Math Xa           Introduction to Functions and Calculus         Fall 2000
   Course Orientation Information and Preliminary Syllabus

Welcome to Math X!
Math X aims to provide you with a deep understanding of both calculus and topics from precalculus. We want you to leave the course with a solid set of mathematical skills and a high degree of mathematical confidence so that you are well-equipped for future studies, whether in mathematics, biology, chemistry, economics, or other disciplines.  In order to do this we will use multiple approaches to problem solving and we'll stress understanding the ideas behind the math formulas that we learn.

The sequence Math Xa/Xb covers all of the material in Mathematics 1a, and so in terms of prerequisites for other courses at Harvard, Math Xa/Xb is considered equivalent to Math 1a.  In fact, in the second semester, Math Xb, we will also cover some of the material from Math 1b as well: geometric series (of particular interest to those pursuing economics) and differential equations (with biological and economic applications).

In Math X we will thoroughly integrate calculus and precalculus material.  We'll start discussing the ideas of calculus immediately, weaving in background material as we need it.  This gives us an entire year to reinforce the most important ideas and skills of calculus and its prerequisites by working with them in numerous contexts.  The span of one year gives us enough time to work on eliminating any mathematical weak spots you might feel you have and to help you to become familiar with and draw on your strengths.

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 time to study more math while she was in college.  Who knows, after Math X you might all want to become mathematicians!

Format of the Course:
Math X is taught in small classes in order to provide an environment where students are active participants and dialogue is promoted between the teacher and the class and between students. Small class size also allows us to tailor the classes to your needs and to offer more individual attention.

There will be twice-weekly math lab sessions. These labs are designed to focus both on problem solving and on conceptual understanding.  If you think of mathematics as a language, you can think of the labs as language labs where you work towards fluency; if you think of mathematics as a science, you can think of the labs as science labs where you work on problem solving. Labs will be coordinated with the classes, allowing you to get new perspectives on topics presented in class by working on carefully chosen problems in a group setting with your fellow students directed by your course assistant.

Regarding the use of calculators, as a general policy, we encourage 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 you won't need to use a calculator.  There will be times when the computations involved will be vastly simplified by using a calculator and so we expect that everyone will have access to a good graphing calculator during the semester.  For the exams we will try to design the problems so that they require minimal calculation, to allow you to spend your time demonstrating your mathematical knowledge, not your calculating ability.

Section Leaders:

Coursehead:
Andy Engelward (contact information given above).  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.

Textbook:
The textbook for this course can be purchased at the Gnomon Copy Center across from Widener Library, next to Toscannini's Ice Cream on Mass. Ave (be aware that there are three different Gnomon Copies in Harvard Square, so be sure to go to the right one!)

Homework:
There is no question 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 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 particular 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 highly encourage you to form study groups with other students in the class so that you can discuss the work with each other.  Early during the semester your Section Leader 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 Xa 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 your 5 lowest homework grades from the semester.  Because of this, please do not try to harass your CA into accepting a late homework - the homework policy is course-wide, 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 somewhat 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.

Additional Resources:
Harvard has many resources to help you learn.  Certainly one of the most important resources to turn to during the semester is your Section Leader, and you should feel free to come to your Section Leader's office hours whenever you want to.  Additional help will be available at the Math Question Center (MQC).  The MQC meets in Loker (basement of Memorial Hall), every evening, except Friday and Saturday, from 8pm to 10pm.  It will be staffed principally by Course Assistants from calculus courses.  The MQC is also an excellent place to meet other students and work in study groups.  Anyone with questions about the MQC should contact John Mackey (jfm@math.harvard.edu).

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 Xa 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.
 

Grading Policy:
The amount that the various parts of the course count towards your semester grade is as follows:
 

* Your higher scoring midterm will be worth 20%, the other will be worth 15%

Lab scores will be based on a combination of attendance and scores on lab quizzes and gateway tests that will be given periodically during labs (there will probably be about four or five of these offered during the semester).  Gateway tests are designed to test basic math skills that you really need to master (during Xa these will mostly involve precalculus topics such as algebraic manipulation and graphing).  Questions on gateways will be relatively short and direct, and you will either pass a gateway test or take variations of the test until you pass.  We will tell you in advance what will be covered on quizzes and gateway tests.

If you do really well on the final exam and your final exam score is substantially higher than all of your other scores, then your section leader may weigh the final exam to count more than 35% of your final semester score.

On behalf of all of the section leaders and course assistants I again want to welcome you to Math X!  We are all looking forward to working with you to make this a great math year for all of you!

Course Outline for Mathematics Xa:

• Introduction to the basic ideas of calculus: rates of change
• Introduction to modeling: functions, graphs, and mathematical language
• Creating new functions from old functions: shifting, shrinking and stretching function graphs
• Linear functions and local linearity: models of quantities that change at a constant rate, constructing and interpreting linear and piecewise linear models
• Introduction to differential calculus: the derivative and its interpretation as a slope function
• Quadratic functions: their graphs and derivatives
• Limits and continuity indepth; intermediate and extreme value theorems
• More differentiation: product and quotient rules
• Exponential functions: models of quantities that change at a rate proportional to themselves, applications to biology and finance, derivatives of exponential functions
• Optimization: analysis of extrema, concavity and the second derivative
• Polynomial and rational functions: their behavior and derivatives
• Inverse functions: finding and interpreting inverses
• Logarithmic functions: working with logs to solve equations, differentiation of logarithmic functions
• Differentiating composite functions: the Chain Rule with applications
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Abbreviated Course Outline for Mathematics Xb:

• Geometric Sums, both finite and infinite, with applications to medicine and economics
• Trigonometry and the calculus of trigonometric functions
• Inverse trigonometric functions and their derivatives
• The definite integral - from a rate of change to total change: approximating sums - Riemann sums
• The definite integral as a limit of a Riemann sum
• The Fundamental Theorem of Calculus and indefinite integrals
• Applications of the definite integral
• An introduction to differential equations with an emphasis on modeling and qualitative analysis
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SECTIONING INSTRUCTIONS: If you have an email account, log onto the Harvard computer system, and instead of typing "pine," type "section." Otherwise, using any Harvard computer, select (telnet to) "fas," and when you are prompted to login, type "section." At the Password prompt, press Enter.

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