Introduction
The goal of Math X is to provide you with a deep understanding of topics from precalculus and calculus as well as a strong sense of how useful mathematics can be and how you can apply mathematics in your work and studies. Our aim is to provide you with a solid set of mathematical skills and a high degree of mathematical confidence when you finish the course so that you will be well equipped for future studies in mathematics, biology, chemistry, economics, or other disciplines. To help you achieve these goals, we use multiple approaches to problem solving, and we stress understanding the ideas behind the mathematical formulas and techniques that you learn.
The Math Xab sequence covers all of the material learned in Math 1a, plus a little extra material to help prepare you to take Math 1b. In terms of prerequisites for the other courses at Harvard, Math Xab is equivalent to Math 1a.
In Math Xb, we continue to integrate topics from calculus and precalculus. We begin the semester by talking about implicit differentiation, and then we review trigonometry and do differential calculus with trigonometric functions. We then begin to develop integral calculus, which allows us to find the net change in a function given only the function's rate of change. Finally, we explore differential equations, a rich field of inquiry which can lead to a mathematical understanding of the interactions between populations of predators and prey, the spread of disease, the pollution in the Charles river, the way a neuron in your brain fires, and the way an apple pie cools.
Course Goals
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You should gain an understanding of the important concepts and techniques associated with functions and calculus.
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You should gain an appreciation of the role of mathematics in the natural and social sciences and in the modern world.
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You should gain skills in mathematical reasoning, in mathematical modeling, in numeric and symbolic computation, and in learning and communicating mathematics.
Textbook
Robin J. Gottlieb. Calculus: An Integrated Approach to Functions and Their Rates of Change, Preliminary Edition. Addison Wesley, Boston, 2002. In Math Xb, we will cover most sections in Chapters 17 through 26, and 31. The textbook is available at the Harvard Coop.
We will generally require that you read the material in the textbook before we discuss it in class, as opposed to reading it after the class. We don't expect that you will understand everything in the material, but we do expect you to come to class prepared to discuss what you have read, with questions about what you do not understand.
Course Procedures
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Classes
Math X is taught in small classes in order to provide an environment where you get to be an active participant, and engaging in dialogue with your fellow students and instructor. To enroll in Math Xb, you must section by computer starting Monday, January 30th, and no later than 12:00 PM on Thursday, February 2nd. Note that you can re-section as often as you like during this period and your previous entry will be erased. You can find directions for sectioning and information about calculus advising at http://abel.math.harvard.edu/sectioning/index.html.
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Workshops
In addition to classroom instruction, you will be working on in-depth problems with other students during weekly 90-minute Workshops led by a former Math X student or an undergraduate Course Assistant. The Workshop problems are directly related to the course material will often be similar to the more difficult exam problems. The Workshop sessions will meet either on Tuesday or Wednesday this semester -- more information will be given in class.
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Problem Sessions
Course assistants will lead weekly problem sessions. The problem sessions are optional, but highly recommended. They will give you a chance to explore the homework problems with the CA and other students, and to review material that has been discussed in class. This semester gateway exams will be given during problem sessions.
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Getting Help
If you have questions about any aspect of the course, feel free to ask your section instructors, either by email or in person during their office hours. Additionally, students in every section of Math X are welcome to ask questions of the course head, either by email or in person during office hours.
The Math Question Center is held from 8 to 10 p.m. on Sunday through Thursday evenings during the term. The MQC is staffed by calculus course assistants, and it is a great place to work on homework with other Math X students and get help from the CAs. For more on the MQC, consult the MQC website.
You are also encouraged to form study groups with other students in the course to work on homework assignments. Working in study groups is one of the most effective strategies you can use to succeed in this or any other mathematics course. All homework assignments must be written up individually. If you do work with a study group, make sure that you are able to explain everything you write up in your homework assignment.
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Absences
In general, if you miss a class, you are still responsible for turning in homework to your course assistant on time. Exceptions will be granted only with the course head's approval.
If you miss a class or lab due to a Harvard-sponsored athletic or extracurricular event, you are responsible for turning in early any homework or lab assignments due during your absence. Exceptions will be granted only with the course head's prior approval.
If you cannot complete or turn in a homework due to illness, let the course head know. The course head may allow you to drop that assignment provided you present appropriate documenation of your illness from University Health Services, your physician, or your Assistant Freshman Dean or Senior Tutor.
Graded Work
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Homework
Generally, there will be homework due each class period, although the assignment due on Wednesdays will be somewhat lighter due to the workshops. Homework is due by 3pm on on its due date. There are no exceptions to this, so be sure to put any homework you want graded in the grader's box by 3pm. Homework will be returned, in general, at the next class period. We will drop your lowest three HW scores.
Some of the problems you do will look different from problems discussed in class or the examples done in the textbook. This is not an accident. We want you to think actively about the material, to be able to apply it in unfamiliar settings, and to interpret it in different ways. Therefore we will not give you a recipe for solving every problem. Your job is to accept this challenge, a challenge we plan to help you meet.
Discussing mathematics with others is a very effective way to learn, and we encourage you to work with other students on the homework. However, any work you submit should be written up on your own; you should be able to explain and develop anything you have done in a group on your own (after all, you won't be able to talk with your study partners during the exams!).
You can find a list of all the homework assignments up to the current week on the course web site.
Because we want you to learn how to communicate about mathematics in this class, we ask that you explain your answers to homework problems clearly. You should try to write your homework problems so that a fellow student could follow your solution, even if he or she is a bit confused about the material. Your homework problems will be graded according to the following rubric.- 3 pts
Work is completely accurate and essentially perfect.
Work is thoroughly developed, neat, and easy to read. Complete sentences are used. - 2 pts
Work is good, but incompletely developed, hard to read, unexplained, or jumbled. Answers which are not explained, even if correct, will generally receive 2 points.
Work contains ``right idea'' but is flawed. - 1 pts
Work is sketchy.
There is some correct work, but most of work is incorrect. - 0 pts
Work minimal or non-existent.
Solution is completely incorrect
- 3 pts
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Participation
We want students to participate in class so that they can learn from each other and learn to communicate about mathematics. We know from educational research that active involvement in learning increases what is remembered, how well it is assimilated, and how the learning is used in new situations. In making statements to peers about their own thoughts on a class topic, students must articulate those thoughts. In listening to their peers, students hear many different ways of interpreting and applying class material, and thus are able to integrate many examples of how to use the information.
Participation is graded on a scale from 0 (lowest) through 4 (highest), using the criteria below. The criteria focus on what you demonstrate and does not presume to guess at what you know but do not demonstrate. This is because what you offer to the class is what you and others learn from. Our expectation is that the average level of participation should satisfy the criteria for a "3".
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Absent. - 1
Present, not disruptive.
Tries to respond when called on but does not offer much.
Infrequent involvement in discussion.
Is off-task during times when class is working individually or in small groups. - 2
Demonstrates adequate preparation: keeps up with the lecture, demonstrates some knowledge of night's reading.
Does not offer to contribute to discussion, but contributes to a moderate degree when called on.
Demonstrates sporadic involvement.
Is on task during times when class is working individually or in small groups, but communicates with other group members only minimally. - 3
Demonstrates good preparation: has clearly done the previous night's reading and thought about the material.
Asks questions which are relevant to what is happening in class.
Demonstrates consistent ongoing involvement in class discussions: responds to other students' points, offers to contribute to discussion without being called on.
During group work communicates with other members of group and contributes to the group's understanding. - 4
Demonstrates exceptional preparation: contributes thoughtful comments or questions about the previous night's reading.
Contributes in a very significant way to ongoing discussion: responds very thoughtfully to other students' comments, asks thoughtful questions, suggests alternative ways of approaching material and helps class analyze which approaches are appropriate, etc.
Demonstrates ongoing very active involvement without dominating class discussions.
During group work is an active participant: asks and answers questions of other group members, helps to keep the other students in the group involved.
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Project
This semester, students will be required to learn an advanced calculus topic and to prepare a lesson on the topic. The topics will be assigned, then the group will learn the topic together and meet with the instructor to assess their understanding. Later in the semester, the group will turn in an outline of what they plan to do for the lesson, and then the lesson will be presented during reading period. More details will be given in class.
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Midterms
Two midterm examinations will be given during Math Xb, each covering roughly one-third of the course material. For more information about the midterm exams, see the exams portion of the course website.
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Final Exam
A cumulative final exam will be given at the end of the term for Math Xb, as scheduled by the FAS Office of the Registrar.
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Gateway Exams
Gateway exams are meant to test your grasp of certain fundamental pre-calculus and calculus techniques and concepts. Each exam consists of eight questions covering the topics listed in the table below. To pass a gateway exam, you must answer at least 7 of the 8 questions correctly. Before each exam, you will be provided with a collection of practice problems on the website (complete with answers) that will be similar to the problems on the gateway exam.
Each gateway exam is first administered at the problem session listed on the table below. If you fail the gateway exam at this time, you must contact your teaching fellow to schedule a retake. You may retake a gateway exam as many times as you need to until you pass. You also must participate in a mandatory problem session to review the gateway material. For each gateway exam that you have not passed by the deadline listed below, your final grade for the whole course will be reduced by one letter. For example, if you have a B in the course but fail to pass one of the gateway exams by the deadline, your course grade will be reduced to a C. However, students who make an earnest effort to pass the gateway exams have never failed to pass the exams.
Note that calculators are not allowed on the gateway exams. Details on each exam can be found on the gateways section of the website.
Gateway Exam Topic Date Deadline for Passing Gateway 1 Differentiation and Advanced Algebra Problem sessions on February 15th or 16th Friday, March 17th Gateway 2 Trigonometry Problem sessions on March 8th or March 9th Friday, May 5th
Grading Scheme
Your course grade will be determined as follows. Note that we will have a "ressurection policy" again in Math X this semester. If your final exam is higher than either of your midterm scores, the final exam score will replace your lowest midterm score.
| Homework and Quizes | 13% |
| Projects | 10% |
| Workshop Attendance | 5% |
| Class Participation | 2% |
| Midterm 1 | 20% |
| Midterm 2 | 20% |
| Final Exam | 30% |
Your numerical score will be converted to a letter grade according to the following scale.
| Score | Grade |
| 90-100 | A |
| 80-89 | B |
| 65-79 | C |
| 50-64 | D |
| 0-49 | E |