Calculus Laboratory Sessions, 1991-92

Most laboratory sessions are designed to emphasize exploration and discovery of calculus concepts; some emphasize applications. During each laboratory session, students are guided by a set of written instructions and supervised by their instructor. In fact, the sessions are intended to be held during regularly scheduled 50-minute class sessions, with written reports to be completed as homework. Laboratory sessions are designed to be worked on by pairs of students, and require a written report of each pair of laboratory partners. Some of the laboratory sessions require use of a very small computer graphing and symbolic package, such as True Basic's Calculus package, or a graphing calculator; others require only pencil and paper.

If you would like copies of some or all of these laboratory sessions, please e-mail me: Email Janet Beery

Please note that since these laboratory sessions were prepared a number of calculus laboratory guides have been published. (For instance, see my references at the bottom of the page.)

Mathematics 105 - Calculus I

Laboratory 1: Graphing Functions

Goals: To become familiar with graphing functions on the computer and to explore the behavior of graphs of ``families" of functions

Laboratory 2: Trigonometric Functions and Absolute Value Functions

Goals: To explore graphs of trigonometric functions, and to discover the effect the absolute value function has on the graph of a function

Laboratory 3: The Slope of a Function, Tangent Lines, and the Derivative

Goals: To define and to find the slope of a function at a point; to discover the relationship between the slope of a function at a point and the tangent line to the graph of the function at that point; to discover under what circumstances a function does not have a slope at a point

Laboratory 4: Derivatives and Graphs

Goal: To determine the relationship between the graph of a function and the graph and values of its derivative

Laboratory 5: Air Traffic Control

Goals: To review interpreting derivatives as rates of change; to discover how to use the derivative to find minimum and maximum values; and to apply a variety of mathematical tools and methods to a ``real life" situation

Laboratory 6: Projectile Motion, or Peace through Calculus

Goals: To practice finding antiderivatives and solving differential equations in a ``real life" setting, and to learn to model simple projectile motion using coordinate equations

Laboratory 7: Bounding Error in Area Approximations

Goals: To discover how to bound the error involved with making area approximations, and to use this technique to bound the error involved with approximating definite integrals

Mathematics 106 - Calculus II

Laboratory 1: Kepler's Third Law of Planetary Motion

Goals: To become familiar with a common technique in experimental science that makes use of the natural logarithm, and to practice rewriting equations with logarithms in them by exponentiating them

Laboratory 2: Mathematical Super Sleuth

Goals: To practice using the law of exponential change and Newton's law of cooling

Laboratory 3: Discovering Integral Formulas

Goals: To recognize patterns, to use them to predict integral formulas, and to check these formulas via differentiation

Laboratory 4: World Population Growth

Goals: To predict future world population using the exponential model and using the logistic model, and to compare the results

Laboratory 5: Approximating Functions Using Polynomials

Goals: To learn to approximate functions using polynomials, to begin to determine how good an approximation is, and to prepare for our work next week with Taylor polynomials and series

Laboratory 6: Peace Through Parametric Equations

Goals: To review finding antiderivatives and solving differential equations in a ``real life" setting, to learn to model projectile motion using parametric equations

Laboratory 7: Graphing Polar Equations

Goals: To explore graphs of polar equations

Laboratory 8: Team Presentations

Goals: To review various topics from the course, to practice presenting mathematics to your peers

References

Most of the above laboratory sessions were adapted from--or at least inspired by--the following sources.

Explorations in Calculus with a Computer Algebra System, Donald B. Small and John M. Hosack, McGraw-Hill, 1991

Learning by Discovery, Anita Solow, editor, Grinnell College, 1990; now published by Mathematical Association of America

The Calculus Reader, David A. Smith and Lawrence C. Moore, Duke University, 1990; now published by D.C. Heath as Calculus: Modeling and Application

To help make the laboratories an integral part of the course:

Make exploration and discovery---especially that using the computer package the students use---a part of each class session.

Let laboratories replace, rather than supplement, more traditional presentations of material.

Use ideas and rules discovered by students in lab immediately and often in other parts of the course (e.g. lecture, homework).

Test students on laboratory work. (Yes, it will be on the exam!)

Encourage students to discuss laboratory results with each other in class after the lab.

Benefits of laboratory sessions in calculus courses:

Students discover and learn at least part of the course material on their own, thereby making it their own intellectual property.

Students can work on more complicated and more difficult explorations and applications than is ordinarily possible---and complete them!

Students learn how to learn and do mathematics on their own.

Students learn to read laboratory instructions.

Students learn to work with, teach, and learn from other students.

Students learn to communicate mathematics aloud.

Students learn to communicate mathematics in writing.


Student reactions to the laboratories

Were the laboratories easy/difficult for you? What was easy/difficult about them? What did you like best about them? What did you like least?

"They were fun. Sometimes difficult."

"I enjoyed all of the labs. The labs were helpful for understanding more depth on a topic."

"I really hated the laboratories and I sometimes got so frustrated because of them."

"Difficult. The write-ups were difficult."

"The writing was easy. Finding the solutions was difficult."

"They were difficult, but they forced you to really understand. The problems themselves were difficult. The writing was the best part."

"Labs were very difficult for me. I have never had to explain math in regular language before. I liked the way I learned so much from them. I disliked the gigantic amount of time they took."

Did the laboratories help you learn the material? Were they essential?

"Yes. Yes. I think they were a painfully essential part of Math 105. Very often an excellent introduction to new concepts and ideas."