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Thursday, November 10
−
Lecture 25 :
Fundamental theorem of calculus.
(Refers to
Section 5.4, 5.5 in your text)
Students who have mastered the content of this lecture know
:
The Fundamental theorem of calculus, the
definition of “antiderivative” and the general antiderivative of a function f(x), the net change of a
function over an interval [a, b].
Students who have practiced the techniques presented in this lecture will be able to
:
State both parts of
the FTC and apply the FTC directly to integrals, compute the definite of integral of a simple function by
first finding an antiderivative of the function, compute the net change of a function over an interval.
The Fundamental theorem of calculus is a theorem which links “differential” calculus to
“integral” calculus. It shows that differentiation (finding slopes of tangent lines to a
curve) and integration (computing areas of regions), two apparently unrelated processes
are actually inverses of each other.
Note: It is important to note that the FTC is actually
two
statements. We often refer to
them as “Part I and Part II of the FTC”. But different authors may choose to order them
differently. So in a different text look carefully to determine how the author prefers to
order them.
25.1
The
Fundamental theorem of calculus
(FTC) – Let
f
be continuous on the interval
[
a
,
b
] and
g
(
x
) be the function defined as
Then:
1)
The function
g
(
x
) is continuous on [
a
,
b
] and differentiable on (
a
,
b
) and for any
x
strictly between
a
and
b
,
g
′
(
x
) =
f
(
x
).
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 Spring '11
 RobertAndre
 Calculus, Derivative, Fundamental Theorem Of Calculus

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