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# Math 409 Spring 2018 Journal

Monday, May 7
The final exam was given, and solutions are available.
Tuesday, May 1
On this redefined day, we revisited the fundamental theorem of calculus and reviewed for the final exam. The slides from class are available.
Monday, April 30
We discussed the fundamental theorem of calculus. The slides from class are available.
Friday, April 27
We proved that continuous functions are integrable. The last page of the slides from class has a practice exercise (not to hand in).
Wednesday, April 25
We discussed the notions of upper sums and lower sums and integrability of a bounded function. The last page of the slides from class has some practice/review exercises (not to hand in).
Monday, April 23
We completed the discussion of Taylor’s formula and began on the theory of the integral. The last page of the slides from class has some review exercises (not to hand in).
Friday, April 20
We proved a version of l’Hôpital’s rule by applying Cauchy’s mean-value theorem, and we began a discussion of Taylor’s formula. The last page of the slides from class has some review exercises (not to hand in).
Wednesday, April 18
We discussed two versions of the mean-value theorem and some applications. As shown on the last page of the slides from class, the assignment due next class is to write solutions to Exercises 4.2.3 and 4.2.5.
Monday, April 16
We proved Rolle’s theorem and began a discussion of the mean-value theorem. As shown on the last page of the slides from class, the assignment due next class is to write a solution to Exercise 4.2.11.
Friday, April 13
We continued the discussion of derivatives and proved the chain rule. As shown on the last page of the slides from class, the assignment due next class is to write a solution to Exercise 4.1.10.
Wednesday, April 11
We continued the discussion of differentiable functions and used the second definition to prove the product rule. As shown on the last page of the slides from class, the assignment due next class is to similarly prove the quotient rule using the second definition of the derivative.
Monday, April 9
We discussed three equivalent definitions of the derivative. As shown on the last page of the slides from class, the assignment due next class is to write a solution to Exercise 4.1.5.
Friday, April 6
We discussed the notion of uniform continuity. The assignment due next class is shown on the last page of the slides from class.
Wednesday, April 4
We discussed the extreme-value theorem and the intermediate-value theorem for continuous functions. The assignment due next class is shown on the last page of the slides from class.
Monday, April 2
I returned the graded exams, and we discussed the concept of continuity. The assignment due next class is shown on the last page of the slides from class.
Wednesday, March 28
The second exam was given, and solutions are available.
Monday, March 26
We reviewed for the exam to be given on Wednesday. The slides from class are available.
Friday, March 23
We discussed Exercise 3.1.1. The slides from class are available.
Wednesday, March 21
We continued the discussion of cluster points, looked at Exercise 2.3.9, and discussed two equivalent definitions of what it means for a function to have a limit at a cluster point of the domain. The assignment due next class is shown on the last page of the slides from class.
Monday, March 19
We discussed the notions of isolated point and cluster point in preparation for considering limits of functions. The assignment due next class is shown on the last page of the slides from class.
Friday, March 9
We discussed convergence of series with terms of mixed signs. The slides from class are available.
Wednesday, March 7
We discussed the relationship between the ratio test and the root test, and we worked on some concrete examples of series. The slides from class are available. There is no assignment to hand in on Friday, but class will meet as usual.
Monday, March 5
We discussed convergence tests for series, especially Cauchy’s root test. The assignment due next class is shown on the last page of the slides from class.
Friday, March 2
We discussed series having positive terms and proved Cauchy’s condensation test. As shown on the last page of the slides from class, the assignment due next class is to finish reading section 2.5 in the textbook.
Wednesday, February 28
We began a discussion of convergence of infinite series. Topics included geometric series and Cauchy’s condensation test. As shown on the last page of the slides from class, the assignment due next class is to read subsection 2.5.1 in the textbook and to write solutions to Exercises 2.4.7 and 2.5.3(a).
Monday, February 26
We discussed Cauchy sequences and proved that a Cauchy sequence of real numbers necessarily converges. As shown on the last page of the slides from class, the assignment due next class is to write solutions to Exercises 2.4.1 and 2.5.1.
Friday, February 23
We worked in groups on interpreting scrambled versions of the definition of limit. As shown on the last page of the slides from class, the assignment due next class is to finish reading sections 2.3–2.4 in the textbook.
Wednesday, February 21
We looked at three equivalent ways of characterizing the limit superior. The assignment due next class is shown on the last page of the slides from class.
Monday, February 19
I returned the graded exams. Solutions are posted. In class, we discussed the Bolzano–Weierstrass theorem and the proposition that every sequence of real numbers has a monotonic subsequence. The assignment due next class is shown on the last page of the slides from class.
Friday, February 16
The first exam was given, and solutions are available.
Wednesday, February 14
We reviewed for the exam to be given on Friday, February 16. The slides from class are available.
Monday, February 12
We continued the discussion of convergence of sequences of real numbers. The slides from class are available.
Friday, February 9
We discussed the interaction of the limit operation with the algebraic, order, and metric structures of the real numbers. Also, we looked at some properties guaranteeing that a sequence converges to zero. As shown on the last page of the slides from class, the assignment due next class is to write solutions to Exercise 1.3.5 and Exercise 2.2.5 and to read subsection 2.2.4 in the textbook.
Wednesday, February 7
We worked an example of applying the monotone convergence theorem to prove that a recursively defined sequence converges. The assignment due next class is shown on the last page of the slides from class.
Monday, February 5
We continued the discussion of properties of sequences and subsequences. As shown on the last page of the slides from class, the assignment due next class is to write solutions to Exercises 2.1.15 and 2.1.20 and to read subsection 2.2.2 in the textbook.
Friday, February 2
We discussed various properties that a sequence might or might not have: increasing, decreasing, monotonic, bounded, convergent, Cauchy. As shown on the last page of the slides from class, the assignment due next class is to write solutions to Exercises 2.1.14 and 2.1.19 (both easy) and to read subsection 2.2.1 in the textbook (about limits and inequalities).
Wednesday, January 31
We discussed in detail Exercise 1.4.6 (about intersections and unions of intervals). As shown on the slide from class, the assignment due next class is to write solutions to Exercises 2.1.1 and 2.1.7 and to read the rest of section 2.1 in the textbook.
Monday, January 29
We discussed the notion of density, the metric structure on the real numbers determined by the absolute value function, and the concept of a sequence. As shown on the last page of the slides from class, the assignment due next class is to write solutions to Exercises 1.3.2 and 1.4.6 and to read the first part of section 2.1 in the textbook, through Example 2.1.8.
Friday, January 26
We looked at the interaction of the supremum with union, intersection, and set difference, and we saw an example of a non-Archimedean ordered field (in which the natural numbers are bounded above). As shown on the last page of the slides from class, the assignment due next class is to write a solution to Exercise 1.2.1 and to read section 1.3 in the textbook.
Wednesday, January 24
We discussed the notions of supremum and infimum; the characterization of the real numbers as the unique complete, ordered field; the notion of cardinality; and an explicit bijection between the set of natural numbers and the set of positive rational numbers. As shown on the last page of the slides from class, the assignment due next class is to write a solution to Exercise 1.1.6 and to read the rest of section 1.2 in the textbook.
Monday, January 22
We looked at some examples of fields and ordered fields. As shown on the last page of the slides from class, the assignment due next class is to write solutions to Exercises 1.1.3 and 1.1.5 and to read subsection 1.2.1 in the textbook.
Friday, January 19
We reviewed the method of mathematical induction and worked through Dedekind’s proof of irrationality of $$\sqrt{2}$$ using the well-ordering property of the natural numbers (equivalent to induction). As shown on the last page of the slides from class, the assignment due next class is to read section 1.1 in the textbook and to write solutions to Exercises 0.3.19 and 0.3.20.
Wednesday, January 17, 2018
At the first class meeting, we reviewed (from Math 220) some of the logic of quantifiers and implications. As shown on the last page of the slides from class, the assignment due next class is to read section 0.3 of the textbook and to write (or type) solutions to Exercises 0.3.8 and 0.3.10.
Thursday, January 4, 2018
This site went live today. Once the Spring 2018 semester begins, there will be regular updates about assignments and the highlights of each class meeting.