Math 130: Calculus I

About Math 130

First-year calculus is usually divided into two parts (terms): differential calculus and integral calculus. Briefly, differential calculus is focused on rates of change ("the slope problem") and integral calculus is concerned with area, in particular the area enclosed by arbitrary curves. The Fundamental Theorems of Calculus show how these different concerns are essentially the two faces of a single coin. However, it requires a substantial amount of work and insight to be able to see this connection.

A single concept is crucial to the analysis of both problems: the notion of a limit. A typical limit problem involves trying to make sense of a ratio where both the numerator and denominator are getting very small (both are nearly 0) or both are becoming very large (becoming "infinite"). We will see that the usual laws of algebra must be extended in some way to make sense of these "nonsensical" expressions.

You can get a sense of the somewhat paradoxical notions involved with limits if you think about a familiar rate of change: the speed of a car. For example, if you travel the 210 miles from Albany to Geneva in 3.5 hours, then your average speed for the trip is 210 miles divided by 3.5 hours or 60 miles per hour. It's unlikely that your speed was exactly 60 mph at each moment of the trip (even if you used cruise control). Rather, if you had looked at your speedometer at each instant along the way, sometimes it might have read 65 mph or more and at other times (like when exiting the thruway) it might have read much lower, say 25 or 30 mph.

The problem is what do we mean by speed "at an instant"? We just calculated an average speed by dividing distance by time. But in an instant, no time passes and no distance is traveled! So there is no change and, consequently, no rate of change. Well, what is a speedometer measuring, you may ask. Good question! It is actually measuring average rates of change over very small time intervals. This is one of the reasons why when you first start to move, the speedometer does not change, or why when you stop the speedometer does not immediately go to 0. Intuitively, the smaller the time period, the closer the average rate of change is to the instantaneous rate of change. This is how the notion of a limit (a ratio with a small numerator and denominator) was born. To make all this work out correctly and consistently requires a careful mathematical treatment that we will develop during the term. This development took over two-thousand years from the time of the Greek mathematicians (or even earlier) to the period in the 17th century of Newton and Leibniz. In fourteen weeks we will not be able to give all the details of this work, but we will consider some of the major ideas involved.

It turns out that this notion of rate of change is intimately related to slope. This is not so surprising; after all both are ratios. Slope is rise over run or the change in y over the change in x, and velocity, for example, is distance over time. If we plot position on a graph with horizontal axis x being time and the vertical axis y being distance, then the change in distance over the change in time (velocity) becomes the change in y over the change in x (slope). We will exploit this connection several times during the term.

For example, we will identify instantaneous rates of change with slopes of curves (not just slopes of straight lines) at specified points (instants). This identification has a myriad of applications, but here's a simple one. Think about the flight of a ball that you throw up in the air. Its velocity is 0 when it reaches its highest point and the ball seems to hang in the air momentarily. "Markspeople" like Annie Oakley and Buffalo Bill would shoot silver dollars that had been tossed in the air. Though this is quite a feat, they made it easier by shooting at the coin when it "stood still" at the highest point in its flight. Similarly, a tennis player will want to hit a serve when the ball is at or near the top of the toss because the ball is almost still there. More generally, the highest point on a graph (of an appropriate function) will occur when the rate of change or slope (or "velocity") of the graph is 0. This point can be determined without ever having to graph the function, once we develop some methods to calculate instantaneous rates of change. This is quite useful. For example, profit is a function of the price at which an item is sold, so we should be able to determine which price produces the highest (maximum) profit.

Goals and Objectives

Most students in Math 130 do not intend to major or minor in mathematics. Rather, the material in the course is useful in a variety of other disciplines in the natural and social sciences. At the same time the careful, critical thinking that permeates all of mathematics is just as useful as the "methods of computation" that most students associate with mathematics. Consequently, there are two types of goals for the course.

On the theory side by the end of the course you should be able to (1) display an understanding of the notion of a limit of a function and why it occurs in the study of rates of change; (2) display a familiarity with the key steps in the development of the derivative and the necessity of using the the limit concept; and (3) display an understanding of the main theoretical results of the differential calculus, especially the Mean Value Theorem. On the practical side, successful students in the course should be able to (1) classify functions (as having limits, as continuous, or as differentiable); (2) classify the critical points of functions and apply this classification to solve extremization problems; (3) interpret the meaning of the derivative; (4) display a knowledge of basic differentiation techniques, (5) justify through written work the application of theorems and techniques to particular problem-solving situations.

Mathematics is different from other disciplines. This is reflected in a few basic principles that I use in constructing this course (which I have adapted from Roger Askey's article).

I won't lie to you, but I may not tell you the entire truth. Mathematics can be complicated in its full exactness and generality.
Some results are more important than others and I will try to emphasize those that are. Your text may take a slightly different view.
Mathematics is not a secret guild where something is true because I say it is or because some computer program says it is. When something simple and important is studied, I will provide reasons ("proof") to justify it that you should understand.
Words and symbols are important in mathematics. I will use them correctly and will try to get you in the habit of doing so, as well.

Prerequisites

Basic algebraic competence is extremely important for calculus. You cannot register for this course unless you have taken Math 100 or successfully passed the Math Placement Test. This means that I assume that you are familiar with the material from the Appendix A and the additional review material in Chapter 1 of our text. You must be familiar with the basic topics below to be ready to start doing calculus.

Topic	See especially
Basic Algebra	Appendix A
Sets of Real Numbers, Intervals, Interval Notation	Appendix A
Solving Inequalities	Appendix A, Example 2
Absolute Value and Distance	Appendix A, Example 3
Coordinates, Distance, and Circles	Appendix A
Equations of Lines	Appendix A
Functions	Chapter 1.1
Representing Functions (especially piecewise functions)	Chapter 1.2
Inverse, Exponential, and Log Functions	Chapter 1.3
Trig Functions	Chapter 1.4 (especially page 40-41)

Also see the Math 130 Prerequisites sheet that contains additional basic material which you should know. If you are concerned about your background, see me immediately! We will not spend any significant time reviewing algebra. You should review basic trig now in advance of our need for it later in the term.

Labs

Once each week there will be a problem solving "laboratory." These labs are an excellent time for you to ask questions about the course material. I may give you some "pre-lab" problems to get you ready for the lab work; come prepared. Bring your text and notes. You should work with a partner or two on the lab problems. Carefully write up the solutions to the problems you work on so that you can use this material to study for exams. The last 15 minutes of some labs will be an 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.

About twice a week, there will be an assignment of problems to hand in for grading. You may discuss 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 others' 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. Plagiarism acts against my obligation as an instructor to insist that students do work that will benefit them and help them think and learn independently. If you are confused about the meaning of this policy, see me. 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. If more than one page is required, please staple them. No late assignments, please.

There will be regular, graded WebWork computer exercises that review the material and concepts we are currently covering. You will get immediate feedback from these exercises that will allow you to assess your progress. Further, if you submit an incorrect answer, you may return to the problem and work on it again until you get it right. Students typically earn 85-100% on this part of the course. Though every student's problems will be similar, they will not be identical to each other.

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

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/WeBWorK and Quiz Scores, Test 1, Test 2, Test 3, Final Exam, Final Exam.
Note: The final is listed twice. Next, I will remove the lowest score. If the final is your lowest grade, it is removed just once. Then I will average the remaining five grades. I reserve the right to consider class participation.

Extra Credit: You may earn extra credit by attending a Mathematics and Computer Science Department seminar. 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 and lab is required. If you must miss a class for some reason beyond your control, talk to me about it in advance. More than three unexcused absences will affect your grade as follows.

Each of the fourth, fifth, and sixth absences will result in a deduction of 3, 4, and 5 points, respectively, from your final grade.
Beyond six absences will likely result in your expulsion from the course.

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

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! Or at least review the notes I place online at the course website. This makes studying for a test easy.
Start homework and WeBWorK assignments early.
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 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 rapid pace through the listed material. We will adjust this schedule based on our actual work.

Week	Topics and Readings
1	Syllabus, Introduction, Preview, The Slope Problem and Instaneous Rates, Introduction to Limits Read: Syllabus, Chapter 2.1, begin 2.2. Review Chapter 1.1, 1.2, 1.3, 1.4 & Appendix A on your own as needed. Topics: Functions, polynomials, rational, trig, exponentials, logs, and inverse funtions (see page 10 for definitions.)
2	Types of Limits, Evaluating Limits, The Limit Definition (hard). Read 2.2-2.3, and 2.7 (p. 112-117). Review Chapter 1.3 on Logs and Exponentials.
3	An Important Trig Limit, Infinite Limits, and Limits at Infinity. Read 2.4-2.6.
4	Continuity, Introduction to Derivatives. Derivatives: Why We Needed Limits. Derivative Functions. Read 2.6 and 3.1.
5	Exam 1 Monday, September 26 in Albright Auditorium @ 7:40/8:45 am (Section 1/Section 2). Basic Derivative Rules and the Derivatives of Exponentials. Read 3.1-3.3.
6	The Product and Quotient Rules. The Trig Derivatives. Derivatives as Rates of Change. The Chain Rule. Read 3.4-3.7. Review Inverse Functions in Chapter 1.3.
7	The Chain Rule, Implicit Differentiation and Derivatives of Logs, Inverse Functions, and Inverse Trig Functions. Read 3.7-3.10.
8	Derivatives of Inverse Trig Functions. Application: Related Rates. Extrema. Read 3.10-3.11, 4.1.
9	Exam 2 Thursday, October 27 in Lab. The Mean Value Theorem. The First Derivative Test. Read 4.6 and 4.2.
10	The Second Derivative and Concavity. Graphing. Read 4.2-4.3.
11	More Graphing. Optimization. Read 4.3-4.4.
12	L'Hopital's Rule. Antiderivatives. Read 4.7 and 4.9.
13	Exam 3 Monday, November 21 in Albright Auditorium @ 7:40/8:45 am (Section 1/Section 2).
14	Antiderivatives and Applications to Motion. Reversing the Chain Rule. Read 4.9, 5.5.
15	If we have time: Looking ahead to Area Under A Curve.
Final	Final Exam: Wednesday, December 14, 2016.

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

At Hobart and William Smith Colleges, we encourage you to learn collaboratively and to seek the resources that will enable you to succeed. The Center for Teaching and Learning (CTL) is one of those resources: CTL programs and staff help you engage with your learning, accomplish the tasks before you, enhance your thinking and skills, and empower you to do your best. CTL has many resources including Teaching Fellows, Study Mentors, Writing Fellows, Q Fellows support you in courses that require math (but see the Math Intern first!), and professional staff help you assess academic needs.

Study Mentors: The CTL resource especially valuable to students either just starting college or adjusting to the demands of their choice of Major is the Study Mentors program. Study Mentors engage directly with each student in the process of adjusting to new academic demands: they help you find the time you need to engage with both your academic and co-curricular activities, accomplish the tasks in front of you, and enhance your reading and study time. To meet with a Study Mentor, one option is to go to the TutorTrac link provided on the CTL webpage and make an appointment. You can also contact Ingrid Keenan, x3832, keenan@hws.edu, or drop in at the CTL office on the second floor of the library.

I encourage you to explore these and other CTL resources designed to encourage your very best work. You can talk with me about these resources, visit the CTL office on the second floor of the library to discuss options with the staff, or visit the CTL website. at http://www.hws.edu/academics/ctl/index.aspx.

Disability Accommodations: If you are a student with a disability for which you may need accommodations, you should self-identify, provide appropriate documentation of your disability, and register for services with the Coordinator of Disability Services at the Center for Teaching and Learning (CTL). Disability related accommodations and services generally will not be provided until the registration and documentation process is complete. The guidelines for documenting disabilities can be found at the following website: http://www.hws.edu/academics/ctl/disability_services.aspx.

Please direct questions about this process or Disability Services at HWS to David Silver, Coordinator of Disability Services, at silver@hws.edu or x3351.