Colorado State University, Pueblo; Fall 2009 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.
Content:
• More discussion of the definition of Q, in particular:
• Definition of equivalence relation, equivalence classes (Z2 as an example), and 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: Send me e-mail (at jonathan.poritz@gmail.com) telling me:
2. 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.)
4. What you intend to do after CSUP, in so far as you have an idea.
5. Past math classes you've had.
6. Other classes you're taking at the moment.
7. The reason you are taking this course.
10. Anything else you think I should know (disabilities, employment or other things that take a lot of time, etc.).
11. [Optional:] The best book you have read recently.
I will only enter your name into my gradebook when I get this e-mail, so you really need to do this assignment ASAP. Please take a moment to be complete (indeed, be as expansive as you can) — the more information I have about you, the better I can adapt the course to your needs and interests.
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?
Content:
• The Nested Interval Property
• The Archimedean Property
• The density of Q in R
• NOTE: Friday is the last day to add classes

Week of August 31:

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

Week of September 7:

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

Week of September 14:

• 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.
• F: Have read § 2.7.
Content:
• group reading of the "Cauchy if and only if convergent" proof
• some properties of series from § 2.7.
• handout on things to know for the in-class part of the midterm (and discussion of whether it will be on Monday or Tuesday)
• distribution of the take-home part of the midterm — here it is

Week of September 21:

• 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.

Week of September 28:

• 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
• 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

Week of October 5:

• 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.
• 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
• F: Have read §§ 3.4
Content:
• finishing discussion of connected sets

Week of October 12:

• 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

Week of October 19:

• 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

Week of October 26:

• The plan for this week:
Content:
• definition of derivative and differentiable
• ramblings about the relationship between continuity and differentiability -- examples of the Heavyside function and then functions f(x)=xnsin(1/x)
• 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)
Content:
• Darboux's theorem
• Two examples of functions which are counterexamples to the converse of the Intermediate Value Theorem
Content:
• Rolle's Theorem and the Mean Value Theorem
• start L'Hospital's Rule

Week of November 2:

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

Week of November 9:

• 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.

Week of November 16:

• The plan for this week:
Content:
• pointwise and uniform convergence of functions
• examples, such as the functions fn=xn on [0,1]
• T:
• 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
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)
Content:
• power series
• some examples

Week of November 23:

• Thanksgiving Break! No classes, of course.

Week of November 30:

• The plan for this week:
Content:
• definition of a metric
• basic examples of metrics, including
• the Euclidean metric on R2
• the taxicab metric on R2
• the sup-norm (L) metric on R2 and C[0,1]
• open balls in different metrics; likewise open sets
Content:
• closure, Cauchy sequences, completeness in metric spaces
• examples: Q in R; C[0,1] in ...
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

Week of December 7:

• 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.
• Monday, December 7th: from 10:30-12:30 we will have the MFAT test in our usual classroom
• Friday, December 11th: from 10:30-12:50 we will have an in-class final specific to our own class, in our usual classroom

Jonathan Poritz (jonathan.poritz@gmail.com)