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Math 409
Spring 2017 Journal

Thursday, May 4
The final examination was given, and solutions are available.
Thursday, April 27
We discussed the idea behind the existence of the Riemann integral of a continuous function, the two parts of the fundamental theorem of calculus, and an example. The notes from class are available.
Tuesday, April 25
We discussed Cauchy's theorem about the existence of the integral of a continuous function along with some properties of integration that follow from the approximation of integrals by sums. The notes from class are available.
Thursday, April 20
We discussed the method of approximating the area under a curve by areas of rectangles, the idea behind the integral as conceived by Cauchy and Riemann. The notes from class are available.
The assignment is to begin reviewing for the final examination, which takes place on the afternoon of Thursday, May 4, from 12:30 to 2:30.
Tuesday, April 18
I returned the graded exams. We discussed properties of differentiable functions, including mean-value theorems. The notes from class are available.
The assignment to hand in next time is Exercise 7 on pages 133–134 in Section 8.5, which is a version of l'Hôpital's rule.
Thursday, April 13
We discussed three ways of saying that a function is differentiable and looked at some examples. The notes from class are available.
There is no assignment to hand in next class.
Wednesday, April 12
I posted the second exam along with solutions.
Tuesday, April 11
The second exam was given. One student has yet to take the exam, so I will post the exam and solutions later.
Thursday, April 6
We worked on some review exercises, contained in the notes from class. Also, we discussed some notions about convergence of sequences of continuous functions (a supplementary topic not covered on the exam to be given on April 11).
Tuesday, April 4
We worked on some exercises, contained in the notes from class, by way of review for the exam to be given on April 11. The assignment is to continue preparing for the upcoming exam.
Thursday, March 30
We discussed uniform continuity and the theorem that a continuous function on a compact set is automatically uniformly continuous. The notes from class are available.
The assignment (not to hand in) is to make a list of the main concepts covered since the first exam (in preparation for the exam to be given on April 11).
Tuesday, March 28
We proved the first of three theorems about properties of continuous functions on intervals. The notes from class are available.
The assignment to hand in next time is Exercises 1 and 3 on page 91 in Section 5.4.
Thursday, March 23
I posted an anonymous feedback form for you to write comments about what you would like me to change or keep the same during the rest of the semester. Please answer the three questions to let me know your thoughts.
In class, we discussed properties of the set of continuous functions, and we worked on some exercises. The assignment to hand in next time is the nutty ionic exercise at the end of the notes from class.
Tuesday, March 21
Welcome back from Spring Break. We discussed the concepts of limits of functions and continuity of functions. The notes from class are available.
The assignment to hand in next time is Exercise 5 on page 87 in Section 5.2.
Thursday, March 9
We discussed three definitions of continuity. The notes from class are available. The assignment for Spring Break is to travel safely.
Tuesday, March 7
We discussed the notion of sequential compactness and the Heine–Borel covering property (which are equivalent for subsets of the real numbers).
The assignment to hand in next time is the Exercise at the end of the notes from class (which we started working on in groups during class).
Thursday, March 2
We discussed notions of open sets and closed sets, interior and closure. We worked on determining the interior and the closure of the sets from the previous assignment; I have added the answers to the notes from class.
The assignment to hand in next time is the exercise at the end of the notes from class (about how interior and closure interact with intersection and union).
Tuesday, February 28
I returned the graded exams. We discussed notions of neighborhoods, interior points, boundary points, isolated points, limit points, open sets. The assignment to hand in next time is the exercise at the end of the notes from class.
Thursday, February 23
The first exam was given, and solutions are available.
Tuesday, February 21
We reviewed for the examination to be given in class on Thursday, February 23. The notes from class are available.
Thursday, February 16
We worked in groups on the following problems:
  1. page 25 #4
  2. page 29 #5
  3. page 39 #10
  4. page 47 #6
  5. page 49 #2
The notes from class are available. The assignment is to review for the exam to be given on Thursday, February 23.
Tuesday, February 14
We proved the Bolzano–Weierstrass theorem by the method of bisection and discussed Cantor’s nested-interval theorem and the notion of a Cauchy sequence. The notes from class are available.
The assignment for next time is Exercise 3 on page 36 in Section 3.2 (about the \(\limsup\) and \(\liminf\) of a sequence of sets).
Thursday, February 9
We proved the squeeze theorem and discussed the notions of \(\limsup\) and \(\liminf\). The notes from class are available.
The assignment to hand in next time is Exercise 7 on page 55.
Tuesday, February 7
We discussed convergence of sequences and the theorem that a bounded monotonic sequence of real numbers converges. The notes from class are available.
The assignment to hand in next time is Exercise 10 on page 43.
Thursday, February 2
We discussed the following notions: boundedness of sequences, monotonicity of sequences, a sequence being in a set ultimately, a sequence being in a set frequently, and null sequences. The notes from class are available.
The assignment to hand in next time is Exercise 4 on page 34 in Section 3.1 and Exercise 4 on page 39 in Section 3.3.
Tuesday, January 31
We finished the proof of the existence of square roots in the positive real numbers, and we began discussing sequences of real numbers. The notes from class are available.
The assignment to hand in next time is Exercise 1 on page 29 in Section 2.8 and Exercise 1 on pages 31–32 in Section 2.9.
Thursday, January 26
We discussed the greatest-integer function \(\lfloor x\rfloor\), the absolute value function \(|x|\), and the square-root function. The proof of the existence of square roots of positive real numbers is in progress. The notes from class are available.
The assignment to hand in next time is Exercise 2 on page 14 in Section 1.4 and Exercise 3 on page 20 in Section 2.4.
Tuesday, January 24
In class, we derived the Archimedean property of the real numbers from the completeness axiom and then proved the density of the rational numbers in the real numbers. The notes from class are available.
The assignment to hand in next time is Exercise 2 on page 18 in Section 2.2.
Thursday, January 19
In class, we looked at two examples (the complex numbers form a field that cannot be ordered; the rational functions with real coefficients form a non-Archimedean ordered field) and introduced the notions of supremum and complete ordered field. So we now know what it means to say that \(\R\) is a complete ordered field (in fact, the only complete ordered field).
I have posted what the notes might have been if the classroom computer had been working.
The assignment to hand in next time is an induction proof of Exercise 13 on pages 7–8 (Bernoulli's inequality in an ordered field). Notice that every ordered field contains a copy of the natural numbers, so the symbol \(n\) is meaningful.
Tuesday, January 17, 2017
In class, we discussed the notions of fields and ordered fields (Sections 1.1 and 1.2 in the textbook). The notes from class are available.
The assignment due next time is Exercises 1 and 2 on page 6 of the textbook.
Thursday, December 15, 2016
This site went live today. Once the Spring 2017 semester begins, there will be regular updates about assignments and the highlights of each class meeting.