MATH 263A - Analytic Geometry and Calculus
Four Quarter Hours

I. PREREQUISITES
MATH 115, Pre-calculus, or equivalent; or level 3 math placement, or permission of Math Department. Students cannot earn credit for both 263A and 163A.

II. COURSE DESCRIPTION
This course is an introduction to calculus, with analytic geometry woven throughout the text as needed to serve as a basis for studying some of the topics in calculus. In addition to the material on analytic geometry, the main concepts discussed are functions and their graphs, limits and continuity, and particularly, the definition and utilization of the derivative. The material in the text is well-suited for liberal arts students, as well as those in mathematics, engineering, and the sciences. It is designed for the average student with a good background in high school mathematics or a college-level pre-calculus course.

III. TEXTBOOK AND SUPPLIES
ISBN 013518911X  Varberg, Dale, and Purcell, Edwin J., Calculus, 7th ed., Upper Saddle River, New Jersey: Prentice Hall, Inc., 1997

...available from EdMap's distance-learning online bookstore.

IV. CALCULATOR
Use of a calculator is optional in this course; a calculator with basic arithmetic and square root functions is sufficient. Programmable calculators may NOT be used.

V. COURSE CONTENT
MATH 263A covers Chapters 1-4 in the Varberg text, but primarily focuses on Chapters 2, 3, and 4. Section 4.5 will be omitted.

Chapter 1, together with the first three sections of Chapter 2 consists of background material for the study of calculus. These "pre-calculus" topics are essential for the study of calculus (which really begins in this textbook in Section 2.4 with the study of limits).

It is assumed that you have been exposed to the main ideas embodied in the "preliminaries" of Chapter 1 and Section 2.3, "The Trigonometric Functions." Depending on your mathematical background, you may or may not feel the need to review Chapter 1 in depth before proceeding to Chapter 2. However, this material is important and should be referred to from time to time as needed.

Throughout your text, important terms are set off in boldface type. You should understand these terms thoroughly, as they constitute basic vocabulary for discussing and understanding the concepts and ideas of calculus. Definitions and results (theorems, properties, rules, etc) which are enclosed in rectangular "boxes" are especially noteworthy and should be understood and essentially memorized (at least in your own words).

In each section, examples are given which provide the solutions of problems related to the basic concepts of that section. These examples also provide techniques for solving similar problems in the problem sets.

In addition to the many worked example problems, each problem set in the text begins with "fill-in-the-blank" items, designed to reinforce the substantial emphasis in your textbook on concepts. Following the suggestion in the Preface, you should respond to these items before proceeding to the later problems. Correct answers to the fill-in-the-blank items are given at the end of the problem sets.

Notes.	(1)	Some problems in the problem sets are marked with the symbol PC or C to indicate that a calculator is useful in solving them. Problems like these will not be on your examination.
	(2)	Section 2.5 gives a precise definition of the meaning of limit. This definition is one of the most important definitions in calculus, and you should study it and try to absorb it. It is frequently referred to as "the , definition of limit." However, you will not be asked on your examination to make any rigorous , proof of limit statements.
	(3)	You will not be required in the examination to prove theorems. It is, however, important that you know what the theorems say and be able to apply them.

VI. EXAMINATION
The course examination for Mathematics 263A in this format is a comprehensive supervised examination consisting of problems covering the material you are responsible for (primarily Chapters 2, 3, 4 as noted above). The problems are similar to those on the Sample Examination. The course examination includes manipulative exercises, plus word or stated problems.

You will have three hours to complete the supervised examination. A hand-held calculator with basic arithmetic and square root functions will not really be needed, but may be used on the examination if you supply your own. Programmable calculators may not be used.

VII. PREPARING FOR THE EXAMINATION

The most effective way for you to prepare for the examination is to concentrate on solving problems (especially the non-proof type problems) from the problem sets of each section. However, for the necessary background to solve problems, you will need to understand concepts and important results in each section. Thus, before you turn to problem solving, it is recommended that you read and study the material in each section.

A large percentage of the problems in the various problem sets involve in their solution computing the derivative of a function; and typically, in computing the derivative of a function, it is convenient to regard the function as composed of combinations (sums, differences, products, and quotients) of other functions. Accordingly, you need to be able to readily apply the general differentiation rules in your textbook for differentiating the sum, difference, product, and quotient of two functions. Also, you should memorize the specialized differentiation formulas introduced in Mathematics 263A.

In addition to the general differentiation formulas cited above, it is useful to have a rule which tells us directly how to find the derivative of a composite function in terms of the derivatives of f and g. Such a rule, called the Chain Rule, is given in Theorem A of Section 3.5. Essentially, this rule states that the derivative of the composite F of two functions f and g is the product of the derivatives of f and g.

The Chain Rule is one of the most important of the differentiation rules. As stated in your text: "The Chain Rule is so important that you will seldom again differentiate any function without using it." Quite obviously, you should read carefully Section 3.5 (and also Section 3.6) concerning the Chain Rule, and follow in detail the worked examples illustrating its use.

Frequently, in computing the derivative of a function, you will need to use the Chain Rule in conjunction with other differentiation formulas. To gain facility in using these formulas properly and efficiently, you should work a considerable number of differentiation problems from the problem sets.

Study especially the examples in Section 3.5 illustrating the use of the Chain Rule in computing the power of a function. Although your text does not state explicitly a general formula for differentiating the power of a function, it is handy to have such a formula. The formula, sometimes called the Power Rule for Functions states: "If g is a differentiable function of x, and r is any real number constant, then

(1) D_x[g(x)]^r = r[g(x)]^r-1D_xg(x)

The Power Rule for Functions is certainly worth memorizing.

By way of illustration, using the Power Rule for Functions in Example 1, page 130 of the textbook, we have immediately,

The Power Rule for Functions, formula (1) above, is easily derived using the Chain Rule as follows:

Let y = [g(x)]r, and u = g(x).

Then, by the Chain Rule,

Observe that in the special case where g(x) = x, the Power Rule for Functions reduces to the ordinary Power Rule D_xx^r = r x ^r-1.

VIII. SAMPLE EXAMINATION
A self-check sample examination is included with this syllabus. There is also an answer key which gives the solutions to all the problems in the sample exam. The solutions to the problems are given in considerable detail, so that you can follow completely the steps and method required to solve each problem. The sample examination should help you make a final determination as to whether or not you are prepared to take the supervised examination. It is recommended that you do not take the sample exam until you feel you have mastered the course material.

When you do feel prepared to take the Sample Examination, take it without using your book or notes. Finally, evaluate your answers using the answer key. Sample Examination

If you did not do well on the sample exam, then review more extensively those troublesome areas before applying for the examination. As mentioned earlier, the problems on the supervised examination will be very similar to those on the sample exam. Hence, it should be helpful to you in preparing for the supervised examination to look carefully at the solutions in the sample exam answer key for those problems you missed.

IX. GRADING SCALE
Your course grade will be determined by your percent score on the examination using the following scale:

88-100% = A to A
75-87% = B to B+
60-74% = C to C+
50-59% = D to D+
Below 50% = F