Math 130 S'08: Calculus I

Course Website: http://math.hws.edu/~mitchell/Math130S08/index.html

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 very large (nearly 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, such as 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. 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. First, 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.

Text

The Department of Mathematics selected the Larson, Hostetler, and Edwards's 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 mechanics of "differentiating" a function, I hope you will still remember what a derivative is and how it is used.

Prerequisites

Basic algebraic competence is extremely important in this course. First, you cannot register for this course unless you have taken Math 100 or successfully passed the Math Placement Test. Further, I assume that you are familiar with the following material from the Appendix D that is online at the Houghton Mifflin Website Appendices for our text or by using the individual links in the table below in the online version of this syllabus. There is also additional review material in Chapter 1 at the beginning of our text. Again, I assume that you are familiar with the basics.

Text Reference	Topic	See especially
Appendix D.1	Inequalities	Pages D2-D3
	Intervals, interval notation	Box on D3
	Solving Inequalities	Page D5 Examples 3 and 4
	Absolute Value and Distance	Pages D6-D7
Appendix D.2	Distance formula	Pages D11-D12
	Circles	Page D13
Appendix D.3	Radians	Page D18
	Trig functions	Page D19 (See Definitions)
	Trig identities	Page D19 Pythagorean, Reciprocal, and Quotient Identities
	Evaluating Trig Functions	Page D20 Know the exact values in the box.
	Trig graphs	D23
Chapter 1	Intercepts	Page 4
	Intersections	Page 6
	Slope, Equations of Lines	Pages 10, 11, 13
	Parallel and Perpendicular Lines	Page 15
	Functions: Notation, Domain, Range, Graphs	Pages 19-22
	Properties of Exponents	See box on page 47

You should 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 amount of time reviewing algebra. You should review basic trigonometry now in anticipation of our need for it as the term develops.

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 may work with a partner or two on the lab problems. You should 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 several 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.

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. Do not simply copy each other's work. Make sure you write up your answers on your own and check your work. These homework assigments should be done neatly. I suggest working out the problems on scrap paper first and then making a final copy to hand in. Use pencil rather than pen to avoid having to cross work out. Remember: Your homework represents you; try to make the best impression possible. Moreover, these problems when neatly and carefully done are an investment.... you can use them to study for exams. Your work will be due at the beginning of the next class. No late assignments, please.

I will also use some announced 10-minute quizzes to check on your progress in the course.

There will be three hour 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: Homework and Quiz Scores Combined, 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 (see note below) into account in determining final grades.

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 piecewise 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 clases.

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.

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.

Math Intern

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

Tips for Success

My best advice is to take good, complete notes during class. Even if you don't understand every detail during a lecture, with some patience you should be able to review each day's lecture and understand everything we did. If you don't, then you should come to see me. Here are a few simple things that you can do to be more successful in the course:

Come to all classes.
Did I mention that you should take good, complete notes? If I think it is important enough to write down, so should you!
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 to see me 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 fairly rapid pace through the listed materials. We may need to adjust this schedule based on our actual work in class.

Week	Topics and Readings
1: January 21-25	Syllabus, Introduction, Preview, The Slope Problem and Instaneous Rates, Finding Limits Read: Syllabus, Chapter 2.1-2.2. [Review Chapter 1.2, 1.3, 1.5 and Appendix D on your own as needed. Key topics: Functions, polynomials, rational functions, trig functions, inverse funtions.]
2: January 28-February 1	Evaluating Limits, The Limit Definition (hard). Exponential & Log Functions. Re-read 2.2, read 2.3-2.4. Review Chapter 1.6.
3: February 4-8	Exponential & Log Functions Limits. Continuity, One-sided limits, and Infinite Limits. Read 2.4-2.5.
4: February 11-15	Introduction to Derivatives. Derivatives--why we needed limits. Derivative Functions and Derivative Formulas. Read 3.1-3.2. Exam 1 (Friday in Class 7:40 am).
5: February 18-22	More on Derivative Formulas. The Chain Rule. Read 3.2 and 3.3.
6: February 25-29	The Chain Rule. Implicit differentiation, Review of Inverse Functions. Read 3.4--3.5 and 1.5.
7: March 3-7	Derivatives of Inverse Functions. Applications: Related Rates. Extrema. Read 3.6 and 3.7.
8: March 10-12	Extrema. Read 4.1. Exam 2 (Wednesday in Class 7:40 am).
9: March 24-28	Extrema. The Mean Value Theorem. The First Derivative Test. Read 4.1 and 4.2, begin 4.3
10: March 31-April 4	The Second Derivative Test. Concavity and Graphing. Read 4.3-4.4.
11: April 7-11	Optimization. Read 4.7
12: April 14-18	Graphing with Asymptotes. Exam 3 Friday in Class 7:40 am.
13: April 21-25	Antiderivatives. Applications to Motion. Read 5.1.
13: April 28-May 2	Laws of Motion: Constant Acceleration. Introduction to the Area Problem. Reversing the Chain Rule. Read 5.1 and 5.5
14: May 5-6	Integration by Substitution. Read 5.5.
Final Exam	Section 130-01: Saturday, May 10, 2008 at 7:00 PM.

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Math 130: Calculus I