Lecture
Outlines and Class Notes
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Chapter 1. Real Numbers and
Limits of Sequences
1.1
lecture outline The Real
Number System
1.1
classnotes
1.2
lecture outline (reposted, Feb. 7 with changes to page 12)
Limits of Sequences & Cauchy Sequences
1.2
classnotes(part1) , 1.2
classnotes(part2)
1.3
lecture outline (reposted,
Jan. 30) The Completeness Axiom And Some of
Its Consequences
1.3
classnotes(part1) , 1.3
classnotes(part2) , 1.3
classnotes(part3)
1.4
lecture outline Algebraic
Combinations of Sequences
1.4 classnotes(part1)
, 1.4 classnotes(part2)
1.5
lecture outline The
Bolzano-Weierstrass Theorem
1.5
classnotes
1.6
lecture outline The Nested Intervals
Theorem
1.6 classnotes(part1) , 1.6
classnotes(part2) (reposted
due to a
mistake on
page 10, 2/28) ,
1.6 classnotes(part3) (Feb.
28) Some
comments on
dense sets and
the density of
the rationals
in the reals
1.7
lecture outline
The Heine-Borel Theorem
1.7
classnotes(part1) , 1.7 classnotes:
Proof of the
Heine-Borel
Theorem
1.8
lecture outline (reposted,
Feb. 8. The change appears in red on slide 5)
Countability of the Rational Numbers; Countable
and Uncountable Sets
1.8
classnotes
Chapter
2. Continuous Functions
2.1
lecture outline (reposted, Feb. 8. The change are on
the last two slides) Limits of
Functions
2.1
classnotes
2.2
lecture outline Continuous Functions
2.2
classnotes(part 1) , 2.2
classnotes(part 2)
2.3
lecture outline Some Properties of
Continuous Functions
2.3
classnotes
2.4
lecture outline The Extreme Value
Theorem and Some of its Consequences
2.4
classnotes(part 1)
(reposted
3/30/2017 to fix a mistake
in the proof of the
Extreme Value Theorem)
2.4
classnotes(part 2)
2.5
lecture outline The Banach space
C[a,b]
2.5
classnotes(part 1) , 2.5 classnotes(part
2) , 2.5 classnotes(part
3)
Chapter
3. The Riemann Integral
3.1
and 3.2 lecture outline Definitions and
Basic Properties of the Riemann Integral
3.1,3.2
classnotes(part 1)
, 3.1,3.2
classnotes(part
2)
3.3
lecture outline Integrals of Uniform
Limits
3.3 classnotes
3.4
lecture outline The Cauchy-Schwarz
Inequality and a New Triangle Inequality
3.4
classnotes
Chapter
4. The Derivative
4.1
lecture outline Derivatives
and Differentials
4.1
classnotes(part1) , 4.1
classnotes(part2)
4.2
lecture outline The
Mean Value Theorem
4.2
classnotes(part1)
(reposted 4/28
to fix a
mistake in the
Local Extrema
theorem on
page 3),
4.2
classnotes(part2)
4.3
lecture outline The Fundamental
Theorems of Calculus
4.3 classnotes , 4.3 classnotes(part
2)
4.4
lecture outline Uniform Convergence
of Sequences of Functions and the Derivative
4.4
classnotes
4.5
lecture outline Cauchy's
Generalized Mean Value Theorem and L'Hopital's Rule
4.5 classnotes
4.6 lecture
outline Taylor's
Theorem