Holy Cross Mathematics and Computer Science
Holy Cross Mathematics and Computer Science
MATH 136 (Calculus 2) is a second-semester college-level calculus course. It assumes that you have a solid grasp of the material covered in our MATH 135 (Calculus 1) course, namely differential calculus: limits, the definition and intuitive meaning of the first and second derivatives of functions of a real variable; derivative formulas for algebraic, exponential and logarithmic, trigonometric and inverse trigonometric functions; applications to straight line motion, related rates, curve sketching, and optimization problems.
If you have taken a year of calculus in high school and scored a 4 or 5 on the AB Calculus AP Exam, you should (almost certainly) start here. You would forfeit one AP credit by taking MATH 135 (Calculus 1) instead. In any case, if your high school calculus course was a solid one and you did well, then MATH 135 (Calculus 1) would be almost entirely review for you.
If you took the BC Calculus AP Exam and scored a 4 or 5, you are probably ``overqualified'' -- you should start in MATH 241 and you will forfeit one AP credit by taking this course.
Calculus 1 and 2 are required for several major programs at the college in addition to the mathematics major: Chemistry, Physics, Biology, and Economics. However, successful completion of MATH 136 alone satisfies the mathematics requirement for the Biology and Economics majors.
If you are taking this course only out of habit (i.e. because you have always taken mathematics), or to fulfill a Common Area Requirement, you are welcome, but you might also want to consider the MATH 110 (Topics in Mathematics) courses that we offer regularly.
If you have any questions about which calculus class is right for you, please feel free to consult with me, with our Chair, Prof. Steven Levandosky, with any other mathematics faculty member, or with your faculty adviser.
Calculus is the mathematics of change. First developed in the 17th century, it has been a major part of mathematics since that time. Calculus is important because it is the basis for a major portion of the science and technology that shape the contemporary world. Many of the techniques used to study motion of objects in physics, kinetics of chemical reactions, growth or decline of populations of organisms in biology, growth of national economies, and many other phenomena in the real world involve calculus. Although it might sound like an exaggeration now to say it, calculus is also one of the crowning achievements of the human intellect. You are in for an exciting journey of exploration as you learn it! Two men, Isaac Newton and Gottfried Leibniz, are given most of the credit for developing the calculus. Their contribution was primarily explaining the relation between finding the rate of change of a function (the derivative) and computing the "total accumulation" of a function over an interval (the definite integral). We will begin this semester by studying their big result -- the Fundamental Theorem of Calculus and then learn a number of applications of this theorem. The topics to be covered this semester are:
The objectives of this course are:
The text book for the course is Calculus (Single Variable, Early Transcendentals version), 3rd edition by Jon Rogawski and Colin Adams, Freeman/Macmillan (available in the H.C. bookstore, and elsewhere). We will cover most of the material in Chapters 5 - 10 this semester.
It is expected that Holy Cross students will have textbooks and other required class materials in order to achieve academic success. If you are unable to purchase course materials, please go to the Financial Aid office where a staff member will be happy to provide you with information and assistance.
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, several times 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 return them with comments, for all members of the group.
Several times during the semester the class will meet in the Haberlin 136 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. We will use Maple to produce graphs of functions, calculate numerical approximations to integrals, and compute symbolic derivatives and integrals of functions. 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!
The other meetings of the class will be structured as lectures 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 11:00 am every Monday, Tuesday, Wednesday, and Friday this semester. If attending class wasn't important, all college courses would be taught as MOOCs or by correspondence, and your tuition would be much lower!
Take notes and use them. This may seem obvious, but in my experience too many students seem to copy down everything on the board diligently, 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. Indeed, one technique that many students find helpful is to recopy the class notes, filling in details that might have been clear the first time, putting in extra examples, explanations or ideas that really helped them understand the material, and so forth.
Use the textbook and class notes actively. Don't just use them to look for worked problems similar to ones on the problem sets. Plan to look over the section to be covered each day before coming to class. That will make the lecture or other class activity more understandable. After the class, if things were not clear, 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. If necessary, make a list of questions to ask in office hours or at the next class. Another 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 -- that process is the real point, and that's what you should take away from studying the example.
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 some other science courses. In this course, however, a calculator with graphing and/or symbolic features will not be required and you will not be allowed to use one on exams when the goal is to make sure you know how to do certain things "by hand". You should obtain a basic scientific (i.e. non-graphing) calculator such as the TI 30X or TI 30XS for use on the exams this semester.
Grading for the course will be based on:
Important Notes:
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):
All education is a cooperative enterprise between teachers and
students. This cooperation works well only when there is trust and
mutual respect between everyone involved.
One of our main aims as a department is to help students become
knowledgeable and sophisticated learners, able to think and work
both independently and in concert with their peers. Representing another
person's work as your own in any form (plagiarism or ``cheating''),
and providing or receiving unauthorized assistance on assignments (collusion)
are lapses of academic integrity because they subvert the learning process
and show a fundamental lack of respect for the educational enterprise.
You will encounter a variety of types of assignments and examination
formats in mathematics and computer science courses. For instance,
many problem sets in mathematics classes and laboratory assignments
in computer science courses are individual assignments.
While some faculty members
may allow or even encourage discussion among
students during work on problem sets, it is the expectation that the
solutions submitted by each student will be that student's own work,
written up in that student's own words. When consultation with other
students or sources other than the textbook occurs, students should
identify their co-workers, and/or cite their sources as they would for
other writing assignments. Some courses also make use of collaborative
assignments; part of the evaluation in that case may be a rating of each
individual's contribution to the group effort.
Some advanced classes may use take-home
examinations, in which case the ground rules will usually allow no
collaboration or consultation.
In many computer science classes, programming projects are
strictly individual assignments; the ground rules
do not allow any collaboration or consultation here either.
It is the responsibility of faculty in the department to
lay out the guidelines to be followed for specific assignments in
their classes as clearly and fully as possible, and to
offer clarification and advice concerning those guidelines
as needed as students work on those assignments.
The Department of Mathematics and Computer Science upholds the
College's policy on academic honesty.
We advise all students taking mathematics or computer science courses
to read the statement in the current College catalog carefully and
to familiarize themselves with the procedures which may be
applied when infractions are determined to have occurred.
A student's main responsibility is to follow the guidelines laid down by the instructor of the course. If there is some point about the expectations for an assignment that is not clear, the student is responsible for seeking clarification. If such clarification is not immediately available, students should err on the side of caution and follow the strictest possible interpretation of the guidelines they have been given. It is also a student's responsibility to protect his/her own work to prevent unauthorized use of exam papers, problem solutions, computer accounts and files, scratch paper, and any other materials used in carrying out an assignment. We expect students to have the integrity to say ``no'' to requests for assistance from other students when offering that assistance would violate the guidelines for an assignment.
In this course, all examinations will be closed-book. No sharing of information with other students in any form will be permitted during exams. On group discussion and computer lab write-ups, close collaboration with the other members of your group is expected. On the individual problem sets, discussion of the questions with other students in the class, with the tutors in the Calculus Workshop, and with me during office hours is allowed, even encouraged. However, your final problem solutions should be prepared individually and the wording and organization of your final problem solutions should be entirely your own work. Moreover, if you do take advantage of any of the above options for discussion of problems with others, you will be required to state that fact in a "footnote" accompanying the problem solution. Failure to follow this rule will be treated as a violation of the College's Academic Integrity policy.
The following is an approximate schedule. Some rearrangement, expansion, or contraction of topics may become necessary. I will announce any changes in class and on the course homepage.
| Week | Dates | Class Topics | Reading (Rogawski/Adams) | |
|---|---|---|---|---|
| 1 | 8/31,9/2 | Areas | 5.1 | |
| 2 | 9/5,6,7,9 | The definite and indefinite integrals | 5.2 - 5.3 | |
| 3 | 9/12,13,14,16 | Fundamental theorem, integration by substitution | 5.4 - 5.7 | |
| 4 | 9/19,20,21,23 | Substitution, exponential growth and decay, Exam I Friday | 5.8 - 5.9 | |
| 5 | 9/26,27,28,30 | First applications: areas, volumes, average value | 6.1 - 6.3 | |
| 6 | 10/3,4,5,7 | Integration by parts, trigonometric integrals, trig substitution | 7.1 - 7.3 | |
| 10/10,11,12,14 | No class -- October Break | |||
| 7 | 10/17,18,19,21 | Partial fractions, strategies, tables | 7.5 - 7.6 | |
| 8 | 10/24,25,26,28 | Improper integrals, probability Exam II Friday | 7.7 - 7.8 | |
| 9 | 10/31, 11/1,2,4 | Center of mass, numerical integration | 8.3, 7.9 | |
| 10 | 11/7,8,9,11 | Differential equations | 9.1 - 9.3 | |
| 11 | 11/14,15,16,18 | More on differential equations | 9.4 - 9.5 | |
| 12 | 11/21, 22 | Sequences and series | 10.1 - 10.2 | |
| No class Wednesday 11/23, Friday 11/25 -- Thanksgiving break | ||||
| 13 | 11/28,29,30, 12/2 | Convergence tests, Exam III Friday | 10.3 - 10.5 | |
| 14 | 12/5,6,7,9 | Taylor series | 8.4, 10.6-10.7 |
A more detailed day-by-day schedule is posted on the course homepage.
The final exam for this course will be given at the established time for MWF 11:00am classes.