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Math S-Xab -
Introduction to Functions and Calculus
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Dr. Derek Bruff
Preceptor in Mathematics
Science Center 231
617-496-0459
bruff@math.harvard.edu
Dr. Thomas Judson
Preceptor in Mathematics
Science Center 429
617-495-5735
judson@math.harvard.edu
Evan Hepler-Smith
hepler@fas.harvard.edu
The goal of Math S-Xab 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.
Math S-Xab covers all of the material learned in Math S-1a. In terms of prerequisites for the other courses at Harvard, Math S-Xab is equivalent to Math S-1a.
Math S-Xab is also excellent preparation for the AP Calculus exam. High school students who take Math S-Xab in the summer often take Calculus AB or BC the following school year.
In Math S-Xab, we integrate topics from calculus and precalculus. We begin the course by discussing the ideas of calculus immediately, integrating the necessary background material into the course as we need it. This gives us the entire term to reinforce the most important ideas and skills of calculus and its prerequisites by working with them in numerous contexts. We have enough time to work on eliminating any mathematical weak spots that you feel you might have and help you to become familiar with and draw upon your strengths. Please work hard, but remember to take time to enjoy yourself this summer.
You should gain an understanding of the important concepts and techniques associated with functions and calculus.
You should gain an appreciation of the role of mathematics in the natural and social sciences and in the modern world.
You should gain skills in mathematical reasoning, in mathematical modeling, in numeric and symbolic computation, and in learning and communicating mathematics.
Upon successfully completing Math S-Xab, you should acquire solid foundation in the following topics and be able to move directly into Math S-1b.
| Functions and Their Graphs | Linear, polynomial and rational, exponential, logarithmic, trigonometric, and inverse functions. Operations on functions. Continuity and limits of functions. The Intermediate and Extreme Value Theorems. |
| Differential Calculus | The definition of the derivative. The derivative as a rate of change, as a linear approximation to a function. Optimization. Techniques of differentiation. Related rates. The Mean Value Theorem. |
| Integral Calculus | The definite integral, Riemann sums, and the Fundamental Theorem of Calculus. An introduction to techniques of anti-differentiation. Some applications of integration. |
Robin J. Gottlieb. Calculus: An Integrated Approach to Functions and Their Rates of Change, Preliminary Edition. Addison Wesley, Boston, 2002.
In Math S-Xab, we cover most sections in Chapters 1 through 25. The textbook is available at the Harvard Coop.
Math S-Xab is taught in a small class so that you have a better opportunity to ask questions and interact with your instructors. Make sure you take advantage of this arrangement. Any questions you ask in class are likely ones that other students want answered as well, so ignore any hesitation you might have and ask questions as the material is presented. You are not penalized for doing this, no matter how trivial or simple you think your questions might seem. Remember, the class is being held for you to learn the material, not just to give you a time to copy notes off of a blackboard, so be sure to get help when you need it and stay involved in your class.
Our course assistant, Evan Hepler-Smith, holds problem sessions on Mondays, Wednesdays, and Fridays from 1 to 2 p.m. in SC 112. The problem sessions are an important part of the course. They focus on working problems and reviewing course material. Even if you are not having difficulty doing the homework problems, you should still make a habit of attending these sessions.
You will find having a graphing calculator helpful in this course. We encourage you to bring your calculator to class with you every day. If you are purchasing a calculator, we recommend a Texas Instruments TI-83 PLUS calculator. You will be permitted to use your calculator on all assignments and exams, except for the gateway examinations and possibly one portion of the final exam.
With the availability of graphing calculators and mathematical software programs, it is now possible to do an amazing number of things almost instantaneously that would otherwise take hours or days to do by hand. Calculators can help you hone your mathematical skills and intuition by reducing time spend on computations and providing detailed graphs of functions. However, you should not rely on technology to such an extent that it keeps you from developing your own skills. It is easy to be misled by a calculator's results if you lack a good understanding of the associated mathematical concepts.
If you have questions about any aspect of the course, feel free to ask your instructors, either by email or in person during their office hours. Also feel free to ask Evan questions at the problem sessions or via email.
There's also the Math Question Center (MQC) on the 9th floor of the Science Center Sundays through Thursdays from 7 to 10:30 p.m. (Take the elevator to the 8th floor and then follow the signs.) The MQC is staffed by the Math S-Xab and S-Ar course assistants, and students in those two courses are encouraged to drop by the MQC to work on or ask questions about their homework.
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.
In general, if you miss a class, you are still responsible for turning in homework assignments to the course assistant on time. Exceptions will be granted only with the instructors' approval.
If you miss a class due to a Harvard-sponsored athletic or extracurricular event, you are responsible for turning in early any homework assignments due during your absence. Exceptions will be granted only with the instructors' prior approval.
If you cannot complete or turn in a homework assignment due to illness, let the instructors know. The instructors may allow you to drop that assignment provided you present them with appropriate documentation of your illness.
The best way to learn mathematics is to do mathematics, which means that homework problems are an essential part of any mathematics course. Attending class and watching your instructors work problems are important, but without attempting the problems on your own, there is little chance that you will learn the material.
It is important not only that you get the right answers on your homework but also that you understand how those answers are obtained. While working on homework assignments, pay attention to the strategies you use to solve the problems and the meaning of your computations. Use the homework assignments to study the material in such a way that you can apply the concepts and techniques to a variety of problems -- even problems that do not look like problems you have seen before.
Homework is assigned to be due most class meetings. Assignments are graded by the course assistant and are typically returned to you at the following class meeting. Make sure to go over your graded homework to make sure you understand any mistakes you make.
Since getting behind in your homework is a bad idea, late homework assignments are not accepted. However, we will drop your lowest three homework grades at the end of the course.
Three midterm examinations will be given during Math S-Xab according to the schedule below. Make-up exams will be administered only if a documented serious illness or personal tragedy prevents a person from taking an exam at the scheduled time.
| Midterm 1 | Friday, July 9th |
| Midterm 2 | Thursday, July 22nd |
| Midterm 3 | Thursday, August 5th |
A cumulative final exam will be given at the end of the term. The final is scheduled for Tuesday, August 17th, at 9 a.m.
Gateway exams are meant to test your understanding of certain fundamental pre-calculus and calculus techniques and concepts. Each exam consists of eight multiple-choice 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 (complete with answers) that will be similar to the problems on the gateway exam.
Each gateway exam is first administered in class on the dates listed on the table below. If you fail the gateway exam at this time, you must schedule a retake with one of your instructors. You may retake a gateway exam as many times as you need to until you pass. For each gateway exam that you have not passed by 5 p.m. on August 13th, 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 two of the gateway exams, your course grade will be reduced to a D. However, students who make an earnest effort to pass the gateway exams almost always pass them eventually.
Note that calculators are not allowed on the gateway exams.
| Gateway 1 | Algebra | Friday, July 2nd |
| Gateway 2 | Differentiation | Tuesday, July 20th |
| Gateway 3 | Exponentials and Logarithms | Tuesday, July 27th |
| Gateway 4 | Trigonometry | Monday, August 9th |
Your course grade will be determined as follows.
| Homework | 15% |
| Midterm 1 | 20% |
| Midterm 2 | 20% |
| Midterm 3 | 20% |
| Final Exam | 25% |
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 |
Page maintained by Derek Bruff
(bruff@fas.harvard.edu).
Last updated on
July 2, 2004.
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