kaphelp.org - Calculus

Calculus

Kenyon Equivalent: MATH 111: Calculus A = 1/2 unit credit

Course Description:
This course is the first of a three-semester calculus sequence at Kenyon. It serves as an introduction to differential calculus, the study of elementary functions and their variations, or rates of change.   Students will learn what a derivative is, how to take derivatives of functions, and when to take derivatives. The course also touches on the notion of an integral. More specifically, topics to be studied include:

1 Functions in Calculus

    * Definition of Function

    * Graphs of Functions
    * Elementary Functions
    * Amount Functions and Rate Functions: The Idea of a Derivative
    * Estimating Derivatives
    * The Geometry of Derivatives

2. The Derivative

    * The Formal Definition of a Derivative
    * Derivatives of Power Functions and Polynomials
    * Limits and Continuity
    * Derivatives of Exponential and Logarithmic Functions
    * Derivatives of Trigonometric Functions
    * The Product and Quotient Rules
    * The Chain Rule
    * Why Continuity Matters
    * Why Differentiability Matters

3. Applications of the Derivative

    * Differential Equations and Their Solutions
    * Limits involving Infinity and l’Hopital’s Rule
    * Taylor Polynomials (Linear and Quadratic Approximation)
    * Newton’s Method
    * Optimization

4. The Integral

    * Areas and Integrals
    * The Fundamental Theorem of Calculus
    * Antidifferentiation by Substitution
    * (If time permits approximating sums are highly recommended as well)

Finally, KAP Calculus is a year-long course while Calculus A at Kenyon is a one-semester course. Students successfully completing KAP Calculus will receive 0.5 units.

What does it really mean to teach a KAP calculus course? To answer this question we refer to the description of KAP as presented on the KAP website

“The Kenyon Academic Partnership (KAP) is an early college program in which twenty-seven central and northern Ohio public and independent secondary schools offer various Kenyon College first?]time courses on their own campuses. The program not only permits students to earn college placement and credit before leaving high school but imitates as closely as possible a college environment in the nature and scope of reading, writing, and laboratory assignments, and the process or atmosphere of a college class.”

For our purposes, it is the italicized portion of this description that is most important. Our goal in developing a successful KAP calculus course is to “imitate” the calculus course typically taught at a Tier 1 liberal arts college and, more specifically, at Kenyon. Such a course must include:

1. The use of a text that emphasizes the conceptual as well as the computational. At Kenyon the text is Ostebee-Zorn’s Calculus from Graphical, Numerical, and Symbolic Points of View”. Other texts are possible but they must be approved by the Director of KAP Calculus (currently Judy Holdener).

2. Daily lessons should be largely student-centered. That is, during class students should be engaged in problem-solving, class discussions, group-work, presentations, etc.   Activities and labs designed to keep students engaged are necessary for a successful calculus course. Students learn mathematics by doing mathematics, not by watching others do mathematics.

3. Technology should make up a significant and ongoing component of the course. “Technology” could mean graphing calculators, but the computer algebra system Maple is preferred. Using technology properly takes practice, and it is perhaps best to start slowly. Here are a few examples of good and bad uses of technology:

Some Good Uses of Technology

1. Calculus students work through a self-paced Maple-based tutorial designed to provide a review of the various types of elementary functions.

2. Visualization of graphs and mathematical constructs (slope fields, Riemann sums, etc)

3. Solving large systems of equations resulting from a “real-world” application as introduced, for example, in a course project. With technology, it is now possible to tackle problems involving long, complicated computations. Such problems were unfeasible in the past.

4. Students check their handwork (Maple’s diff and int commands allow for easy computation of derivatives and integrals.)

5. Maple-based projects involving an element of design, and perhaps even a competition.
Some Bad Uses of Technology

1. Students use the technology to compute derivatives for them; they rely on their calculator rather than memorizing basic differentiation rules.

2. Students redo computations done by hand on the computer (but not necessarily to check their work!).

3. Uses involving technology as an “add-on” to the course. For example, at the end of class you include a five-minute coverage of how to do everything done during class on the computer.

4. KAP Calculus should include a significant writing component. This means that there should be at least one significant paper (about 5 pages in length) each semester, and homeworks/projects should require written explanations on a regular basis. KAP teachers should also require proper write-ups of computations. For example, equal signs serve as verbs in mathematical sentences; they can not be omitted! Random mathematical expressions floating in space mean nothing.

5. Supplemental projects. A Kenyon calculus course usually involves 3 projects per semester. The projects vary in nature and difficulty, but they usually have two-week deadlines. Sometimes the projects are computer-based (students work through a Maple file), sometimes the projects involve problem-solving to be done by hand, and sometimes the project results in a paper.   In all scenarios, the projects are designed to develop presentation skills as well as problem-solving skills. Projects also tend to involve a more in-depth coverage of the material, or perhaps a more comprehensive approach to several different topics in the course (as opposed to daily homework which tends to cover the topic at hand).

6. Large amounts of problem-solving on a daily basis! Homework should be assigned on a daily basis, even if it cannot always be collected and graded. In-class problem-solving is quite effective in introductory calculus courses.

http://kap.kenyon.edu/whoweare.html

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