12/19/20: Final Exam due at 3:30 p.m.
12/19/20: Final Exam due at 3:30 p.m.
12/17/20, 3:30 p.m.: Final Exam is posted on the Homework and
Exams page. You will have 48 hours to work on it.
12/16/20: The last class meeting. Discussion of Midterm
2. [Class notes]
12/14/20: Continuing our problem
sessions. [List of
problems; Class notes]
12/11/20: No class meeting: everybody is working at home on their
midterm. Midterm Exam II due at the end of class, 1:10 p.m.
12/10/20, 1:10 p.m.: Midterm Exam II is posted on the Homework and
Exams page. You will have 24 hours to work on it.
12/09/20: Homework is due on Canvas by the beginning of the
class. Lots of fun (problem
session). [Class notes]
12/07/20: Lots of fun (problem
session). [Class notes]
12/04/20: Newton's method. No fun yet, but some is coming our
way... [Class notes. Rudin:
Exercise 5.25]
12/02/20: Homework is due on Canvas by the beginning of the class. A
higher-derivative test for relative extrema. The contraction mapping
theorem. [Class
notes. Rudin: Sections 9.22-9.23 and Exercise 5.22]
11/30/20: Higher-order derivatives. Taylor's
theorem. [Class notes. Rudin:
Sections 5.14-5.15]
11/27/20: Thanksgiving Break: no classes.
11/25/20: L'Hôpital's rule. [Class
notes. Rudin: Sections 5.13]
11/23/20: Darboux's theorem. Cauchy's mean value
theorem. [Class notes. Rudin:
Sections 5.9, 5.12]
11/20/20: Homework is due on Canvas by the beginning of the
class. Relative extrema and critical points. Rolle's theorem and the
mean value theorem. Derivatives and monotone
functions. [Class notes. Rudin:
Sections 5.7-5.8, 5.10-5.11]
11/18/20: Differentiabily and continuity. The derivative: the sum,
difference, product, and quotient rules. The chain
rule. [Class notes. Rudin:
Sections 5.1-5.6]
11/16/20: Monotone functions. The set of discontinuities of a monote
function. The derivative:
definition. [Class notes. Rudin:
Sections 4.28-4.31, 5.1]
11/13/20: Homework is due on Canvas by the beginning of the
class. Infinite limits and limits at
infinity. Discontinuities. [Class
notes. Rudin: Sections 4.25-4.27, 4.32-4.34]
11/11/20: Continuity and compactness. Uniform continuity. The extreme
value theorem. [Class
notes. Rudin: Sections 4.13-4.16, 4.18-4.21]
11/9/20: Inverse images of open and closed subsets under continuous
maps. Continuous images of connected sets. Intermediate value
theorem. [Class
notes. Rudin: Sections 4.8, 4.22-4.24]
11/6/20: Homework is due on Canvas by the beginning of the class. More
on limits of
functions. Continuity. [Class
notes. Rudin: Sections 4.2-4.7, 4.9-4.12]
11/4/20: The proof of Riemann's theorem on rearrangements of
conditionally convergent series. Limits of
functions. [Class notes. Rudin:
Sections 3.54, 4.1-4.2 (just the Corollary)]
11/2/20: Rearrangements and Riemann's theorem (no proof yet, but an
important proposition about the series ∑
ak± is
proven). [Class notes. Rudin:
Sections 3.52-3.55, skipping the proof of 3.54]
10/30/20: Homework is due on Canvas by the beginning of the
class. Example on the ratio test. Summation by parts. Dirichlet's
test. [Class notes. Rudin:
Sections 3.34-3.35, 3.41-3.43]
10/28/20: More on lim inf and lim sup. The root and ratio
tests. [Class notes. Rudin:
Sections 3.33-3.37 (skipping the proof of 3.37)]
10/26/20: Limit superior and limit inferior. Infinite limits
(divergence to ±∞). [Class
notes. Rudin: Sections 3.15-3.19]
10/23/20: Homework is due on Canvas by the beginning of the
class. Alternating series. Absolute and conditional convergence. The
comparison test. [Class notes. Rudin:
Sections 3.43, 3.45-3.46, 3.25]
10/21/20: One more special
sequence: .
Introduction to series: definition, partial sums, the sum. The Cauchy
criterion for series. The harmonic series. The nth term divergence
test. Sums of series. Series of nonnegative terms. The geometric
series. The Euler number as a
series. [Class notes. Rudin:
Sections 3.20 (c), 3.21-3.23, 3.28, 3.47, 3.24, 3.26, 3.30-3.32]
10/19/20: Some special sequences. The Euler number e as a
limit. [Class notes. Rudin:
Section 3.20, except (c)]
10/16/20: No class meeting: everybody is working at home on their
midterm. Midterm Exam I due at the end of class, 1:10 p.m.
10/14/20: Monotonic sequences. Discussion of homework problem on the
connectedness of a convex subset of ℝn. Discussion of
topics related to the exam moved to office
hours. [Class notes. Rudin:
Sections 3.13-3.14]
10/12/20: Sequential compactness implies the Bolzano-Weierstrass
property. Cauchy sequences. [Class
notes. Rudin: Sections 3.8-3.12]
10/9/20: Homework is due on Canvas by the beginning of the class. The
algebra of sequences and limits in ℝ, ℂ, and
ℝn. Subsequences and sequential
compactness. [Class notes. Rudin:
Sections 3.3-3.7]
10/7/20: Connected subsets on the real line are the intervals,
continued. Sequences and their limits in metric
spaces. [Class notes. Rudin:
Sections 2.7, 3.1-3.2]
10/5/20: The Cantor set. Connected sets. Connected subsets on the real
line are the intervals. [Class
notes. Rudin: Sections 2.44-2.47]
10/2/20: Homework is due on Canvas by the beginning of the class. The
Bolzano-Weierstrass property implies compactness. ℝn:
Nested intervals, the Heine-Borel
theorem. [Class notes. Rudin:
Sections 2.38-2.42]
9/30/20: Compact sets are closed and bounded. Closed subsets of
compact sets. The Bolzano-Weierstrass property. ℝn:
Nested intervals, the Heine-Borel
theorem. [Class notes. Rudin:
Sections 2.34-2.37]
9/28/20: Basic topology in metric spaces: closure, dense sets, unions
and intersections of open and closed sets, compact sets (definition).
[Class notes. Rudin: Sections
2.22, 2.24-2.27, 2.31-2.33]
9/25/20: Homework is due on Canvas by the beginning of the
class. Metric spaces. Basic topology in metric
spaces. [Class notes. Rudin:
Sections 2.15-2.21, 2.23, 2.26]
9/23/20: A useful lemma on the modulus. Decimal representations. The
Euclidean space
ℝn. [Class
notes. Rudin: Sections 1.22, 2.14, 1.34-1.38]
9/21/20: The existence of the square root of 2. Complex
numbers. [Class notes. Rudin: Sections 1.21, 1.24-1.33]
9/18/20: Homework is due on Canvas by the beginning of the
class. Supremum and infimum. Complete ordered fields. The complete
ordered field of real numbers. The Archimedean property and
consequences. [Class
notes. Rudin: Sections 1.7-1.11, 1.19-1.21]
9/16/20: More on countable sets. The rationals as a countable
set. Fields and ordered fields. [Class
notes. Rudin: Sections 2.13 and 1.12-1.18]
9/14/20: Using the Schröder-Bernstein theorem. Countable
sets. [Class notes. Rudin: Sections 2.8, 2.12]
9/11/20: Comparing cardinalities. Partial and total orders. The
Schröder-Bernstein theorem. [Class notes. (Baby) Rudin:
Chapter 1 through Section 1.6 (note that Rudin's order is
equivalent to what is usually called a total
order). Wikipedia
(If you find an error in Wiki, correct it!)]
9/9/20: Introduction. The syllabus and rules of the online class
meetings are discussed. Equivalence of sets and
cardinality. [Syllabus. Class
notes. (Baby) Rudin: Sections 2.1-2.6, 2.9-11]
Last modified: (2020-12-16 14:42:41 CST)