# 21-122: Integration, Differential Equations, and Approximation

### Description

This course is a continuation of the ideas in 21-115 Differential Calculus and 21-116 Integral Calculus. These courses introduce the ideas of differentiation and integration, respectively.

The first half of the course extends the range of functions for which we can compute integrals. We introduce three new techniques for use in different situations. When combined with the Method of Substitution and Integration by Parts, these techniques allow us to integrate a wide variety of functions.

The second half of the course will be an introduction to differential equations, i.e. equations that involve the derivative of a function. Determining the original function generally involves computing an integral. Differential equations are ubiquitous in the natural sciences and social sciences, because they are useful in modeling the behavior of systems over time. We will discuss how to write a mathematical model for a physical system, and also how to compute solutions for two fundamental types of equations.

[...]

In previous courses of the calculus sequence, functions were dealt with just as they were presented. As a result, all of the answers obtained were exact. Sometimes, however, an approximate answer is all that is needed.

In Calculus of Approximation, we investigate techniques that may be used when an exact answer is not necessary. There are two issues involved here. The first is finding a convenient way to approximate the desired information. The second is measuring the errors we create by our approximations. We will deal with both of these topics in this course.

### News

Wednesday 13 December: Just to make sure everone is clear: The final exam will be a comprehensive exam. All the material from the semester will be fair game. You should study the review material for Exams #1, #2 and #3 as well as the review material linked specifically for the final exam.

Tuesday 12 December: I was out sick yesterday and didn't get to post any updates, but there are a number of things you ought to know:

1. Solutions to the three midterm exams are now available. They are in a box on the floor outside my office door (WEH 6214).
2. I will be holding office hours today from 3:00-5:00pm.
3. I'll be holding a review session tomorrow, December 13, from 6:30-8:00 in DH 2210.
4. Pulak Goswami will hold a review session immediately after mine, from 8:30-10:00pm, also in DH 2210.

Thursday 7 December: I've posted a reveiw page for the Final Exam. I'll update it with a link to old exam problems, office hours for next week, and times and places for various and sundry review sessions.

Thursday 7 December: It looks like I'll have to push back my office hours today also. I'll hold office hours today from 3:30-5:20. Again, sorry about the change.

Wednesday 6 December: A reminder about FCE's: The On-Line Faculty Course Evaluation (FCE) system will remain open until 8AM Monday, December 11th.

Wednesday 6 December: I should be back in my office by 5:00 this afternoon, so you can expect me to be there from 5:00 to 6:00 at least. See you then.

Wednesday 6 December: I'm going to have to leave early today, and my office hours may be canceled. I may be back later on, though, so they may only be postponed. I'll post an update here when I have a better handle on things. Sorry about that.

Monday 4 December: Every semester there are students who have final exam conflicts (i.e. two exams scheduled at the same time or three exams beginning in a 25 hour period). I'll be scheduling a makeup exam to be given on Monday December 18. If this affects you, please contact me as soon as possible and let me know whether you are available during the morning or afternoon (or both) exam periods that day. Additionally, if you require (and are entitled to!) extra time on the exam, please contact me so that arrangements can be made.

Friday 1 December: The reading and homework assignments for Week #15 have been posted. Follow the link from the Schedule Page.

Thursday 30 November: Here's an update for this week's homework assignment - For problem 11.12.32 only do part (a). Skip part (b), since we did not end up covering the Section 11.11 The Binomial Series.

Tuesday 28 November: The page of course statistics has been updated to include Exam #3.

Tuesday 28 November: Students planning to take tomorrows makeup exam should meet me in my office, WEH 6214, at 7:25 tomorrow morning. The exam in only available for students who missed the original exam for some reason.

Monday 27 November: A makeup for Exam #3 will be given this Wednesday morning at 7:30am. There will be no penalty for students with legitimate and documented absences.

Monday 27 November: The reading and homework assignments for Week #14 have been posted. Follow the link from the Schedule Page.

Monday 27 November: Here's a Taylor series demo.

Saturday 18 November: Pulak Goswami will be holding a review session today from 6:30-8:00 in DH 2315. Additionally, Carlene Ulish, our SI leader, will be holding a review session on Sunday from 2:00-3:30 in WEH 5403.

Friday 17 November: Solutions to the homework for Week #12 are available outside my office door (WEH 6214). They may be useful while studying for Monday's Exam #3.

Wednesday 15 November: I've added some old exam problems to the review page for Exam #3, wich will take place on Monday 20 November (i.e. next Monday).

Friday 10 November: The reading and homework assignments for Week #12 have been posted. Follow the link from the Schedule Page.

Monday 6 November: The reading and homework assignments for Week #11 have been posted. Follow the link from the Schedule Page.

Monday 6 November: Lost and Found: A cell phone after this mornings class. I'll hang on to it for a while, and if no one comes forward to claim it, I'll sell it on Ebay leave it at the information desk in the University Center.

Thursday 2 November: I will have to cancel part of my office hours today. I will be otherwise occupied from 1:30-2:30, but I should be available for the second hour, from 2:30-3:30.

Friday 27 October: The reading and homework assignments for Week #10 have been posted. Follow the link from the Schedule Page.

Thursday 26 October: Office hours update: I'm in my office now, and open for business. Turns out that meeting is next week. Integration and differential equations is one thing, but reading a calendar? Thats hard!

Thursday 26 October: Here are statistics for the course, up to an including Exam #2.

Thursday 26 October: Sorry I had to cancel my office hours yesterday. I had to leave unexpectedly after class. More bad news: I will also have to cancel part of my office hours today. I will be otherwise occupied from 1:30-2:30, but I should be available for the second hour, from 2:30-3:30.

Tuesday 24 October: I've corrected the duedate for this weeks assignment: it is due in the future, not the past. Also, I removed the problems from Section 9.3, which you had already turned in with a previous assignment, from this weeks homework.

Monday 23 October: The reading and homework assignments for Week #9 have been posted. Follow the link from the Schedule Page.

Saturday 21 October: I've posted the mid-semester grades. You should be able to see them on the SIO website. If you have recently transferred into the course from Analysis I, and are alarmed by your mid-semester grade, let me know, and we can talk about your situation.

Saturday 21 October: I've had a few questions about the logistic equation. You have not had any homework from Section 9.5, so I did not put that on the review sheet. You have, however had some home made homework problems that did cover the logistic equation. You should review those, and the old exam problems concerning the logistic equation.

Friday 20 October: I've added information about the review session (6:00-7:30pm on Sunday in DH 2315) to the review page for Exam #2. Also, the formula sheet is the same as it was for Exam #1. There's probably a bunch of stuff you won't need, but it won't hurt you and it will save me cutting and pasting when the Final rolls around. The trig identites might be helpful.

Wednesday 18 October: You may be interested to know that Academic Development is holding another group workshop on Time Management. It's scheduled for for Tuesday 24 October from 8:00-9:30.

Monday 16 October: I've posted a review page for next Monday's Exam. Follow the link from the Schedule page. There is no homework assignment for this week, but I will post the reading assignment pretty soon.

Friday 13 October: Another Cell phone was left I WEH 7500 after the 8:30 class this morning. I'll take it to the information desk in the University Center. Whoever left it can pick it up there.

Wednesday 11 October: Dr. Handron's office hours will be shortened today, to 4:00-5:00pm. Sorry for any inconvenience.

Wednesday 11 October: As announced in class, I've added two problems to this weeks homework. You can find them on the homework page, along with a link to the Dfield applet.

Friday 6 October: The reading and homework assignment for Week #7 have been posted. Folow the link from the Schedule Page.

Wednesday 4 October: Someone seems to have left a cell phone in WEH 7500 after the 8:30 lecture this morning. If it's yours, please let me know, and we can make arrangements for you to pick it up.

Wednesday 4 October: In class today I used the Java applet Dfield to draw direction fields and graph approximate solutions. You may want to take a look at this program. You'll probably want to use it for some of next week's homework assignment.

Tuesday 3 October: Here are a whole slew of statistics for the course, up to an including Exam #1. The low cumulative average in Nakul Shankar's sections is related to the division by 0 error in the corresponding homework averages.

Tuesday 3 October: The homework assignment for week #6 has now been posted. Folow the link from the Schedule Page.

Friday 29 September: I have determined the grade cutoffs for Exam #1. They are as follows: A (75-100), B (65-74), C (55-64), D (40-54). I'll post a more detailed collection of statistics as soon as I get a chance to do so.

Friday 29 September: If you have questions or concerns about the grading of your Exam #1, please write a note on the coversheet explaining the situation and submit it to me. I'll make sure the appropriate grader(s) review it.

Friday 29 September: Today in class I returned the exams for all sections except B and C, which had yet to have their grades recorded. If you are registered in on of these two sections, you should contact your TA to see when and where you can pick up your exam.

Sunday 24 September: I will give you a reference table to use during Exam #1. It will have the error estimates for the trapezoid rule, the midpoint rule, and Simpson's rule as well as some trig identities. I've added a link to the table from the review page, so you can see exactly what information you will have.

Friday 22 September: I've added some old exam problems to the review page for your studying pleasure. I don't have solutions available for them, but you can ask questions about them. Possibly at the review session Monday evening.

Thursday 21 September: I've also corrected a typo on the Exam #1 Review page. The exams will be held during the normal class time (though at a different location). The review page initially indicated that the exams would be an hour later than the usual class time, which in most definitely not the case.

Thursday 21 September: The Schedule page was a bit out of kilter overnight, but it seems to be corrected now. One errant mouseclick and everything goes haywire!

Wednesday 20 September: I've posted a review page for Exam #1. You can also follow the link from the Schedule page.

Wednesday 20 September: With midterm exams coming up, you may want to know about Academic Development's group workshop on Time Management. It's scheduled for for Monday 25 September from 8:00-9:30, which, I note, overlaps substantially with the 21-122 Exam #1 Review Session. Which one should you go to? How should you manage your time? I report, you decide!

Tuesday 19 September: One of your classmates pointed out an interesting relationship among the Simpson's rule, midpoint rule and trapezoid rule approximations for numerical integration. If you compare the formulas carefully, you will note that

S2n = ( Tn + 2Mn )/3.

This makes intuitive sense when you consider our discussion of the error estimates: The error ET is about twice the size of EM, and has opposite sign, so the weighted average on the right should "cancel" "most" of the error.

Friday 15 September: The reading and homework assignments for Week #4 have been posted. Follow the link from the Schedule page.

Tuesday 12 September: Academic Development, the folks who provide the SI sessions and peer tutoring, are sponsoring several group workshops focusing on study skills. "For Freshemen Only" seems to cover a wide range of topics. They have a couple other sessions that address time management.

Tuesday 12 September: I though I'd make a few comments about the web resources on the publisher's site. The link for Trigonometric Substitution seems to be a dead link. Not too much to say about that. The link for Partial Fractions has been timing out on me all morning. I'll try again later.

The link for "Strategies for Integration" is basically a summary of the techniques we've covered. It uses non-standard terminology that seems a little silly to me, and in some cases insulting. For example, instead of "polynominal", the site uses the term "straights". The site seems to be bases in Ireland, so perhaps this terminology is common there. All in all, I wouldn't really recommend this page.

Instead, I would recomment a page pointed out to me by one of your classmates. Paul's Online Math Notes, posted by Paul Dawkins at Lamar University, seem to be well written, and thorough. For this course, you'll be especially interested in his notes for Calculus II.

Friday 8 September: The reading and homework assignments for Week #3 have been posted. Follow the link from the Schedule page.

Wednesday 6 September: The math department at Saint Louis University has a helpful guide for Success in Math. It might be worth your time to take a look at it.

Wednesday 6 September: I've posted my office hours, and those of the TA's on the staff page.

Wednesday 6 September: I want to point out to you some resources you have available to you in addition to class, recitations and office hours. The Supplemental Instruction (SI) sessions will begin tomrrow (6:30-7:30pm in Cyert Hall B6A). Additionally, Academic Development will be running "Walk-in Tutoring" sessions for calculus at the following places and times: Mudge Library, Sunday thru Thursday from 8:30-11:00 pm; Donner Reading Room, Sunday thru Thursday from 8:30-11:00 pm; West Wing Tech Lounge: Monday & Wednesday from 8:30-11:00 pm.

Friday 1 September: The reading and homework assignments for Week #2 have been posted. Follow the link from the Schedule page.

Thursday 31 August: I'll be holding office hours today from 1:30-3:30.

Wednesday 30 August: The publisher's web resources for Section 7.1, Integration by parts look useful, though not too different from lecture and your text. You should take a quick look and see if they seem useful to you.

Wednesday 30 August: There are now additional copies of the syllabus outside my office door (WEH 6214). You can pick one up if you like. I'll bring the rest with me on Friday.

Monday 28 August: I've just received this message about the air conditioning in WEH 7500: "The air handler in Wean Hall classroom 7500 is shutdown due to a broken chilled water coil. An interim repair is being attempted until the coil can be replaced. During this time and until further notice, air conditioning in the room will not be working." The replacement part is expected to arrive on Tuesday, and air conditioning will hopefully be restored a few days later.

Monday 28 August: The reading and homework assignments for Week #1 have been posted. Follow the link from the Schedule Page. You will have to take a short tour of the Policies page before you can view the Exercises.

Monday 28 August: The publisher of our textbook has made some web resources available. Some of them look quite helpful. A good place to start would be the Author's Welcome.

Tuesday 15 August: Welcome to 21-122 Integration, Differential Equations, and Approximation.