Math 131: Calculus II

Course website: http://math.hws.edu/~mitchell/Math131S09/index.html where I will post (all) course documents, including solution sets. Please bookmark and use this site.

About Math 131

First-year calculus is usually divided into two parts (terms): differential calculus and integral calculus. Briefly, differential calculus is concerned with rates of change ("the slope problem") and integral calculus is concerned with area, in particular the area in the plane enclosed by arbitrary curves. The Fundamental Theorem of Calculus shows how these different concerns are essentially the two faces of a single coin. Having completed a "tour" of differential calculus, you are now prepared to focus on integral calculus.

To prove the Fundamental Theorem of Calculus requires us to connect the process of antidifferentiation to the notion of area under a curve. You know how to find the areas of some regions: squares, rectangles, triangles (using triangles you can figure out the area of any polygon), and circles. While you might be able to justify the area formula for a rectangle, it is unlikely that you could give a satisfactory proof for the area formula of a circle. In fact, there is a more fundamental problem here: What is area?

In differential calculus, the motivating problem was the paradoxical notion of an "instantaneous" rate of change. What do we mean by speed "at an instant?" Average rates or speeds are familiar ideas: divide distance by time. But in an instant, no time passes and no distance is travelled. So there is no (rate of) change! We resolved the paradox by using average rates of change over smaller and smaller time intervals. In the end, the instantaneous rate of change (or derivative) was defined to be the limit of average rates of change. But note, the concept of an "instantaneous" rate had to be defined through a fairly long process and was not "obvious."

In the same way, area, though familiar, is not an obvious notion. Some area formulas are familiar, but what is area? What are its defining characteristics? We will start with this problem and see that its solution has a wide variety of applications. In learning how to find area, we will also learn how to find or define the length of a curve, the volume of a solid, the work done by a force applied over a distance, and so on.

As with derivatives, limits will be crucial to the solution of the "integral" or area problem. This time the "paradox" will be that we add up lots of (i.e., an infinite number in the limit) of small (0-sized in the limit) areas to obtain the area of a figure. There are lots of questions to resolve: How do you add an infinite number of things? How do you divide a region up into smaller pieces whose area you know? Answers to such questions are what will motivate our "definition" of area.

Goals and Outcomes

You should think of Calculus I and II as a tightly linked sequence of questions and ideas. There are several goals for the Math 130 and 131 sequence. On the theory side these include: (1) displaying familiarity with the key steps in the development of the derivative and the integral by using two different notions of the limit, (2) displaying a familiarity with the fundamental relation between the integral and the derivative, and (3) displaying an understanding of the main theoretical results of the subject (e.g., the Mean Value Theorem and the Fundamental Theorems of Calculus). On the practical side, objectives include (1) classifying functions (e.g., those having limits, continuous, differentiable, or integrable), (2) classifying the critical points of functions and applying this classification to solve extremization problems, (3) interpreting the meaning of the derivative and the integral, (4) displaying a knowledge of basic differentiation and integration techniques, (5) adapting known techniques (e.g., finding area under curves) to new situations (e.g., finding volumes or lengths of curves), (6) justifying through written work the application of theorems and techniques to particular problem-solving situations.

Text

The Department of Mathematics selected the Larson, Hostetler, and Edwards' calculus text for this course for a variety of reasons. The text is concise and yet still readable. It focuses on the concepts of calculus, not just the algebraic and formulaic manipulations. The central ideas or theorems of the course require justification or "proof." Though this course will not be overly theoretical, there will be several arguments or proofs that I want you to understand. I will help you with them, but carefully read through them in the text. It's a very "readable" text, so please spend lots of time with it! Long after most of you have forgotten the "rules of integration," I hope you still remember what the process of integration is and how it is used.

Labs

Every Thursday our section will meet for a problem solving laboratory. Attendance is required. These labs are an excellent time for you to ask questions about the course material. Bring your text and notes. Most students find lab sessions very helpful. For the last 10-15 minutes of some labs, there may be an open book, open note quiz based on the work you have done that day and which you must complete on your own. Selected Problems from each lab may be collected and graded.

Assessment

Homework, reading, and practice exercises will be assigned at the beginning of each class. I encourage working in small groups on practice problems. This can be very helpful in understanding the material.

Once or twice a week, there will be an assignment consisting of selected problems to hand in for grading. In the past, I have insisted that graded assignments are to be your own work without collaboration. But this term, I will try something different. You may work on graded homework assignments with others. Verbalizing your questions, explaining your mathematical ideas, and listening to others will increase your understanding. BUT you are not free to simply copy other's work or to make your work available in this way to others. You will be heavily penalized for such instances. Make sure you write up your answers on your own. I reserve the right to change this policy if it is abused. (Note: Other faculty will have different policies about homework.) Your work will be due at the beginning of the next class. It should be neat and done in pencil. No late assignments, please.

I may also use a few announced 10-minute quizzes to check on your progress in the course.

There will be three in-class tests and a final exam. The dates are listed in the outline below. Tests will be cumulative but will concentrate on more recent material. It is impossible to construct fair makeup exams in mathematics. For your own protection, my policy is that there are no makeup examinations. If for some extraordinary reason you find you are unable to take an exam, let me know as soon as possible, certainly well before the exam is administered.

Your course grade will be calculated as follows. First I will make a list of your grades:

Combined Homework and Quiz Scores, Test 1, Test 2, Test 3, Final Exam, Final Exam.
Note: The final exam is listed twice. Next, I will toss out the lowest score. (If the final is your lowest grade, it is removed just once.) Then I will average the remaining five grades. I also reserve the right to take class participation and attendance into account in determining final grades.

Extra Credit: There is an opportunity to earn extra credit by attending Mathematics and Computer Science Department seminars. You must write up a half-page reaction to the seminar to receive credit. (You may recieve a maximum of three such credits.)

Attendance and Courtesy

Because of the nature of this course, its assignments, and its assessment, your attendance and participation is crucial. Mathematics is learned by regular, sustained, attentive effort over an extended period. Only when such effort has been invested will the concentrated study for an exam have any benefit. Therefore, attendance at class is required. Unexcused absences may adversely affect your grade; certainly more than three absences will lower your grade. I use the following split function to determine how your grade is affected by the number of absences, where x is the number of classes missed. See what happens if you miss 0, 3, or 5 classes.

Final Grade(x) = Computed Grade + 1.5 - 0.5x, if 0 <= x <= 3
                 Computed Grade + 6.0 - 2.0x, if x > 3.

If you must miss a class for some reason beyond your control, talk to me about it in advance.

Please help me out the first week or two of class by reminding me of your name when you ask a question in class or lab. It will help me (and your classmates) to learn your names more quickly.

Finally, common courtesy demands that you be on time for class and that you do not leave the room during class (unless you are ill). This will help you, your classmates, and me to give our full attention to the course.

Expectations and Prerequisites

Calculus I is a prerequisite for this course. If you did not pass Math 130 or its equivalent with a grade of C- or better, this course is inappropriate for you and you should see me immediately. I assume you understand how to use the notions of limit and derivative and are familiar with their applications and interpretations. You should be familiar with basic antidifferentiation, as in Section 5.1 of the text. To be successful in Math 131, you must be able to build on a solid understanding of differential calculus. If you are concerned about your background, see me immediately!

I expect that you will put in at least two to three hours of work outside of class for each hour in class, including lab. This should take the form of reading the text, reviewing class notes (recopy if necessary), finishing lab problems and checking answers online, doing practice problems, and then doing assigned homework problems. This effort will be rewarded by making the exams rather easy.

Math Intern

The Math Intern, David Brown, is available on the third floor of Lansing Hall for extra help in the late afternoon and the evenings. His office hours are listed at the beginning of this document. Please utilize this extra resource.

Tips for Success

Here are a few simple things that you can do to be more successful in the course.

Come to all classes.
Take complete and careful notes.
Do the readings and practice problems carefully.
Keep the answers to problems and notes on the readings in a journal.
Review the notes from the last class before coming to class again. In fact, I suggest recopying the notes! This makes studying for a test easy.
Come in for help whenever you need it, preferably before you get too far behind.
Visit the Math Intern for help in the evenings.
Review solution sets online at the course website.
Review some of the lab questions once a week.
Ask questions in class. Ask questions about the homework and readings.

Outline of Weekly Readings

This assumes a fairly rapid pace through the listed materials. We may need to adjust this schedule based on our actual work in class.

Week 1: Introduction to the Area Problem, Brief Review of Antiderivatives, Summation Notation. 5.2. On you own: Review 5.1 and the Mean Value Theorem (page 214; make sure you understand the figure).
Week 2: Reimann Sums and the Definite Integral. Chapter 5.2-5.3.
Week 3: The Fundamental Theorem of Calculus. Integrals using Substitution. Chapter 5.4-5.5.
Week 4: The Natural Log Function. Chapter 5.7.
Friday, February 13: First Hour Test @ 7:40 AM.
Week 5. Integrating Inverse Trig Functions. Application: Area Between Curves. Chapter 5.8, 7.1.
Week 6: Applications: Volume by Slicing, Arc Length. Chapter 7.2, 7.4
Week 7: Application: Work. Techniques of Integration: Integration by Parts. Chapter 7.5, 8.2. (On your own: Review of Basic Integration Rules. Chapter 8.1.)
Week 8: Trig Integrals (briefly) and Trig (Triangle) Substitution. Chapter 8.3, 8.4, and Handout.
Thursday, March 12: Second Hour Test in Lab.
Week 9: Partial Fractions (briefly) and Improper Integrals. Chapter 8.5 and Chapter 8.8.
Week 10: Sequences, Series, and Convergence. Chapter 9.1-2.
Week 11: The Divergence Test, Integral Test, and Comparison Test. Chapter 9.3-9.4.
Week 12: Alternating Series Test, Absolute and Conditional Convergence, the Ratio Test. Chapter 9.5-9.6.
Friday, April 17: Third Hour Test @ 7:40 AM.
Week 13: Power Series: Interval and Radius of Convergence, Differentiation and Integration. Chapter 9.8.
Week 14: McLaurin and Taylor series (excluding error estimation). Chapter 9.10.
Week 15: Catch-up and Review.
Final Exam: May 11, 2009 at 1:30 PM.

A Note about the Center for Teaching and Learning (CTL)

Hobart and William Smith Colleges encourages students to seek the academic collaboration and resources that will enable them to demonstrate their best work. Students who would like to enhance their study skills, writing skills, or have other academic inquiries should contact the CTL. You may visit the CTL web site to learn more about the services and programs that are available: http://www.hws.edu/academics/ctl/

If you are a student with a disability for which you need or may need accommodations, you should self-identify and register with the Coordinator of Disability Services at the Center of Teaching and Learning. You will then be required to provide for review documentation of your disability to that office. Disability related accommodations and services generally will not be provided until the registration and documentation process is complete. Guidelines for documenting disabilities and information pertaining to registration with the CTL can be found at our website: http://www.hws.edu/disabilities

If you have a question about this process or Disability services at HWS, please contact David Silver (Coordinator of Disability Services - CTL) at silver@hws.edu or x3351.