This is a five (5) hour introductory Calculus course designed primarily for engineering majors and certain other technical majors. As mentioned above, the text is Calculus by Stewart, 4^th edition. The student is assumed to be capable and versed in the standard Pre-Calculus topics, functions, graphing, solving equations, exponential, logarithmic and trigonometric functions. No prior exposure to Calculus is assumed. Since this is a five hour class, the student should expect at least 4 hour exams and a final exam. Of course, the testing and evaluation for each class is entirely at the discretion of the instructor. Regarding the use of technology in the classroom, no departmental policy is presently in place. Some sections require the use of calculators and some classes prohibit them. The student is advised to consult with the instructor about this topic.
 

The student should emerge from the class with
 

1) A basic introduction to limits and continuity for functions of a single variable.

2) The ability to differentiate the elementary functions and apply those derivatives to solve problems.

3) An awareness of the foundations of the Riemann Integral and some of its applications.
 

A specific section by section suggested syllabus and appropriate comments are shown below
 
 

Chapter 1- Optional- This is a Pre-Calculus review chapter and may be briefly discussed or assigned at the discretion of the instructor. Incoming students should be familiar with the topics in this chapter. Most faculty skip this chapter.
 

Chapter 2

Section 2.1- Optional. Motivational devices for limits. In this section, the author uses the problems of tangents and velocities to motivate the notion of a limit. If the instructor wishes, he may employ this idea.
 

Sections 2.2 and 2.3 - Limits and limit laws. The approach is non-rigorous here. Most faculty present the non-rigorous approach before the epsilon-delta definition.
 

Section 2.4 - Epsilon-delta definitions. The instructor should use their own discretion in this section. This material is more abstract than most of the material in a beginning textbook. If you are new to teaching, you might wish to see the course coordinator (Paul Britt e-mail to tennisguy@home.com or 388-4738) or consult with some of your colleagues about this topic.
 

Section 2.5 - Continuity. Continuity, the basic hypothesis for most Calculus theorems, is often misunderstood by the beginning student. The instructor should explore the topic in full so that confusion might be avoided later in the course.
 

Section 2.6 - Limits at infinity and horizontal asymptotes. This is a fairly basic subject.
 

Sections 2.7, 2.8 and 2.9 - Tangents, velocities and derivatives. This is the true introduction to the derivative as a limit of a difference quotient. The student should be made aware of the derivative as an instantaneous rate of change, a tangent line slope and the velocity of a particle. Too many students emerge from beginning classes with an ability to compute difficult derivatives with little or no idea about the nature of the derivative. The instructor is advised to stress interpretation of what a derivative tells us. This is the section where that instruction begins.
 

Chapter 3

Sections 3.1 and 3.2 Differentiation of polynomials, exponentials and the product and quotient rules. Many textbooks have begun to introduce transcendental functions early in the course. The thinking is that if a topic is introduced early and used often, the students will be more inclined to realize its importance. Thus, the differentiation of exponentials in section 3.1.

Section 3.3 Optional - Rates of change. I prefer to cover this topic. This section gives examples of derivatives in use in other fields. The students should know that derivatives are not just used in math class. This discussion also continues the exposition about interpretation of the derivative. The students are exposed to the idea that a derivative is not simply a formula.

Section 3.4 Derivatives of trig functions. The material is standard. The students may not recall as much trig as the instructor deems appropriate.

Section 3.5 Chain Rule - This topic seems to be troublesome for the student. I suspect the problem begins in their algebra courses. They cannot differentiate composed functions very easily because they do not really understand composition itself.

Section 3.6 Implicit Differentiation - A standard presentation.

Section 3.7 Higher Derivatives - A standard presentation.

Section 3.8 Derivatives of logarithms - A non-standard presentation. The standard presentation is in a later section.

Section 3.9 Optional - Hyperbolic Functions - The decision is left to the instructor.

Section 3.10 Related rates - This topic gives the students some trouble because they do not understand the chain rule. The student has to be made aware that differentiation of rates involves differentiating with respect to time.

Section 3.11 Linearization and differentials - The tangent line. The first degree Taylor approximation to f (x). The student should understand the geometric interpretations of dy and  delta y.
 

Chapter 4

Section 4.1 Maximum and minium values of functions - An extremely important, but fairly easy topic.

Section 4.2 The Mean Value Theorem - The building block for the proofs of many other basic Calculus theorems. This theorem is open to a very interesting physical, as well as a geometric interpretation.

Section 4.3 Derivatives and Graphs - Another section in which the student is taught interpretation of derivatives. This section includes increasing, decreasing, concavity and the first and second derivatives tests. A very important section.

Section 4.4 L'Hopital's Rule - A fairly standard presentation.

Section 4.5 Curve sketching - The summary section for elementary graphing techniques.

Section 4.6 Optional - Calculus and calculators - A fairly nice treatment of some pros and cons of calculator usage.

Section 4.7 Optimization problems - The student will encounter some difficulty in this section. Expect to spend more than one day on this topic.

Section 4.8 Optional - Economic applications - Coverage left to the discretion of the instructor.

Section 4.9 Newton's Method - The material on Newton's Method should not be considered as optional. This method of finding roots, while not robust, does offer the students reinforcement of the geometric interpretation of the derivative and the ideas of convergence.

Section 4.10 Antiderivative - Fairly standard topic.
 

Chapter 5

Section 5.1 Optional - Areas and distances - This section seeks to motivate the idea of a Definite Integral using areas and distances. If the instructor is pressed for time, this section may be omitted and the general integral from section 5.2 can be taught.

Section 5.2 Definite Integral - Fairly standard presentation.

Sections 5.3 and 5.4 Fundamental Theorem of Calculus - Be aware that students often emerge from this class knowing only the second form of the theorem (Integral Evaluation Result). The first form, while easily forgotten, shows the beautiful interplay between derivative and integral.

Section 5.5 Substitutions - Students have trouble with this topic. Since the idea of substitution is so important in Calculus II, the instructor should be very sure that his students receive adequate practice in this topic.

Section 5.6 Optional - Logarithm defined as an integral - This section revisits the exponential and logarithm functions and defines them in the traditional, rigorous manner.
 

Chapter 6

Section 6.1 Areas between curves - A fairly standard presentation.

Section 6.2 Volumes of solids of revolution and slicing problems - Many student have a great deal of difficulty with these topics. The instructor should illustrate his examples with pictures (to the best of his ability). Be advised that even with marvelously drawn examples, the volumes by slicing represent some of the more challenging problems in Calculus I. The students do not "see" well in three dimensions.

Section 6.3 Volumes by shells - As per the above, copious illustrations may make this topic easier for your students.

Section 6.4 - Work - The instructor and the students are advised to be careful with the units. Sometimes the book might accidentally trick you.

Section 6.5 - Mean Value Theorem for Integrals - A fairly standard topic. A geometric interpretation may aid the students.
 

Chapter 8

Sections 8.1 and 8.2 Optional - Arc length and area of a surface of revolution - Fairly standard presentation.

Section 8.3 Optional - Physics applications - This section incorporates fluid pressure, mass and center of mass problems. Again the warning about units in appropriate. Remember, there is a difference between mass-density and weight-density.
 

SUMMARY
 

The guidelines above are just that, guidelines. I am available to discuss the course with you. If you have any questions, please stop by to chat.
 
 

Paul W. Britt Calculus Coordinator Locket 325