MATH 192, Section 1 Calculus II Fall 2019
Syllabus
Welcome! We are about to take the first steps on a journey through some of the most beautiful mathematical lands. The focus of Calculus II is integral calculus, a field with innumerable appli-cations to engineering, physics, chemistry, astronomy, computer science, economics, biology, and pretty much every branch of mathematics known. Beyond its usefulness, integral calculus is simply gorgeous in its elegance and simplicity, and I hope that you will come to appreciate the subject’s beauty as much as I do.
I’ll have more to say about what we’ll be studying in a little while, but let’s take care of the nuts and bolts of the course first.
• Who’s the teacher? Patrick Bahls...please feel free to call me “Patrick.” Please note that as your “teacher,” I consider myself more of a facilitator than a lecturer...I expect that you all will be active agents in your learning, generators of new ideas. Think of me as a “guide on the side” and not a “sage on the stage.”
• Where and when will class meet? We will meet in Rhoades Hall, Room 212, from 9:30 to 10:45 on Monday, Wednesday, and Friday mornings.
• What text will we be using? We will work through Chapters 6, 8, 9, and 10 ofCalculus (2nd Edition), by William Briggs, Lyle Cochran, and Bernard Gillett (Pearson, 2015; ISBN-13: 978-0-321-95435-0). Please procure a copy of this text (new or used, bought or rented) as cheaply as you can, and let me know if you have any problems in so doing. Note that the bookstore will have copies for sale or rental, but you may certainly look elsewhere if you’d like. Please note that we will not be using MyMathLab, the electronic companion to this text.
• What should I bring to class? You should always have something to write with (an old-fashioned pen or pencil will do just fine) and take notes on. You will be required to use the LATEX typesetting environment (more about this below) to submit weekly problem set write-ups and take-home exams, and it wouldn’t be a bad idea for you to get in the habit of bringing a laptop with LATEX at the ready. I’ll let you know on a day-to-day basis if you’ll need anything else for a particular class meeting.
• When and where are your office hours? I will generally be free on Monday, Wednes-day, and Friday after our class meets (11:00 a.m. through 1:00 p.m.) and Tuesdays and Thursdays from 10:00 a.m. to noon. I will also have availability at a number of other times; please feel free to make an appointment if you’d like to guarantee a meeting with me. Incidentally, all of my office hours will be held in my Honors Program office, 140 Karpen Hall.
• How can I get a hold of you? My campus phone number is 828-232-5190 and my e-mail addresses are [email protected] and [email protected]. E-mail’s typically the best way to get a hold of me.
https://sites.google.com/a/unca.edu/pbahls/,
and there’s a link on that website to our class page. There, you can access just about any course-related information you need (a schedule of homework assignments, LATEX resources, a copy of this syllabus,etc.). We willnot be using Moodle for this class!
Finally, I should mention my social media policy. I am happy to be your friend on Facebook (I’m old, so I don’t have the Instagram, and I only very rarely tweet or twoot or whatever you kids call it these days), but it is my policy not to actively friend my students. I will, however, warmly accept your friend requests. I only ask that if you do wish to be my friend on Facebook, please think carefully about the sorts of things you’d like your professor to be able to see about you when you update your status, post pictures,etc. Cute pet pictures are always welcome and will be liked; videos in which you show off your alcohol tolerance are not advisable.
A few words about accessibility, equity, and inclusion. I would like to make every attempt to address any learning disabilities or other accessibility issues you might have that could affect your participation in our course. To that end, if you believe your participation may be so affected, I encourage you to get in touch with the Office of Academic Accessibility by writing to [email protected]. Furthermore, I would like to make our classroom as equitable and inclusive as possible. To that end, at the very least, I ask that you ensure that I address you in the manner in which you wish to be addressed. This means not only that I invite you to inform me of the name you wish to go by, but that you also, for instance, inform me of the gender(s) (if any) with which you wish to be identified so that I might make appropriate use of pronouns. I invite you to inform me of any other accommodations I might be able to make in order to improve our class’s accessibility and your experience in our class.
A few words about sexual misconduct and sexual harassment. All members of the university community are expected to engage in conduct that contributes to the culture of integrity and honor upon which the University of North Carolina, Asheville is grounded. Acts of sexual misconduct, sexual harassment, dating violence, domestic violence and stalking jeopardize the health and welfare of our campus community and the larger community as a whole and will not be tolerated. The university has established procedures for preventing and investigating allegations of sexual misconduct, sexual harassment, dating violence, domestic violence and stalking that are compliant with Title IX federal regulations. To learn more about these procedures or to report an incident of sexual misconduct, go to titleix.unca.edu. Students may also report incidents to an instructor, faculty or staff member, who are required by law to notify the Title IX Office.
Unlike the computation of derivatives, which is little more than the rote application of a handful of shortcut formulas, the computation of antiderivatives often requires ingenuity, cleverness, cre-ativity, and artistry, in addition to mathematical acumen. The first several weeks of the semester will have us learn various techniques for computing antiderivatives.
Meanwhile, we will be learning how touse antiderivatives. To this end, we’ll see how antideriva-tives help us to compute areas, volumes, work, electrical charge, centers of mass, dimension, eco-nomic cost,etc.
As we undertake our study of antiderivatives and their uses, please keep in mind that the subjects we’ll be studying took brilliant mathematicians thousands of years to develop. The topics we will address are the results ofcenturiesof arduous work, featuring frequent dead ends and many missteps. Therefore it’s entirely understandable if you experience occasional difficulties and make a few mistakes as we go. Please remember that we areall in this together, and that the best way for us to succeed in mastering the material we’ll be studying is for us to work together. I cannot emphasize enough thatmathematics is a social endeavor: we learn mathematics best by doing it and by doing it in the company of others. Thus, I have taken care to design our course so that it is highly collaborative, encouraging your engagement with one another (and with me) both in and outside of class.
What do you expect me to get from this course? I hope that after this course is over you should be able to
• explain to a peer the concepts integral, sequence, and series, and demonstrate high-level computations involving these concepts;
• demonstrate the way in which integral calculus can be used to solve problems from various math-based fields such as physics, engineering, chemistry, and economics;
• craft clear and well-composed explanations of mathematical ideas (using the LATEX type-setting environment); and
• explain mathematical ideas in oral presentations to your peers.
You should go back over this list every so often and ask yourself (and me!), “am I making progress in developing these skills?” If at any time you feel the answer is “no,” please come and talk to me about it; one or both of us might be able to change our ways to help you better reach these goals.
How will our class be organized? Though the topics we’ll deal with will change from day to day, a typical class will feature each of the following:
• (perhaps) a presentation of a solution to a homework problem, • a brief overview of the reading assigned for that day’s class,
• problems, projects, and activities designed to help us all gain familiarity with the topic of the day, and
• a short quiz on that topic.
I’ll have more to say about each of these as I answer the following question:
What kind of work will I be responsible for in this course? Your grade will be based upon the following activities:
• completing daily quizzes,
• completing three take-home exams, and
• delivering a collaborative presentation on a relevant mathematical topic of your choice (at the end of the semester).
Daily quizzes. Before coming to class on a given day, you will be expected to (1) read the section of the text assigned for that class meeting and (2) craft rough solutions to a small number of problems corresponding to that reading. To encourage active engagement with the reading, we will end each class period with a quiz on the reading for that day. The quizzes will bebrief, no more than 3 to 5 minutes in length, and they should be easy, if you’ve done the reading for that day.
I would like to propose that quizzes, altogether, be worth 15% of your overall grade.
In-class problem presentations. Further encouragement to do the readings in advance (seri-ously...do them, y’all) will be given by the requirement that at least once during the semester you each present a solution to one of several problems assigned for each reading. These presentations do not have to be polished, but they should demonstrate a firm understanding of the problem and its solution. Using LATEX (yes, yes...more on that below) for these presentations is encouraged but not required. We will generally have time for at least one presentation per day.
I would like to propose that your presentation of at least one problem’s solution be worth 10% of your overall grade.
Homework write-ups. As noted above, I will assign several problems from each section you’re expected to read. For one (or, in rare cases, two) of these problems you will be expected to write up a clear, complete, correct solution using LATEX. These write-ups will be due to me (in a single .pdf file per week) via email by 11:59 p.m. on a designated Friday, roughly 2-4 problems per week...totally doable! See the schedule of readings, problems, and write-up due dates near the end of this document for more info.
I would like to propose that your homework write-ups be worth 30% of your overall grade.
Take-home exams. Though we will not have in-class exams, you will have three take-home exams during the semester, handed out in, roughly, the 5th, 9th, and 14th weeks of the term. You will have a week to complete and submit solutions to each exam, using LATEX.
I hope that it goes without saying that these exams are to be completed individually and without consulting any mathematical resources other than your own notes. (You may, however, come to ask me for guidance or clarification, as needed...but I promise to not be very helpful!) For the first two exams, after I have graded them I will offer you the chance to revise your solutions and the opportunity to earn up to half credit back on any problems that you miss. For instance, if you were to initially receive 3 out of 5 points on a problem whose solution you revised to near perfection, you could receive 1 point (1 = 2 points missed÷2) back for that revised solution.
Though I do not want to belabor the point, I will mention thatNOTHING makes me grumpier as a teacher than academic dishonesty. If I see evidence of cheating on take-home exams (and I’ve been at this for 20 years; I can see it if it’s there), rest assured that I will take action on the matter. ‘Nuff said? ‘Nuff said.
I would like to propose that your take-home exams be worth 30% of your overall grade (10% apiece).
related to the content of our course. You might choose, for example, to compute a particularly challenging integral...or to demonstrate an application of integration to a field you’re all keenly interested in...or maybe you’d like to supplement our understanding of any one of the topics we’ve talked about in class with a deeper dive into the literature. I will give you a great deal of latitude, as long as you clear your group’s topic with me before working on it.
Once you’ve settled on a topic, your group will work to put together a 10-12 minute presentation on the topic, which you will deliver at the end of the semester. These presentations will take place on our final day of class (Monday, December 2) and during our scheduled final exam time slot,
from 8:00 a.m. to 10:30 a.m. on Monday, December 9. I’ll share a feedback form for these presentations nearer to the midpoint of the semester, so that you can plan your presentation accordingly.
I would like to propose that your collaborative presentations be worth 15% of your overall grade.
The astute reader will note that I’ve listed the percentage values for the various components of our course as “proposals.” This is because I feel strongly that you all should have a hand in designing this course (it is, after all, our course and not merely mine), and so later this semester I will lead us in a discussion on the matter of assessment and grades...we might decide, as a class, on different point values; those listed above are meant as a starting point for negotiations.
Okay...can you please say a few more words about LATEX? Certainly! LATEX is a variation of an earlier (and harder-to-use) software distribution called TEX, the purpose of both of which is to give the user complete control of the typesetting environment, enabling them to manipulate not only ordinary things as spacing, indentation, pagination, etc. but also special characters, symbols, and so forth that pertain to technical fields like mathematics. In LATEX for instance, you can typeset simply beautiful mathematical expressions like
f(x) = sin(x)⇒F(x) = Z
f(x)dx= Z
sin(x) dx=−cos(x) +C
or
d dx
ecos2(x)
=−2 cos(x) sin(x)ecos2(x).
You will find that LATEX will be more than up to the challenge of this class. (Note: nearly every document I’ll use in this class, including this syllabus, was created using LATEX.)
Using LATEX well takes practice, and it’s most difficult at first. We will spend a full class period (Friday, August 23) near the beginning of the semester on a crash course on the software, and I urge you to dive into it in order to surmount the learning curve as quickly as you can. As I’ve mentioned above, I willREQUIREyou to complete all of your solutions to problem set and exam problems in LATEX and I encourage you to use it in in-class presentations as well. Once you get used to it, you’ll be producing indescribably beautiful mathematical works with relatively little effort!
Tentative schedule of class activities and readings. Please note that the reading listed for a given date is the reading to be completed for the NEXT class meeting. (See the next list for the problems I would like you to complete for each given section.)
Please note also that this schedule is subject to change, should we need to adjust course to account for inclement weather, lingering conversations,etc.
• Monday, August 19. First day of class! We will get to know one another and talk a bit about the class.
• Wednesday, August 21. This day will be dedicated to reviewing some of the concepts from Calc I that will be useful to us as we begin our work this term, including techniques for computing derivatives, Riemann sums, and u-substitution.
• Friday, August 23. Today we’ll have a crash course on LATEX! You should come to class on this day with your computer, some sort of LATEX platform (e.g., Overleaf or Texmaker) at the ready. Read Section 6.1 for Monday!
• Monday, August 26. We will examine the concept of net change and how it relates to integration. Read Section 6.2 for Wednesday!
• Wednesday, August 28. We look at another highly useful (and generalizable) application of integration: finding the area between two curves! Read Section 6.3 for Friday, please! • Friday, August 30. We begin talking today about how to use integrals to find the volume
of various solid objects. No new reading for next week! • Monday, September 2. Labor Day: no class!
• Wednesday, September 4. We continue to slice ‘n’ dice to find volumes! Please read Section 6.4 for Friday!
• Friday, September 6. A new method for finding volumes, bycylindrical shells! No new reading for Monday...
• Monday, September 9. ...we do some more problems involving cylindrical shells, high-lighting the difference between this method and the slicing method. Please read Section 6.5 for Wednesday.
• Wednesday, September 11. Another geometric application today: finding the length of a given curve segment! Please read Section 6.6 for Friday!
• Friday, September 13. We look at yet another geometric application: finding the surface area of a solid of revolution! Please read Section 6.7 for Monday.
• Monday, September 16. We look at a few different applications of integration to physics and engineering. No new reading for next class!
• Wednesday, September 18. We continue to look at physical applications. For Friday, please complete the review problems on exponential and logarithmic functions which I handed out in class today.
• Friday, September 20. We spend some time in review of exponential and logarithmic functions. Please read Section 7.4 for Monday’s class.
• Monday, September 23. What do integrals have to do with exponents and logs? We’ll find out today! Please read Section 7.7 for Wednesday.
• Wednesday, September 25. Exponential functions give an easy entry point into hyper-bolic functions, about which we’ll talk today. For Friday, please read Section 8.1.
Section 8.2 for Monday. Tentatively, I will hand out Exam 1 on this date, and it will be due by midnight on Friday, October 4.
• Monday, September 30. Integration by partsis our first “nontrivial” means of computing integrals. Please read Section 8.3 for Wednesday.
• Wednesday, October 2. Trig integrals can be fun! We’ll do a bunch of them today! Meanwhile, read Section 8.4 for Friday.
• Friday, October 4. Trigonometric substitutions are tricky, and we’ll gain some practice with them today. Please read Section 8.5 for Monday.
• Monday, October 7. Ouch...I apologize in advance for today’s class; “I love partial fractions!” saidno one,ever. Moving on, please read Section 8.6 for Wednesday.
• Wednesday, October 9. Today we’ll review all of our methods for explicit solution of integrals. Go ahead and read Section 8.7 for Friday.
• Friday, October 11. What if you can’t solve an integral explicitly? We’ll talk about numerical approximations today! Please read Section 8.8 for Wednesday.
• Monday, October 14. Fall Break: no class!
• Wednesday, October 16. Today and Friday we’ll talk about improper integrals, which are not as inappropriate as they sound! No new reading for Friday.
• Friday, October 18. I will not be present on this day, but I will be sure that you have the tools needed to further explore improper integrals (don’t tell your mother, please...)! No reading for Monday.
• Monday, October 21. We’ll spend the day reviewing, catching up, working out practice problems, and so forth. A good catch-up day!
• Wednesday, October 23. In case Monday wasn’t enough, let’s play a little more catch-up...it’s been a whirlwind of a semester, and I want to make sure we’re all feeling confident moving into the home stretch! Please do read Section 9.1 for Friday, though!
• Friday, October 25. What’s an infinite sequence? What’s an infinite series? What do they have to do with one another? And why do we even care?!?! We’ll find out today! Please read Section 9.2 for Monday! Tentatively, I will hand out Exam 2 on this date, and it will be due by midnight on Friday, November 1.
• Monday, October 28. We begin our study of sequences in earnest! No new reading for Wednesday...
• Wednesday, October 30. More on sequences! Please read Section 9.3 for Friday! • Friday, November 1. All right, now it’s onto infinite series! No new reading for Monday. • Monday, November 4. We continue talking about series, including a more careful
ex-amination of the idea ofconvergence. Please read Section 9.4 for Wednesday.
• Wednesday, November 6. Let’s come up with some tests for convergence! Also, please read Section 9.5 for Friday!
• Friday, November 8. I will not be present on this day...in my absence, please use this as a “catch-up” day! Do, however, make sure you read Section 9.5 for Monday.
• Monday, November 11. Today we’ll talk about a few more tests for convergence: the Root, Ratio, Comparison, and Limit Comparison Tests! Be sure to read Section 9.6 for Wednesday!
• Friday, November 15. Today all of our work with sequences and series comes together, as we discover how series enable us to do amazing things with approximation! Please read Section 10.2 for Monday!
• Monday, November 18. More on the use of polynomials and power series in approxima-tion. Please read Section 10.3 for Wednesday!
• Wednesday, November 20. Now that we’ve seen how useful power series can be, let’s see how they behave! Also, read Section 10.4 for Friday, please!
• Friday, November 22. Taylor series are one of the oldest and most useful of power series, and we’ll spend today talking about them. Finally, please read Section 10.4 for Monday. • Monday, November 25. Wut, wut...we’ll spend a little while talking about Taylor
series...but then we’re done! No more readings for the semester!
• Wednesday, November 27and Friday, November 29. Thanksgiving: no classes! • Monday, December 2. On this last day of class four groups will give their collaborative
end-of-semester presentations! Tentatively, I will hand out Exam 3 on this date, and it will be due by midnight on Monday, December 9.
• Monday, December 9 (8:00 a.m. to 10:30 a.m.). This is our designated final exam time, when we’ll meet in our usual classroom for the remaining groups’ end-of-semester presentations.
Problems and due dates for solution write-ups. Below are the due dates for all write-ups, as well as the problems I would like you to complete for practice, presentation, and/or written submission. The problems listed with an asterisk are those for which you must submit LATEXed write-ups. Please recall that these solutions are due to me, in the form of LATEXed .pdf files, sent via email, no later than 11:59 p.m. on the given date.
Notes: I will notrequireyou to solve any problems other than those listed with asterisks; however, the only way to truly learn math is to do math, and I strongly encourage you to complete as many of the problems listed below as you can. Moreover, you may present solutions toany of the problems listed below (starred or not) in class.
• Friday, August 30. Sections 6.1 (8,12∗,26,36,44,58) and 6.2 (8,14,16,26,34,48,52∗,64) • Friday, September 6. Section 6.3 (12,20,30∗,40,44∗,56)
• Friday, September 13. Sections 6.4 (6∗,20,28,42,56∗) and 6.5 (10∗,14,28)
• Friday, September 20. Sections 6.6 (8,14,16,24∗) and 6.7 (12,16∗,20,24,28∗,34,50,58) • Friday, September 27. Sections 7.4 (12∗,22,34) and 7.7 (12,20,26∗,30,38,56,82∗)
• Friday, October 4. Sections 8.1 (8,12,16,20,24,28,32∗,36,40,44,48∗), 8.2 (10,16,22∗,30, 38,42,46∗,58), and 8.3 (16,24,28,32,42,48∗,50,68)
• Friday, October 11. Sections 8.4 (8,16,24∗,32,40,48∗,56,62,66,80), 8.5 (16,24,32,40∗,48, 60∗,68,80), and 8.6 (12,20,32∗,40,56,60)
• Friday, October 18. Section 8.7 (12,16,24∗,40)
• Friday, October 25. Section 8.8 (6,14∗,22,30,38,46,56∗)
• Friday, November 1. Sections 9.1 (12,16,18,22∗,24∗,30,36,44,54,62,76∗) and 9.2 (12,20,22, 28,42,50∗,64,78,88∗)
• Friday, November 8. Sections 9.3 (8,16,22∗,56∗,60,68∗,72,78,94)
• Friday, November 22. Sections 10.1 (10,18∗,32,42∗,50) and 10.2 (10,18,24∗,30,38,46,56, 66,72∗)
• Friday, November 29. Sections 10.3 (16,26,32∗,36,50,66,70) and 10.4 (12,20∗,28,38,54,60∗)