MATH 2400 LECTURE NOTES
PETE L. CLARK
1.
First Lecture
1.1.
The Goal: Calculus Made Rigorous.
The goal of this course is to cover the material of single variable calculus
in a
mathematically rigorous way
.
The latter phrase is important:
in most calculus
classes the emphasis is on techniques and applications; while theoretical explana
tions may be given by the instructor – e.g. it is usual to give some discussion of
the meaning of a
continuous function
– the student tests her understanding of
the theory mostly or entirely through her ability to apply it to solve problems.
This course is very different: not only will
theorems
and
proofs
be presented in
class by me, but they will also be presented by you, the student, in homework and
on exams.
This course offers a strong foundation for a student’s future study of
mathematics, at the undergraduate level and beyond.
As examples, here are three of the fundamental results of calculus; they are called –
by me, at least – the three
Interval Theorems
, because of their common feature:
they all concern an arbitrary continuous function defined on a closed, bounded
interval.
Theorem 1.
(Intermediate Value Theorem) Let
f
: [
a, b
]
→
R
be a continuous
function defined on a closed, bounded interval. Suppose that
f
(
a
)
<
0
and
f
(
b
)
>
0
.
Then there exists
c
with
a < c < b
such that
f
(
c
) = 0
.
Theorem 2.
(Extreme Value Theorem) Let
f
: [
a, b
]
→
R
be a continuous function
defined on a closed, bonuded interval. Then
f
is bounded and assumes its maximum
and minimum values. This means that there exist numbers
m
≤
M
such that
a) For all
x
∈
[
a, b
]
,
m
≤
f
(
x
)
≤
M
.
b) There exists at least one
x
∈
[
a, b
]
such that
f
(
x
) =
m
.
c) There exists at least one
x
∈
[
a, b
]
such that
f
(
x
) =
M
.
Theorem 3.
(Uniform Continuity and Integrability) Let
f
: [
a, b
]
→
R
be a con
tinuous function defined on a closed, bounded interval. Then:
a)
f
is uniformly continuous.
1
b)
f
is integrable:
∫
b
a
f
exists and is finite.
Except for the bit about uniform continuity, these three theorems are familiar re
sults from the standard freshman calculus course.
Their proofs, however, are
not.
Most freshman calculus texts like to give at least
some
proofs, so it is often
the case that these three theorems are used to prove even more famous theorems
1
The definition of this is somewhat technical and will be given only later on in the course.
Please don’t worry about it for now.
1
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PETE L. CLARK
in the course, e.g. the
Mean Value Theorem
and the
Fundamental Theorem
of Calculus
.
Why then are the three interval theorems not proved in freshman calculus?
Because
their proofs depend upon fundamental properties of the real numbers that are not
discussed in such courses.
Thus one of the necessary tasks of the present course is
to give a more penetrating account of the real numbers than you have seen before.
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 Fall '11
 Clark
 Math, Calculus, ﬁeld axioms, pete l. clark

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