Math 1b Calculus, Series, and Differential Equations Harvard University Fall 2005 http://my.harvard.edu/course/math1b Good morning. It is my privilege and honor to welcome you to Math 1b for the fall semester. If this is your first course here, let me welcome you to Harvard as well. This meeting is to lay the groundwork for what I hope will be a successful term. We will advertise the course a little, go over the syllabus, requirements, and resources, and explain the setup for this course. Today at least, you won’t need to take notes. All the information is on the paper syllabus. I’ve prepared a presentation as well, partly because my handwriting is atrocious and partly because I only get to do this twice a year. Both of these will be placed on the course web site. So for the moment, sit back and enjoy the blinking lights. :-)

Course Goals Learn the techniques of calculus for analyzing functions Learn how to model complex situations with mathematics To read, write, and critique mathematical arguments Let’s talk about the goals for the course. Calculus is the study of functions. Have you ever met a function outside a math class? Everything is a function! The temperature in this room is a function of time. Profit is a function of quantity sold, which is a function of price, which is a function of cost (among other things). Modeling is the key idea that allows us to do math and make money! This is not a proof class. But we will focus more on concepts than in your last math class. The intellectual pursuit of mathematics involves an understanding of the principles behind the machinery.

Course Head Dr. Matthew Leingang leingang@math.harvard.edu (not leingang@fas) Science Center 323 Office Hours: Tues., Weds., 1–3pm. I’m Dr. Leingang, and I’m a preceptor in the mathematics department. This is junior faculty teaching position, so I do a lot of calculus, some teacher training and student advising. I’ve been teaching calculus since I got my Ph.D. five years ago. If you would like to contact me the best way is through e-mail….Leingang@fas is my cousin who’s a junior. He’s a great guy but doesn’t really want to talk to you about your math problems. My office hours are TW 1-3pm. These are times when you are free to drop in and we can talk about the course or anything else. No appointment is necessary, just come on in. If you need to see me and you cannot make the OH, e-mail and we can schedule an appointment. And, if you come by and the door is open, I can usually spare a few minutes no matter what. --- Many of our international instructors spend some time on the first day talking about their native land. It helps in the bonding process for the students to understand the teacher. So in that vein…

Get to Know Indiana Any Hoosiers in the house? Maybe they all went to Princeton. Indiana is the Crossroads of America, and has the more interstates going through it than any other state.

Get to Know Indiana Unfortunately, none of them go through my hometown of Evansville. What goes on in Indiana?

Get to Know Indiana Basketball

Get to Know Indiana Corn

Get to Know Indiana Soybeans So have we bonded now? I’ve lived on the East Coast for the last ten years, so I am as much a Bostonian as a Hoosier. Last year I was grading the Math 1a exam while the Red Sox were winning their first world series title in 86 years. Sorry you missed it! Hopefully you’ll only have a few months to wait for the next one.

Other teaching staff Faculty members or Teaching Fellows in mathematics Majority of instruction takes place in section, three hours/week All sections cover same topics and have the same workload What’s unique about calculus at Harvard compared to other science courses here is that we teach in sections. This means that we won’t meet in a lecture hall like this except at exam time. The majority of the time you will be learning in a small group of 20-25 students lead by a teaching fellow. We believe that mathematics is best learned in more intimate dicussions like this.

Sections Choose your section by computer: MWF9, 10, 11, 12, T10, 11:30 We offer calculus at just about any time you want to take it, but some sections are only going to appear if we get a large enough group.

Sections Choose your section by computer: MWF9, 10, 11, 12, T10, 11:30 With sufficient enrollment To section from a UNIX prompt: ssh section@ulam.fas.harvard.edu More information on flyer and at http://math.harvard.edu/sectioning/ You choose your section by using a ultra-modern 20th century (joke) computer program, which requires you to log in with a a terminal. If you have a Mac, this is really easy and requires no extra software. If you have a Windows machine, this is does require a download but it still pretty easy. There are step-by-step instructions on our department’s web site.

Problems you will be able to solve after this course 発問 Hatsumon Problems you will be able to solve after this course If these problems sound intriguing to you, you should be here!

Hatsumon – Volume What is the volume of a Krispy Kreme Donut? Let’s talk about Donuts. Have you ever wondered how much sugar is in one of these? Well if you knew the density of sugar in donut batter you can use calculus to find this out by finding the volume of a donut-shaped region in space.

Hatsumon – Microbiology A population of bacteria reproduces asexually. How can you predict the growth of the population over time? Bacteria are a primitive population--ideally, all they do is eat and reproduce. Given (essentially) unlimited food, how fast does such a population grow? Now consider another population that competes for food: like fish, or people. What changes?

Hatsumon – Fluid Dynamics A hole is punched near the bottom of a tank How long does it take for the tank to drain? Take a tank of water or some other blue liquid and punch a hole in the bottom of the tank. At first, the water above the hole pushes water out quickly, but then it slows down. How long does it take to drain?

Hatsumon – Organismal Bio Owls eat mice. What will happen to the owl population when the mouse population is halved by starvation? How long until we notice a change? Back to populations. Suppose you have two populations where one is the predator of the other. Lots of owls means fewer mice, which means fewer owls, which means more mice. How do these populations evolve over time given their dependence on each other?

Hatsumon – Numerical Analysis What is the 173rd digit of π? The 1000th? The millionth? Here’s a fun one (ever seen the movie π by the way? More realistic than Good Will Hunting, but not much.)

All of these problems (and many more) can be solved with Calculus! “Over three centuries of constant use have not completely dulled this incomparable instrument.”—Nicholas Bourbaki

Course Topics Those are some the the problems we will solve in Math 1b. If you are looking for the names of things we’re going to do, here they are.

Techniques and Applications of Integration Basic Problem: find the area under a curve Extends to many concepts besides area: Volume Work Flux The first part of the course you may have seen before in your Calc I (Math 1a, AB) class. The integral was invented to measure area. However, it is infinitely extensible. Volume, work, flux, revenue, anything which can be expressed as a product of other quantities can be computed with an integral.

Differential Equations describe the way quantities change with respect to other quantities (for instance, time) The laws of science are easily expressed by DE F=ma (more difficult when F depends on position, or on time) Newton’s Law of Cooling Population Dynamics Since functions are everywhere, so are differential equations. F=ma: if you remember the calculus you got before you got here, you know that a is the second derivative of position. You might have F a function of position, or of time, or both. This gives you a differential equation for the position function of time. NLC: the rate of change of temperature is proportional to the difference between the current temp and ambient temp Populations: Owls eat mice. You have mice which grow kind of like bacteria, and owls which die if they have no mice to eat. Interactions between owls and mice result in more owls, fewer mice. This becomes two DE; one for each species

Infinite Series Approximate many complicated functions by simple polynomials Solve differential equations Our digital world would be impossible without them—how does your calculator know sin(0.1) to 8 digits? Infinite series are often portrayed as an “out there” topic that doesn’t relate to the rest of calculus, but it is integrally (pun) related to the study of functions. Infinite power series allow us approximate complicated functions by polynomials. This makes their derivatives and antiderivatives as easy as those for polynomials. Anything digital uses series to approximate the transcendental functions.

Course Expectations That’s the content. Here’s the dirty work--what all’s going to happen in the course. If this is your first college math course, you may find this a little bit more work than your last math course. That’s natural. But if you hit the ground running it will become easier.

Pre-class Reading Assignments Reading assignments the night before class Go to course web site to answer questions Your teaching fellow will use this information to prepare a customized class The purpose of these assignment is twofold: You will get an extra pass through the material so that by the time you sit down to do the homework you have seen it at least twice. Also, your section leader will read your responses before class and use it to prepare the section with your confusions in mind.

Homework Assigned each day, due the next No late homework! Please keep up. Drop one week’s worth in computation of final grade Homework will test your ability to use the techniques taught in class but also your understanding of the concept. Some will be quite easy; some will harder. Some may be nearly impossible, and I do this not because I like to torture students but because I want to show you what is possible. I like to think of homework as weight training: it’s part of how you prepare for competition. Repetition is important! It’s important to keep up. You won’t be able to do it all overnight. So please keep up! We understand that sometimes there are more important things (conflicts, sickness, emergencies) than HW. So we willd rop one week’s worth.

Exams Technique Test: October 6 and 13 (second is optional; maximum counts) Two midterm exams: Thursday, October 27 and December 1 Final Exam (cumulative) tentatively scheduled for January 14 As far as training goes, this is what for. We’ll have two midterms which we all take together, on these dates. Please clear your calendar now. If you have a conflict, contact me as early as possible, no later than a week before the exam.

Breakdown of your course grade 20% Midterm I 25% Midterm II 35% Final 15% Homework 10% Technique Test 5% Pre-class Reading Assignments 100% Total The technique test comes up first and is not actually a midterm but more like a quiz on techniques of integration. You’ll get two chances, so don’t sweat it too much.

Grading Scale Could be adjusted for equity (“on the curve”)—up, but not down Pluses and minuses will also be determined in final analysis 90% A 80% B 65% C 50% D Not exactly on the curve--only in the way that can help you.

Texts Single Variable Calculus—Concepts and Contexts by James Stewart, ISBN 0-534-41022-7 Available for purchase at the Coop We provide the ISBN in case you want to try your luck online. Keep in mind you need the book right away. There is a large version that includes multivariable. By this if you think you might take Math 21a. I cannot recommend you buy the 2nd edition. There are additional problems and reorganization which will make synchronization with the 3rd hard. Do this at your own risk. There will also be some additional material to be distributed on the website.

Texts Schaum’s Outlines: Precalculus by Fred Safier, ISBN 0-07-057261-5 Optional Cheap Available for purchase at the Coop This is an optional resource for students who would like to brush up on some of the precalculus prerequisites---functions, trigonometry, logarithms, etc. It's slim and cheap.

Prerequisites

Logical Prerequisite: Calculus Derivatives Definition of the Integral Fundamental Theorem of Calculus (Integration by Substitution) Math 1a or 8 on HMPT2 You should have had a good differential calculus class before coming into Math 1b, one that covered the definition of the integral and the Fundamental Theorem of calculus. Math~1a is certainly acceptable. Also, a set of scores on the Harvard math placement exams in which the first number (precalculus) is at least 20 and the second number (differential calculus) is at least 8 gives a Math~1b placement.

Logical Prerequisite: Precalculus Functions Graphs of “famous” functions and manipulating them Trigonometry Logarithms 20 on HMPT1 We find that students often come into this course not remembering much precalculus. We understand that many of you took that a long time ago, and so this prerequisite is often relaxed. This means, however, that you assume the responsibility for refreshing yourself on the important aspects of trigonometry and logarithms that will be taken for granted in the course. The precalculus primer may prove to be a resource.

For those who have taken AP Calculus BC Math 1a and 1b together cover the Calculus BC syllabus Math 1b does more than what’s on the BC, with different emphasis You will still find lots to learn in Math 1b You may consider yourself a master of calculus and are here to get an easy A. What does Yoda say? “You must unlearn what you have learned.”

For those who have taken AP Calculus AB Some of your classmates will have seen some of this material before We are committed to supporting all qualified students We will not teach this course as its own prerequisite!

Resources Your section’s problem session Math Question Center (Sunday–Thursday, 8–10pm, Loker Commons) Your TF’s office hours My office hours Problem Sessions: held by course assistants. Your CA will schedule his or her session to best fit the schedules of your section, but you are free to go to any problem session you wish. the Math Question Center: in Loker Common 8:00--10:00 pm every night except Friday and Saturday. Course Assistants and mathematics graduate students staff this center. Besides math help, there is usually food. Office Hours: These are times set aside by your section leader to talk informally with students. Your TF will publicize his or her office hours frequently.

Math Warm-Up Series Brush-up on some precalculus topics (trig, logs, algebra) Advice on study skills and course selection http://www.math.harvard.edu/mwus/

Other Activities Online Placement Exam http://math.placement.fas.harvard.edu/

Other Activities Math Department advice is ongoing; see http://www.math.harvard.edu/sectioning

First Assignments Read Section 5.1 and 5.2 of the text Select a section by computer answer questions online (to appear later this week)

Conclusion I hope you take and enjoy Math 1b! Web site reminder: http://my.harvard.edu/course/math1b (Gratuitous baby shot)