Math 421 — Advanced Calculus I,

Homework Assignments & Course Schedule

Here is a link to the current week, below.

Here is a link back to the course syllabus/policy page.

In the following all sections and page numbers refer to the required
course textbook, * Understanding Analysis*, by Stephen Abbott.

This schedule is subject to change, but should be accurate at any moment for at least a week into the future.

For each day, please read the section(s) named in *the plan*, **before
that day** -- we will have discussion in class on those sections for which
you will have to have read the book.

*The plan for this week:**M:*Mostly bureaucracy and introductions.

*Content:*- Some discussion of the subfield of mathematics called "real analysis".
- Mention the notation for integers (
**Z**) and rationals (**Q**) -- critique of the book's definition of**Q**. - Proof that the square root of 2 is irrational.

*T:*Have read §§1.1&1.2.

*Content:*- More discussion of the definition of
**Q**, in particular: - Definition of
*equivalence relation*,*equivalence classes*(**Z**as an example), and_{2}*representatives of equivalence classes*, so also of**Q**. - Some bits of set theory ("element of", set inclusion, complement, "infinite set")
- The natural numbers
**N**as something "out there" or as something that must be constructed (starting from 1 and using the successor function an infinite number of times). - Induction -- both as a principle (which is a consequence of the
Well-Ordering Principle) applied to the previously existing
natural numbers and as a straightforward consequence of the
successor-function approach to the construction of
**N**. - De Morgan's laws, notation and meaning of infinite unions and intersections.

**HW0:**(at jonathan.poritz@gmail.com) telling me:*Send me e-mail*- Your name.
- Your e-mail address. (Please give me one that you actually check fairly frequently, since I may use it to contact you during the term.)
- Your year/program/major at CSUP.
- What you intend to do after CSUP, in so far as you have an idea.
- Past math classes you've had.
- Other classes you're taking at the moment.
- The reason you are taking this course.
- Your favorite mathematical subject.
- Your favorite mathematical result/theorem/technique/example/problem.
- Anything else you think I should know (disabilities, employment
or other things that take a lot of time,
*etc.*). - [Optional:] The best book you have read recently.

- More discussion of the definition of
*W:*Have read §1.3.

*Content:*- the Axiom of Completeness (or Least Upper Bound Property)
- Bounds, upper bounds, lower bounds, least upper bounds and greatest
lower bounds,
*sup*and*inf* - Difference between
*sup*and*max* - Characterizing the
*sup* - Discussion: should we first construct
**R**and then prove the AoC as a theorem?

*F:*Have read §1.4.

*Content:*- The Nested Interval Property
- The Archimedean Property
- The density of
**Q**in**R**

*NOTE:*Friday is the last day to add classes

*The plan for this week:**M:*Have read §§1.4&1.5. Hand in**HW1**: 1.2.{1,2,3,8,11}, 1.3.{6,7,9}

*Content:*- bijections, cardinality
- countable and uncountable sets, the equivalence class
_{}

*T:*Read §1.5.

*Content:*- more cardinality, esp.:
**Q**is countable **Q**is countable**R**is uncountable — the Nested Interval Property and Cantor's diagonal argument

- more cardinality, esp.:
*W:*Read §1.6.

*Content:*- discussion of HW1
- there are no infinite cardinals less than
_{}— this is problem 1.4.7

*F:*

*Content:*- discussion of issues related to HW2
- there are many cardinals greater than
_{}: Cantor's Theorem

*The plan for this week:**M:*Have read §2.1. Hand**HW2**: 1.4.{5,7,9,10,11}, 1.5.{3,4,9*[hint: part***(c)**is**hard**]

*Content:*- examples from the book of infinite sums with counter-intuitive values when the terms are grouped in certain ways — discussion of how this seems to happen only when there are an infinite number of sign changes
- definition of a
**convergent sequence**and its**limit**, first in terms of ε and*N*and then in terms of the**ε-neighborhood**of a real number*a*

*T:*Have read §2.2.

*Content:*- discussion of HW2
- proving sequences converge or diverge
- student presentation of the existence of an uncoutable antichain in
**P(N)**

*W:*Have read §2.2.

*Content:*- more discussion of HW2
- more convergence and divergence examples and a general strategy
- student presentation of the fact that the collection of
*finite*subsets of**N**is countable

*F:*Have read §§2.2, 2.3.

*Content:*- more discussion of (pieces of) the Algebraic Limit Theorem
- discussion of (pieces of) the Order Limit Theorem
- example limit computation, (half of) Exercise 2.3.5.
- start the Monotone Convergence Theorem

*NOTE:*Monday is the last day to drop classes without a grade being recorded

*The plan for this week:**M:*Have read §§2.3, 2.4. Hand**HW3**: 2.2.{2,7,8}, 2.3.{1,3,4,7,8}

*Content:*- finish the Monotone Convergence Theorem
- definition of
*series*, and*convergent*or*divergent*for series - examples of series --
*harmonic*and*geometric* - skipping the "Cauchy Condensation Test"

*T:*Have read § 2.5.

*Content:*- discussion of convergence of decimal and binary expansions of real numbers -- they do converge, any sequence of digits does express an actual real number
- definition of a
*subsequence*of a sequence - proving the Bolzano-Weierstrass Theorem
- describing the limit in the Bolzano-Weierstrass Theorem in terms of a binary expansion.

*W:*Have read § 2.6.

*Content:*- definition of a
*Cauchy sequence* - proof that all Cauchy sequences are bounded
- proof that a convergent sequence is Cauchy
- proof that a Cauchy sequence is convergent -- so now we know that a sequence is Cauchy if and only if it converges.

- definition of a
*F:*Have read § 2.7.

*Content:*

*The plan for this week:**M:*Have read §§ 2.7, 2.8. Hand**HW4**: 2.4.6, 2.5.3, 2.6.{1, 3}.

*Content:*- some properties of series from § 2.7.
- brief discussion of term rearrangement in infinite series
- last discussions (review) before in-class midterm part

*T:*Have read § 3.1.

*Content:*- comments on in-class part of test
- the Cantor set, other famous fractals, fractal dimension

*W:*Have read § 3.2.

*Content:*- open and closed sets (book and class definitions)
- statements about unions and intersections of open and closed sets
- limit points
- closure

*F:*Take-home part of the midterm**is due today**.

*The plan for this week:**M:*Hand**HW5**: 2.7.9, 3.2.{3, 11}.

*Content:*- starting to go over the take-home midterm -- problem-solving
strategies
*revisited*(hopefully) - preparation for the in-class midterm

- starting to go over the take-home midterm -- problem-solving
strategies
*T:*In-class portion of midterm I is**today**.*W:*Have read § 3.3.

*Content:*- more discussion of the take-home midterm
- starting
*post mortem*on the in-class midterm

*F:*

*Content:*- discussion of both parts of the midterm
- more on open and closed sets, limit points, and closure

*The plan for this week:**M:*Have read §§ 3.1-3. Hand**HW6**: 3.2.{12, 13}*[and anything else you owe me!]*

*Content:*- handed out and discussed in part a solution set to the take-home part of of the midterm
- compact sets, definition with sequences
- open covers, an equivalent definition of compactness using covers
- the Heine-Borel Theorem

*T:*Have read §§ 3.3

*Content:*- more on compact sets, open covers,
*etc.*

- more on compact sets, open covers,
*W:*Have read §§ 3.4

*Content:*- example of an open set containing all of
**Q**but not equalling all of**R** - discussion of
*separated sets*,*disconnected*and*connected*sets. - some ideas about
*path-connected*sets

- example of an open set containing all of
*F:*Have read §§ 3.4

*Content:*- finishing discussion of connected sets

*The plan for this week:**M:*Have read §§ 4.{1,2}. Hand**HW7**: 3.3.{3, 4, 5, 7, 9}, 3.4.{8, 9}

*Content:*- motivation: sets of discontinuity of real-valued functions
- examples of Dirichlet and Thomae

*T:*

*Content:*- functional limits

*W:*

*Content:*- divergence criterion for functional limits
- sequential criterion for functional limits

*F:*Have read § 4.3.

*Content:*- combinations of continuous functions

*The plan for this week:**M:*Have read §§ 4.{4,5}.

*Content:*- purely topological characterization of continuous functions: A function is continuous iff the inverse image of every open set is open.

*T:*Hand**HW8**: 4.1.{1, 2, 3, 6}, 4.3.{3, 5, 7}

*Content:*- the continous image of a compact set is compact
- the continous image of a connected set is connected

*W:*

*Content:*- the Intermediate Value Theorem
- the Extreme Value Theorem
- start uniform continuity

*F:*Have read § 4.6.

*Content:*- more uniform continuity
- a continuous function is uniformly continuous on a compact domain
- some remarks on sets of discontinuity

*NOTE:*Friday is the last day to withdraw (with a**W**) from classes

*The plan for this week:**M:*Have read §5.1.

*Content:*- definition of
*derivative*and*differentiable* - ramblings about the relationship between continuity and
differentiability -- examples of the Heavyside function
and then functions
*f(x)=x*^{n}sin(1/x)

- definition of
*T:*Have read §5.2. Hand**HW9**: 4.3.11, 4.4.{2, 6, 9 or 11}, 4.5.{3, 7}

*Content:*- some last discussion of the HW — particularly
*Lipschitz*functions and the relationship to uniformly continuous functions - proof that differentiable implies continuous
- discussion of the usual differentiation rules — Sum Rule, Constant Multiple Rule, Product Rule, Quotient Rule, and Chain Rule — from calc I, dressed up in fancy clothes
- statement and proof of the Interior Extremum Theorem (essentially Fermat's Theorem from calc I)

- some last discussion of the HW — particularly
*W:*Have read §5.3.

*Content:*- Darboux's theorem
- Two examples of functions which are counterexamples to the converse of the Intermediate Value Theorem

*F:*Have read §5.4.

*Content:*- Rolle's Theorem and the Mean Value Theorem
- start L'Hospital's Rule

*The plan for this week:**M:*

*Content:*- student presentation of the proof of Darboux's Theorem
- problem-solving practice: discussed exercise 5.2.8(c)

*T:*Hand**HW10**: 5.2.{3, 8}, 5.3.{1, 3, 5}.

*Content:*- student presentation of the proof of the Generalized Mean Value Theorem
- problem-solving practice: finished discussion of exercise 5.2.8(c)
- more problem-solving practice: discussed exercise 5.3.2.
- distribution of the take-home part of the second midterm — here it is

*W:*

*Content:*- student presentation of the proof of L'Hospital's Rule
- problem-solving practice: discussed §5.4 very interactively
- here is a review sheet for the in-class part of the second midterm

*F:*Have read §§5.{4,5}

*Content:*- finishing discussion of the book's example of a continuous, nowhere-differentiable function.

*The plan for this week:**M:*one-on-one discussions of students' progress on the take-home midterm*T:*The take-home part of the second test is due at the end of the day.

*Content:*- Starting review for the in-class part of the midterm.

*W:*

*Content:*- handed out and discussed in part a solution set to the take-home part of of the midterm
- tried to come to some agreement about what kinds of problem were most likely to appear on the in-class part of the second test

*F:*The in-class part of the second test will happen today.

*The plan for this week:**M:*Read §6.2

*Content:**pointwise*and*uniform*convergence of functions- examples, such as the functions
*f*on [0,1]_{n}=x^{n}

*T:*

*Content:*Have read §6.3- more on pointwise and uniform convergence
- proved that if the pointwise and uniform limits both exist then they must be equal
- summarized convergence for derivatives as well: pointwise convergence plus uniform convergence plus uniform convergence of the derivative makes the limit differentiable with derivative equalling the limit of the derivatives

*W:*Have read §6.4

*Content:**series*of functions- the Cauchy criterion and the Weierstrass M-Test
*uniform convergence on compact sets*- examples, including Fourier series (series of
*sin*'s and*cos*'s with higher and higher frequencies and various constant coefficients)

*F:*Have read §6.5

*Content:*- power series
- some examples

**Thanksgiving Break!**No classes, of course.

*The plan for this week:**M:*Read §8.2

*Content:*- definition of a
*metric* - basic examples of metrics, including
- the Euclidean metric on
**R**^{2} - the taxicab metric on
**R**^{2} - the sup-norm (
**L**^{∞}) metric on**R**^{2}and**C[0,1]**

- the Euclidean metric on
- open balls in different metrics; likewise open sets

- definition of a
*T:*Have read §§8.2.

*Content:*- closure, Cauchy sequences, completeness in metric spaces
- examples:
**Q**in**R**;**C[0,1]**in ...

*W:*Have read §8.4

*Content:*- Perhaps a construction of the real numbers

*F:*Hand**HW11**: 6.2.{7, 8, 14, 15} and any one of 8.2.{2,4,5}

*Content:*- final review
- discussion of format and content of upcoming final

**Exam week**, no classes.- Hand in any outstanding homework assignments.
- here is a review sheet for the final
- If you have a notebook with class or reading notes, I would be happy to see it and therefore to give you class credit; similarly, I am happy to give you credit for improved (written) solutions to HW problems that you may have missed during the term, or any extra problems you may have done.
**Friday, December 11th:**from 10:30-12:50 we will have an in-class final specific to our own class, in our usual classroom