Math 1b Second Semester Calculus Fall 99
Course Information, Syllabus and General Philosophical Ramblings
Welcome to the last offering of math 1B in this millenium! The semester will be broken into three roughly equal parts, during which we will study three major topics:
We will start the semester with what should be a relatively new topic for everyone, infinite series, which will include power series and Taylor series. Taylor series are extremely useful for a number of applications. In the same way in which someone might approximate a particular number, such as 
, by a finite decimal expansion, such as 3.1415, it is possible to approximate whole functions, such as sin(x), by finite Taylor polynomials, such as 
. These approximations are used in a wide number of fields, including engineering, biology and physics. Along the way to working with Taylor series we will explore the idea of an infinite sum - whether it is possible to add together an infinite number of positive numbers and yet end up with a finite number.
After our work with series, we will then revisit a topic which should be familiar to all of you, calculus. Needless to say calculus has become a foundation stone in science, and calculus has played a significant role in many scientific advances of the past several hundred years. Although people often think that calculus is a static subject, there is still research going on in some areas of the field. In any case, the way in which calculus is taught has changed dramatically over time, especially over the past two decades which have seen the advent of graphing calculators and mathematical software programs such as Matlab and Mathematica.
Today there is less of a need to memorize numerous methods of approaching specific types of integrals, as these can often be worked out in a split-second using a calculator such as the TI-92. As part of our course, we will still study the basic techniques for handling many types of integrals. The point now is not so much to be able to handle any integral which might come along, as to appreciate the ingenious methods of integration themselves, and to probe how they work. In studying the basic techniques we will take away the mystery of solving integration problems. In the same way people still learn addition and multiplication tables to build mastery and confidence in basic arithmetic long after the pocket calculator was introduced, there will always be a reason to learn the basic tools and techniques of integral calculus to build mathematical confidence and increase our own mathematical sophistication. Besides, it can be a lot of fun!
While we're discussing calculators, as a general policy, we will not forbid the use of graphing calculators during the semester as long as they are used as tools to help you learn and explore math, as opposed to crutches which keep you from developing your own understanding. However, it will be vital for you to develop confidence in your mathematical abilities independent of a calculator, and so we will design the course so that for the most part there will be no need to use a calculator to do the homework problems. We will not allow the use of any calculators on the exams as this puts people with different types of calculators at a disadvantage, as well as ends up testing how well you can use a calculator instead of how well you've learned the basic concepts. We will make sure that the problems on the exams require minimal calculation, to allow you to spend your time demonstrating your mathematical knowledge, not your calculating ability.
Remember to enjoy yourself this semester - often students take math because they feel that they have to take it. Although many of you might think you know exactly what you will be doing with your life, it is rarely that simple. You might think that your current plans do not require you to have to know much math, but who knows, you might end up working in a different subject one day and find yourself thankful to have learned the math you are about to learn. My wife majored in history, took the minimum number of mathematical courses required as an undergraduate, and then later on became a molecular biologist. She is now extremely thankful for the math that she did learn, as it quite useful for much of the biological modeling she does, and yet at the same time she regrets that she didn't take the time to study more math while she was in college. Who knows, after math 1b you might all want to become mathematicians!
General Course Information:
Course Head: Andy Engelward, Sci Ctr 435, 495-4744, engelwar@math.harvard.edu. Although it is likely that you will only see me during our large orientation meeting at the beginning of the semester, I am here to help ensure that the class runs smoothly for you. My main responsibility is to coordinate all of the sections of the class, so that they all run uniformly. To this end, you should feel free to contact me at any point during this semester if any issues arise, such as a family emergency, which might cause you difficulty in keeping up with the class. In general, you should contact your section leader first, to let them know what is going on.
Prerequisites: Before beginning this class you are expected to have had the equivalent of a first semester course in calculus. You should have experience with such concepts as limits, derivatives, logarithmic and exponential functions, Riemann sums and the fundamental theorem of calculus. If some time has passed since you've worked with these topics, then it is up to you to review the appropriate material - there will be very little time to do much review in class during the semester as there are a number of new topics to cover. Although it is expected that you will take the initiative to review any material you need to on your own, you should always feel free to talk to your section leader or course assistant for help.
Classes and Problem Sessions and Course Assistants: Math 1b is taught in sections which meet three hours per week. The philosophy behind the sections is that it is far better to work on math in smaller groups than in one huge, impersonal lecture setting. This gives you a better opportunity to ask questions in class, and interact with the teacher. Make sure you take advantage of this arrangement, and try to get the most out of these smaller groups. Any questions you ask in class will likely be ones that other students are having as well, so get over any hesitation you might have, and ask questions as the material is presented. Remember, the class is being held for you to learn the material, not just for you to copy notes off of the blackboard.
You will also be attending a problem session once per week led by a Course Assistant (a "CA"). Course Assistants grade homework and hold weekly problem sessions. They also attend classes alongside of you, so you will get to know them well during the semester. The problem sessions are an integral part of the course and will be devoted primarily to working problems as well as to amplify and review the material. You are strongly urged to attend these problem sessions - even if you find you're not having difficulty doing the homework problems, you should still make a habit of going to these sessions. A schedule of all of the problem sessions will be posted within the first two weeks on the math bulletin board on the first floor of the Science Center, as well as on the course web page.
Question Center: In addition to class, problem sessions, and office hours, the math department operates a Question Center in Loker on Sunday through Thursday evenings from 8pm to 10pm (it is generally assumed that you will have other things to do on Friday and Saturday night!) The Question Center will be staffed by Course Assistants and by graduate students and other teaching staff. You should feel free to drop in any time you want a bit of help, or if you just want to solidify your basic math understanding by doing some review problems.
Homework: There is no question but that the best way to learn math is by doing math. Homework exercises are an essential part of any math course. I know from personal experience that if you just go to a math class and watch the teacher work problems, but don't try doing any problems on your own, then there is very little chance you will actually learn what is going on. It is also extremely unlikely that you will do well on the exams without working through homework problems ahead of time! While doing homework, don't just crank through computations and write down answers - think about the problems posed, your strategies, the meaning of your computations, and the answers you get. The main point is not to come up with the specific answers to the specific problems you're working on, but to develop an understanding of what you're doing so that you can apply your reasoning to a wide range of similar mathematical situations. It is unlikely that later on in life you will see the exact same math problems you're working on now - so learn the material in such a way that you are prepared to use your general math knowledge in the future, not just particular formulas for very specific questions.
We encourage you to form study groups with other students in the class so that you can discuss the work with each other. Your Section Leader or Course Assistant will provide names and contact information for everyone in your section in order to help facilitate discussion. Although we encourage you to work together with your classmates, all work submitted must be written up individually. Make sure that even if you do work in groups, that you come away with the ability to explain everything you end up writing up in your homework. You're cheating yourself of learning if you just copy down someone else's answers.
Problems will generally be assigned in each class and due at the next class. Assignments will be graded by your Course Assistant and will typically be returned at the following class meeting. We will then post solutions to the homework on the math 1b web site (we might also post them on reserve in Cabot Library if students want this as well). Check the solutions so that you can learn from your work. To make it possible to post solutions as soon as possible, and in light of the fact that getting behind in a class is one of the most uncomfortable things you can do to yourself, homework must be turned in on time. We will make a general policy for the course of dropping the 5 lowest homework grades if you're in a MWF class, and 3 if you're in a T/Th class (i.e. just about 1 1/2 weeks worth of homework over the semester). Because of this, please do not try to harass your CA into accepting a late homework - the homework policy is a course-wide policy, and it would be unfair if certain CAs were lenient when others weren't. If you want to gripe about homework, please come see me instead! Although it may seem as though you're doing homework all the time at first, most students have been thankful in the end that the course was designed to keep you working throughout the semester - it means less last minute catching-up for you when you're getting ready for tests.
There will be times when problems for homework will look a bit different from problems discussed in class and in the text. This is not the result of your teacher not covering all the material in class. The point is that we want you to get used to thinking about the math and learning how to apply it in unfamiliar settings. If you don't get used to this now, then there will be little chance that you will feel confident about applying your math understanding to general situations later on in life, and the class will not have been as useful to you as it could have been. Test and exam questions will be similar to the problems which you have worked on in class and on homework, but only up to a point - they certainly will not simply be copies of problems you have already seen. Be prepared to spend time thinking during tests, not just spending your time busily write down memorized formulas.
Textbook - Our main course book this semester will be Anton, Calculus: Brief Edition, A New Horizon, available at the Harvard Coop. Towards the end of the semester we will supplement the material on differential equations with a small course packet, listed in the syllabus as Math 1b Supplementary Materials. We will announce when this course packet is available, and where you can buy it.
Tests: There will be several opportunities during the semester for you to show off your math knowledge. Tests are uniform across all of the math 1b sections. Because of this, we need to schedule them at a common time, something which is practically impossible to do early during the day. As a result the two midterms are scheduled in the evening. It is your responsibility to let your section leader know as soon as possible of any potential conflicts. It is also generally the case that it is your responsibility to try to resolve any conflicts - there are only two of these evening tests during the semester, and they should take precedence over other obligations that you might have.
First Test: Wednesday, Oct. 27th, from 7:30 to 9:30 pm in Science Center auditoriums C and D
Second Test: Wednesday, Dec. 1st, from 7:30 to 9:30 pm in Science Center auditoriums C and D
Final Exam: tentatively scheduled for Friday, January 21st, rooms TBA
Grading: The weights of the various parts of the course are as follows:
Test #1: 20% Test #2: 20% Homework: 20% Final Exam: 40%
Typically we don't set absolute point value levels ahead of time (i.e. 92 and above equals A, etc.). The reason for this is to take into account the fact that the course and the tests vary somewhat from year to year, and it would be unfair to penalize the class if it turned out that scores on a particular test were lower one semester from another. We will indicate after each test a rough range of grade equivalence, so that you can keep track of how you are doing in the course.
And finally, a syllabus! Note that this is a tentative syllabus, if there are any changes, your section leader will let you know.
Math 1B - Second Semester Calculus - Tentative schedule for Fall 99
First Topic: Infinite Series, Power Series, Taylor Series
Sept. 27 (28) 29 (30) Oct. 1: (MWF classes are in bold, T/TH are in parentheses)
Introduction to semester, in Anton, §7.4 Summation notation for series, §11.3 Partial sums of series, convergence, §11.4 Divergence, integral tests and p-series
Oct. 4 (5) 6 (7) 8:
§11.6 Comparison test, ratio test, §11.7 Alternating series, §11.8 Introduction to power series
Oct. (12) 13 (14) 15 18: (Columbus day holiday on Monday, Oct 11)
§11.8 Power series, continued, §11.5 Maclaurin Series, Taylor series, §11.10 Differentiating/integrating power series, Calculating new Taylor series from previous ones
Oct. (19) 20 (21) 22 25 (26) 27:
§11.9 Convergence of Taylor series, Review for first Exam,
Second Topic: Integration Techniques, Applications of Definite Integral
Brief introduction to new topic (§7.5 and §7.6) before first exam (will not be included on first midterm)
First Midterm - Wednesday Oct. 27nd, 7:30 - 9:30 pm, Science Center auditoriums C and D
Oct. (28) 29 Nov. 1 (2) 3:
§7.3 Integration by substitution, §7.8 Definite integrals by substitution, §9.2 Integration by parts, §9.5 Partial fractions
Nov. (4) 5 8 (9) 10:
§9.8 Improper integrals, §8.2 Volume integrals (by rotation and "pop-up"), § 8.4 Lengths of plane curves
Nov. 12 15 (16) 17 (18): (Veteran's day on Thursday, Nov. 11)
§8.5 Area of a surface of revolution, §8.6 Work problems, §8.7 Fluid pressure and force problems
Third Topic: Differential Equations and Systems of Differential Equations
Nov. 19 22 (23) 24 (Thanksgiving break) 29 (30) Dec.1
§10.1 First-order differential eq.s, with simple applications, separation of variables, integrating factors, §10.2 Direction fields (Euler's method), Review for second midterm
Second Midterm - Wednesday Dec. 1st, 7:30 - 9:30 pm, Science Center auditoriums C and D
Dec. (2) 3 6 (7) 8:
§10.3 Modeling with differential equations, general applications, logistic growth, radioactive decay, exponential growth, §3.0 to §3.3 in the Math 1b Supplementary Materials packet, Qualitative analysis of differential equations, equilibrium solutions
Dec. (9) 10 13 (14) 15:
§3.4 in Math 1b Supplementary Materials More qualitative analysis, §3.5 Systems of differential equations
Dec. (16) 17 20 (21):
§3.6 in Math 1b Supplementary Materials Systems of differential equations continued, §3.7 Analyzing systems of differential equations, §3.8 Power series solutions to differential equations
Reading Period, Jan. 3rd - 14th, Final Exam on Friday, Jan 21st.
Coursewide review sections, office hours TBA