Holy Cross Mathematics and Computer Science
Holy Cross Mathematics and Computer Science
AP Calculus -- MATH 136 -- is the recommended starting mathematics course at Holy Cross for students who have taken year-long AP calculus courses in high school and have mastered the material well enough to score 4 or above on the Calculus AB Advanced Placement exam (or an AB subscore of 4 or above on the BC Advanced Placement exam). With that Advanced Placement score, you earn one semester course credit, and this is almost certainly the right course for you to take if you want to continue with mathematics at Holy Cross.
Even if you did not take an AP exam, if you took a year of a strong high school calculus and earned a grade of B+ or better, this might be the correct place for you to start. Look at the topics listed in the day-by-day course schedule at the end of this syllabus. If you feel pretty confident about the topics through mid-October, you are probably in the right place. The topics after that point will be new for most students for whom this is the right starting point at Holy Cross.
The department tries to see that every student ends up in a course that is appropriate for her/his abilities and background. There are situations where MATH 136 might be too much of a review for an ambitious and well-prepared student. If you scored a 4 or higher on the BC exam, or if your high school calculus course was much stronger than average (see topics in the course schedule at the end of this syllabus -- if all these look familiar, then your course was much stronger than average) and you did very well in it, you should consider starting with MATH 241 -- Multivariable Calculus.
There are other situations where students may be better off starting with Calculus for the Physical and Life Sciences 1 (MATH 131), the Intensive Calculus for Science (MATH 134) course, or even the Calculus for Social Science (MATH 125) rather than MATH 136. We expect a certain amount of movement between these courses, and we are prepared to be flexible even after the end of the formal add-drop period. Be aware that if you decide to take the option of ``moving back,'' and you will lose any advanced placement course credit you may have earned via an AP exam score because you will be repeating a large portion of the material from your high school course.
If you have any questions about which calculus class is right for you, please feel free to consult with me, with our chair, Prof. Catherine Roberts, or with any other member of the Mathematics section of our department.
The course will cover the following topics. (Also see the detailed course schedule at the end of this syllabus.)
The primary text book for the course is Single Variable Calculus: Concepts and Contexts, 4th edition by James Stewart (available in the H.C. bookstore -- among other places). We think you will find reading and studying this book to be challenging, but ultimately very rewarding.
In order for students to get as much as possible out of this or any course, regular active participation and engagement with the ideas we discuss are necessary. And working through questions in a group setting is a good way to develop a deeper understanding of the mathematics involved, if you approach the enterprise as a truly collaborative effort. That means asking questions of your fellow students when you do not understand something they say or think they do see, and being willing to explain your own ideas carefully when you see something that someone else does not. Looking to the future as well, working effectively in a group is also a valuable skill to have.
So with these points in mind, regularly throughout the semester the class will break down into groups of 3 or 4 students for one or more days, and each group will work individually for most of those class periods on a group discussion exercise. I will be responsible for designing and preparing these exercises, and I will be available for questions and other help during these periods. At the conclusion of each discussion, the class as a whole may reconvene to talk about what has been done, to sum up the results, to hear short oral reports from each group, etc. Each group will keep a written record of their observations, results, questions, etc. which will be handed in. I will make copies of these, and return them with comments, for all members of the group.
Other meetings of the class will be structured as lectures or computer demonstrations using the Maple computer algebra system when that seems appropriate.
A good "work ethic" is key. You do not need to be a "math genius" to master this material and do well. But you will need to put in a consistent effort and keep up with the course.
Come to class. Unless you are deathly ill, have a genuine family emergency, etc. plan on showing up here at 1:00 pm every Monday, Tuesday, Wednesday, and Friday this semester. If attending class wasn't important, all college courses would be by correspondence, and your tuition would be much lower!
Read the textbook. Don't just use it to look for worked problems similar to ones on the problem sets. You will find alternate explanations of concepts that may help you past a "block" in your understanding. Reading a math book is not like reading a novel, though. You will need to read very carefully, with pencil and paper in hand, working through examples in detail and taking notes. Make a list of questions to ask in office hours or at the next class. One thing to bear in mind while reading your text is that the result of an example is often secondary to the process used in obtaining the result.
Take notes and use them. This may seem obvious, but in my experience too many students diligently copy down everything on the board, and then never look at their notes again. Used intelligently, your notes can be a valuable resource as you work on problem sets and prepare for exams.
Set up a regular study schedule and work at a steady pace. It's not easy to play catch-up in a mathematics course, since every day builds on the previous one. You should expect to budget at least 6 hours in a typical week for work outside of class. The best way to use your time is to do a few problems, a little reading from the book, and reviewing of class notes every day.
Most importantly, if you are having difficulty learning something, get help as soon as possible. You can do this by asking questions during class (any time something isn't clear), seeing me during office hours, or attending the Calculus Workshop in the evenings Sunday-Thursday.
It is certainly true that a decent graphing calculator is a good investment for this and your other science courses. However a calculator will not be required and you will not be allowed to use one on exams where the goal is to make sure you know how to do certain things "by hand". The department has a supply of "basic" (i.e. non-graphing) calculators that will be provided for use on exams.
Grading for the course will be based on
I will be keeping your course average in numerical form throughout the semester, and only converting to a letter for the final course grade. The course grade will be assigned according to the following conversion table (also see Note below):
If you ever have a question about the grading policy or your standing in the course, don't hesitate to ask me.
The following is an approximate, evolving schedule. Some rearrangement, expansion, or contraction of topics may become necessary. I will announce any changes in class and via a revised schedule posted on the course homepage.
| Date | Class Topic | Reading (Stewart) |
|---|---|---|
| 9/2 | Course introduction | Chapter 1 |
| 9/4 | Review of the ``library of functions'' | Chapter 1 |
| 9/7 | Review of the ``library of functions'' | Chapter 1 |
| 9/8 | Rates of change | 2.1 |
| 9/9 | Limits and continuity | 2.2-2.4 |
| 9/11 | Limits involving infinity | 2.5 |
| 9/14 | The derivative | 2.6-2.7 |
| 9/15 | Meaning of the derivative | 2.8 |
| 9/16 | Computing derivatives (review) | 3.1-3.4 |
| 9/18 | Maple demo: differentiability = local linearity | |
| 9/21 | Implicit differentation | 3.5 |
| 9/22 | Derivatives of inverse functions | 3.6-3.7 |
| 9/23 | Related rates | 4.1 |
| 9/25 | Maxima and minima | 4.2 |
| 9/28 | Curve sketching with calculus | 4.3-4.4 |
| 9/29 | L'Hopital's Rule | 4.5 |
| 9/30 | Exam I (material through 9/23) | |
| 10/2 | Optimization | 4.6 |
| 10/5 | Antiderivatives | 4.8 |
| 10/6 | Riemann sums, the definite integral | 5.1-5.2 |
| 10/7 | Evaluating integrals | 5.3 |
| 10/9 | Maple for integration | 5.3 |
| 10/12,13 | Columbus Day Break -- NO CLASS | |
| 10/14 | The Fundamental Theorem of Calculus | 5.4 |
| 10/16 | Integrals by substitution | 5.5 |
| 10/19 | Integrals by parts | 5.6 |
| 10/20 | Integrals of trig functions | 5.7 and supplemetary materials |
| 10/21 | Integrals by trig substitution | 5.7 |
| 10/23 | Integrals by partial fractions | 5.7 |
| 10/26 | Tables and computer algebra systems | 5.8 |
| 10/27 | Improper integrals | 5.10 |
| 10/28 | More on areas; volumes by slices | 6.1-6.2 |
| 10/30 | Volumes by slices and shells | 6.2-6.3 |
| 11/2 | Average value | 6.5 |
| 11/3 | Applications in physics | 6.6 |
| 11/4 | Exam II (9/25 - 10/27) | |
| 11/6 | Maple demo: Arc length, numerical approximations | 6.4 |
| 11/9 | Differential equations | 7.1 |
| 11/10 | Direction fields and Euler's Method | 7.2 |
| 11/11 | Separation of variables | 7.3 |
| 11/13 | Exponential growth and decay | 7.4 |
| 11/16 | Other population models | 7.5 |
| 11/17 | Sequences | 8.1 |
| 11/18 | Series | 8.2 |
| 11/20 | Maple demo: modelling populations | 7.5 |
| 11/23 | Convergence | 8.3 |
| 11/24 | Power series | 8.5 |
| 11/25,27 | Thanksgiving Recess -- NO CLASS | |
| 11/30 | Taylor polynomials | 8.7 |
| 12/1 | Taylor series | 8.7 |
| 12/2 | Exam III (10/28 - 11/24) | |
| 12/3 | Maple demo: Taylor polynomials | 8.7 |
| 12/7 | Applications of Taylor polynomials | 8.8 |
| 12/8 | Semester wrap-up |
The final exam for this course will be given Friday, December 18 at 8:30 a.m.