The College of the Holy Cross
AP Analysis is the recommended starting mathematics course at Holy Cross for students who taken most 1-year calculus courses in high school. If you received a score of 3 or above on the AB Calculus Advanced Placement exam, you have been placed into this course automatically and given college credit for one semester of mathematics. Even if you did not take an AP exam, if you took a year of high school calculus and earned a grade of B or better, this is probably the correct place for you to start. If you scored a 3 or higher on the BC exam, or if your high school calculus course was stronger than average and you did well in it, you may want to consider starting with Mathematics 41 -- Analysis 3 (multivariable calculus). It is also possible, though not recommended, to start with Analysis 1 (Mathematics 31). Be aware that if you decide to take that option, you will be repeating a large portion of the material from your high school course, and you will lose your advanced placement credit. If you have any questions about which calculus class is right for you, please feel free to consult with me, or with any other member of the mathematics 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 Calculus by Deborah Hughes-Hallett, Andrew Gleason, et al. (available in the H.C. bookstore). We think you will find reading and studying this book to be challenging, but ultimately very rewarding. It is definitely not a standard calculus book --
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 will 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.
Some of the other meetings of the class will be structured as lectures when that seems appropriate.
Roughly once a week during the semester (mostly on Wednesdays or Fridays, see schedule), the class will meet in the Haberlin 408 PC laboratory for ``math lab'' classes. Each of these sessions will lead to a lab writeup assignment. (The due date will be announced when the assignment is given out.) We will be using an excellent, extremely powerful, general-purpose mathematical software system called Maple (the developers of the program come from the University of Waterloo in Canada, which explains the name). We will use Maple to produce graphs of functions, calculate numerical approximations to integrals, and compute symbolic derivatives and integrals of functions. But in fact Maple is so powerful that we will be using only a small fraction of what it can do. If you take additional mathematics courses at Holy Cross, you will probably use many other features of this same program. And programs like this are now a standard tool in many areas of science, engineering, even finance. Being able to use them effectively is also a valuable skill to have!
Grading for the course will be based on
The following is an approximate schedule. Some rearrangement, expansion, or contraction of topics may become necessary. I will announce any changes in class.
| Date | Class Topic | Reading (H-H, et. al.) |
|---|---|---|
| 9/2 | Course introduction; What is a function? | S 1.1, 1.2 |
| 9/3 | The ``library of functions'' | S 1.3,1.4,1.6 |
| 9/5 | Lab 1: Getting started with Maple | |
| 9/8 | The ``library of functions'' continued | S 1.9, 1.10 |
| 9/9 | Rates of change | S 2.1 |
| 9/10 | Formal definition of the derivative | S 2.2 |
| 9/12 | Lab 2: Differentiability = local linearity | S 2.2 |
| 9/15 | Computing derivatives (review) | S 4.1,4.2,4.3 |
| 9/16 | Computing derivatives, continued | S 4.4, 4.5, 4.6 |
| 9/17 | Lab 3: Applications of derivatives | |
| 9/19 | Critical points | S 5.1, 5.2 |
| 9/22 | Applications of derivatives | S 5.5, 5.6 |
| 9/23 | Taylor polynomials | S 10.1 |
| 9/24 | Lab 4: The error in Taylor approximations | S 10.5 |
| 9/26 | Exam I | (to 9/22) |
| 9/29 | The error formula | S 10.5 |
| 9/30 | Taylor series | S 10.2 |
| 10/1 | Lab 5: Taylor series | S 10.4 |
| 10/3 | New Taylor series from old | S 10.3 |
| 10/6 | Applications | |
| 10/7 | Total change of a function | S 3.1 |
| 10/8 | Lab 6: Approximating total change | S 3.2 |
| 10/10 | Riemann sums, the definite integral | |
| 10/13,14 | Columbus Day Break -- NO CLASS | |
| 10/15 | Fundamental Theorem of Calculus | S 3.4, 6.1 |
| 10/17 | Integrals by substitution | S 7.2, 7.3 |
| 10/20 | Integrals by parts | S 7.4 |
| 10/21 | Tables of integrals | S 7.5 |
| 10/22 | Integrals -- choosing a method | |
| 10/24 | Setting up Riemann sums | S 8.1 |
| 10/27 | More on Setting up Riemann sums | |
| 10/28 | Volumes by slices | S 8.2 |
| 10/29 | Lab 7: Arc length, numerical approximations | S 8.2 |
| 10/31 | Exam II | (to 10/27) |
| 11/3 | More applications of integrals | S 8.5, 8.6 |
| 11/4 | What is a differential equation? | S 9.1 |
| 11/5 | Slope fields | S 9.2 |
| 11/7 | Lab 8: Slope fields in Maple | S 9.3 |
| 11/10 | Separation of variables | S 9.4 |
| 11/11 | Growth and decay | S 9.5 |
| 11/12 | Other population models | S 9.6, 9.7 |
| 11/14 | Lab 9: Mathematical modelling | |
| 11/17 | Lab 9 continued | |
| 11/18 | 3D space, coordinates, distance | 2nd text |
| 11/19 | Lab 10: 3D plotting | " |
| 11/21 | Functions of several variables | " |
| 11/24 | Graphs of z=f(x,y) | " |
| 11/25 | More on graphs; critical points | " |
| 11/26,28 | Thanksgiving Recess -- NO CLASS | |
| 12/1 | Lab 11: Contours and critical points | " |
| 12/2 | The graphs z = Ax2 + Bxy + Cy2 | " |
| 12/3 | Spare day | |
| 12/5 | Exam III | (to 12/1) |
| 12/8 | SAC and semester wrap-up |
The final exam for this course will be given Tuesday, December 16 at 8:30 a.m.