Math 10860 - Honors Calculus 2

Spring 2020

Organization of the course from March 23 on

Zoom meeting information (all times South Bend time)

https://notredame.zoom.us/j/844811786

Mondays 11.30-12.20

Wednesdays 11.30-12.20

Thursdays 4.30-6.00

Fridays 11.30-12.20

Drop in anytime to ask questions, discuss examples, or just say hello!

LECTURES

I will be recording short videos (``mini lectures'') and posting them on YouTube, with links provided here, along with information about how the videos correspond to lectures/course notes. You should watch the videos at whatever time is convenient for you.

No doubt there will be some typos in the videos, since you aren't in front of me to correct me in real time! As I spot errors I'll make comments on the YouTube page --- feel free to comment on errors, too.

I will host a number of Zoom meets throughout the week, to allow for personal contact, discussion, and questions and answers. Specifically, I'll be available during each of the regularly scheduled times of the class (MWF 11.30am-12.20pm EDT) and also Thursday 4.30pm to 6pm EDT. Details of how to check in to these meetings will be send by email. These meetings will not be recorded.

  1. Welcome! (2 minutes)

    2. (Monday March 23: 46 minutes)
  2. A basic example of substitution (9 minutes, Monday March 23, Section 13.3)
  3. A cautionary substitution example (7 minutes, Monday March 23, Section 13.3)
  4. A first trigonometric substitution (11 minutes, Monday March 23, Section 13.4)
  5. General trigonometric substitutions I and II (9 minutes, Monday March 23, Section 13.4)
  6. General trigonometric substitution III (10 minutes, Monday March 23, Section 13.4)

    7. (Wednesday March 25: 53 minutes)
  7. A fully worked example of a trigonometric substitution (10 minutes, Wednesday March 25, Section 13.4)
  8. A more involved trig example --- enter at your own risk! (very optional)
  9. Integrals of powers of sin and cos (13 minutes, Wednesday March 25, Section 13.4)
  10. A ``magic bullet'' trigonometric substitution (5 minutes, Wednesday March 25, Section 13.4)
  11. Some ``magic bullet'' examples (8 minutes, Wednesday March 25, Section 13.4)
  12. Partial fractions, first example (9 minutes, Wednesday March 25, Section 13.5)
  13. Partial fractions in general, steps 1 and 2 (8 minutes, Wednesday March 25, Section 13.5)

    14. (Friday March 27: 49 minutes)
  14. Partial fractions in general, step 3 (11 minutes, Friday March 27, Section 13.5)
  15. Partial fractions in general, step 4 (11 minutes, Friday March 27, Section 13.5)
  16. Partial fractions, final comments (5 minutes, Friday March 27, Section 13.4)
  17. Taylor polynomials, introduction (8 minutes, Friday March 27, Section 14.1)
  18. Taylor polynomials, examples (14 minutes, Friday March 27, Section 14.1)

    19. (Monday March 30: 46 minutes)
  19. Properties of Taylor polynomial (15 minutes, Monday March 30, Section 14.2)
  20. Uniqueness of Taylor polynomial (13 minutes, Monday March 30, Section 14.2)
  21. Taylor polynomial of arctan (14 minutes, Monday March 30, Section 14.2)
  22. Preview of the remainder term (4 minutes, Monday March 30, Section 14.3)

    23. (Wednesday April 1: 52 minutes)
  23. Taylor's theorem, integral remainder form (17 minutes, Wednesday April 1, Section 14.3)
  24. Taylor's theorem, Lagrange remainder form (13 minutes, Wednesday April 1, Section 14.3)
  25. Remainder term for sin, part 1 (9 minutes, Wednesday April 1, Section 14.4)
  26. Remainder term for sin, part 2 (2 minutes, Wednesday April 1, Section 14.4)
  27. Remainder term for the exponential (11 minutes, Wednesday April 1, Section 14.4)

    28. (Friday April 3, 46 minutes)
  28. Remainder term for arctan (13 minutes, Friday April 3, Section 14.4)
  29. Summary of Taylor polynomials (11 minutes, Friday April 3, Section 14.4)
  30. Introduction to infinite sequences (11 minutes, Friday April 3, Section 15.1)
  31. Convergence and divergence (11 minutes, Friday April 3, Section 15.2)

    32. (Monday April 6, 51 minutes)
  32. Calculating limits from the definition (14 minutes, Monday April 6, Section 15.2)
  33. Limit Theorems (15 minutes, Monday April 6, Section 15.2)
  34. Limits of rational functions (7 minutes, Monday April 6, Section 15.2)
  35. Limits and functions, part 1 (9 minutes, Monday April 6, Section 15.3)
  36. Limits and functions, part 2 (6 minutes, Monday April 6, Section 15.3)

    37. (Wednesday April 8, 48 minutes)
  37. Limits and continuity, part 1 (15 minutes, Wednesday April 8, Section 15.3)
  38. Limits and continuity, part 2 (8 minutes, Wednesday April 8, Section 15.3)
  39. Monotonicity and convergence (15 minutes, Wednesday April 8, Section 15.4)
  40. Application of monotonicity convergence criterion (10 minutes, Wednesday April 8, Section 15.4)

    41. (Wednesday April 15, 52 minutes)
  41. Subsequences and Bolzano-Weierstrass (13 minutes, Wednesday April 15, Section 15.4)
  42. Cauchy sequences --- theory (17 minutes, Wednesday April 15, Section 15.4. Note: ends abruptly!)
  43. Cauchy sequences --- example (7 minutes, Wednesday April 15, Section 15.4. Note: starts abruptly! Also, unfortunately the top of the board is cut off slightly. The example I'm working with is the sequence a_n = sum(k from 1 to n) 1/k^2.)
  44. Introduction to summability (15 minutes, Wednesday April 15, Section 16.1)

    45. (Friday April 17 --- no lecture, exam 2)

    (Monday April 20, 55 minutes)
  45. Basics of summability (17 minutes, Monday April 20, Section 16.2)
  46. Comparison test (12 minutes, Monday April 20, Section 16.2)
  47. Limit comparison test (9 minutes, Monday April 20, Section 16.2)
  48. Ratio test (17 minutes, Monday April 20, Section 16.2)

    49. (Wednesday April 22, 53 minutes)
  49. Integral test (12 minutes, Wednesday April 22, Section 16.2)
  50. Leibniz' alternating series test (9 minutes, Wednesday April 22, Section 16.2)
  51. Proof of Leibniz (12 minutes, Wednesday April 22, Section 16.2)
  52. Absolute convergence (11 minutes, Wednesday April 22, Section 16.3)
  53. Infinite commutativity (9 minutes, Wednesday April 22, Section 16.3)

    54. (Friday April 24, 47 minutes)
  54. Introduction to Taylor series (12 minutes, Friday April 24, Section 17.1)
  55. First example of using Taylor series (7 minutes, Friday April 22, Section 17.1)
  56. A first cautionary example of convergence of functions (9 minutes, Friday April 24, Section ???)
  57. A second cautionary example of convergence of functions (8 minutes, Friday April 24, Section 17.2)
  58. Uniform convergence will save the day (11 minutes, Friday April 24, Section 17.3)

    59. (Monday April 27, 52 minutes)
  59. Uniform convergence and continuity (9 minutes, Monday April 27, Section 17.3)
  60. Uniform convergence and integrability (11 minutes, Monday April 27, Section 17.3)
  61. Uniform convergence and differentiability (11 minutes, Monday April 27, Section 17.3)
  62. Introduction to power series (12 minutes, Monday April 27, Section 17.4)
  63. Radius of convergence summary and examples (9 minutes, Monday April 27, Section 17.4)

    64. (Wednesday April 29, 51 minutes)
  64. Continuity of power series (5 minutes, Wednesday April 29, Section 17.4)
  65. Differentiability of power series (8 minutes, Wednesday April 29, Section 17.4)
  66. Integrability of power series (6 minutes, Wednesday April 29, Section 17.4)
  67. Power series and Taylor series (5 minutes, Wednesday April 29, Section 17.4)
  68. Some quick applications of power series (11 minutes, Wednesday April 29, Section 17.4)
  69. The general radius of convergence 1 and the general radius of convergence 2 (7 minutes and 1 minute, Wednesday April 29, not covered in notes)
  70. A hint at complex power series (8 minutes, Wednesday April 29, not covered in notes)

TEXTBOOK

Along with the recorded lectures, the course notes gives you another avenue to engage with the material. Usually, the course notes will go into each topic in a little more detail and with a little more generality than the lectures. I will continue to update the course notes regularly, and I will indicated which sections correspond to which videos (as well as sometimes referring to specific pages of the course notes during the videos).

Here is the link to the full course notes. In case it is easier to download smaller files, here are the notes section-by-section:

HOMEWORK

Homework will be posted here each Friday, and will be due by 11pm the following Friday.

Homework should be emailed to math10860homework at gmail.com!

As I did towards the end of last semester, I will identify a subset of the homework problems that I consider to be ``core'', and these are the problems that should be turned in. Other problems labelled optional are, of course, highly recommended! The first homework, posted below, is due Friday March 27, and is a shortened version of old Homework 7.

If you have familiarity with LaTeX, or any other math-oriented word processing software, I strongly encourage you to use it for your homework! As a general rule, typing math helps you focus on brevity and clarity of presentation; but at present, there is also the practicality that it will be much easier to transmit homework (between you and me, between me and the grader) if it starts out in electronic form.

If typing homework is unfeasible, then you should write it NEATLY, using plenty of blank space, and scan it to pdf --- this can be done easily, for example, on Google drive. Sending the homework by pdf will allow me and the grader to easily make notes on the file. To make the digital copy legible, consider using pen rather than pencil.

However you produce the homework, you should email it to math10860homework at gmail.com by the deadline.

TUTORIALS

There will be a weekly tutorial meeting at the usual time, Thursday 11am-11.50am Easter Daylight Time, held via zoom, and focused, as always, on discussion of course material/upcoming homework. Details of how to check in to the workshop will appear here later.

QUIZZES

There will be a short quiz during most workshops (shorter than previous quizzes). You should do the quiz on a blank sheet of paper and email a pdf scan of it to Sarah, during the first fifteen minutes of the workshop. More details on this process will be available closer to the first tutorial on March 26.

PI DAY EXTRA CREDIT

If you haven't already passed it on to me, please email your Pi Day extra credit to me --- at math10860homework at gmail.com --- by 11pm on Monday, March 23. If it could be typed, that would be great; if not, a legible pdf scan works.

EXAMS

The final will take place on Monday May 4. Details can be found here. The actual final exam (with solutions) is here.

The second midterm will happen on Friday April 17. Details, along with some practice questions, can be found here; solutions to the practice problems are here. Here is the actual exam, with solutions.

RESOURCES