that f(c) approaches f(x):
Putting it all together…Slide5
Deriving the Theorem
Definition of Derivative
Rule for Integrals
For some
c
between
x
and
x
+
h
Because
f iscontinuousSlide6
The Fundamental Theorem of
Calculus, Part 1
If is
continuous on [
a
, b], then the function
has a derivative at every point
x
in [a, b], and
…the definite integral of a continuous function is a
differentiable function of its upper limit of integration…Slide7
The Fundamental Theorem of
Calculus, Part 1
Every continuous function is the derivative of some other
function.
The processes of integration and differentiation are
inverses of one another.
Every continuous function has an antiderivative.A Powerful Theorem Indeed!!!Slide8
The Fundamental Theorem of
Calculus, Part 1
Evaluate each of the following, using the
FTC.Slide9
The Fundamental Theorem of
Calculus, Part 1
Find
dy
/
dx
if
The upper limit of integration is not
x
y is a composite of:
and
Apply the Chain Rule to find
dy
/
dx
:Slide10
The Fundamental Theorem of
Calculus, Part 1
Find
dy
/
dx
ifSlide11
The Fundamental Theorem of
Calculus, Part 1
Find
dy
/
dx
if
Chain
RuleSlide12
Deriving
More of the
Theorem
Let
If
F
is
any
antiderivative
of f, then F(x) = G
(x) + C for someconstant C.
Let’s evaluate
F
(
b
) –
F
(
a
):
0Slide13
The Fundamental Theorem of
Calculus, Part 2
If
is continuous at every point of [a, b], and if F is anyantiderivative of on [a, b], then
This part of the Fundamental Theorem is also called the
Integral Evaluation Theorem
.Slide14
The Fundamental Theorem of
Calculus, Part 2
Any definite integral of any continuous function
can be
calculated without taking limits, without calculating Riemann sums, and often without major effort all we need is an antiderivative of !!!
Another Very Powerful Theorem!!!Slide15
The Fundamental Theorem of
Calculus, Part 2
The usual notation for
F
(
b
) – F(a) is
A “note” on notation:
or
depending on whether
F
has one or more terms…Slide16
The Fundamental Theorem of
Calculus, Part 2
Evaluate the given integral using an
antiderivative
.
Antiderivative:
How can we support this
answer
numerically
???Slide17
The Fundamental Theorem of
Calculus, Part 2
Evaluate the given integral using an
antiderivative
.
Antiderivative:Slide18
The Fundamental Theorem of
Calculus, Part 2
Evaluate the given integral using an antiderivative.
Antiderivative:Slide19
The Fundamental Theorem of
Calculus, Part 2
Evaluate the given integral using an antiderivative.
Antiderivative:Slide20
The Fundamental Theorem of
Calculus, Part 2
Evaluate the given integral using an antiderivative.
Antiderivative:Slide21
The Fundamental Theorem of
Calculus, Part 2
Evaluate the given integral using an antiderivative.
Antiderivative: