A Humorous Calculus Review
The videos in the That's Calculus! series are not ordinary math videos. They have humor, character development, even an occasional plot. They also have an occasional blackboard, but even a blackboard becomes something out of the ordinary when performance artist Josh Kornbluth stands in front of it.
According to one definition, learning a concept consists of making mental connections to other concepts and ideas. The more connections, the more complete the learning. The creators of the That's Calculus! videos, Dartmouth professors Dorothy Wallace and Marcia Groszek, suggest connections through visual imagery, metaphor and humor, to help students learn mathematical concepts and remember them.
The videos in this series are not intended to replace classroom lessons or textbooks, nor to be a student's first introduction to the concepts they explore. They are designed to supplement high school or college classes, home schooling, or independent study. Whether used concurrently with the study of a given topic, as a review and synopisis at the end of a unit of study, or later as a reminder, the That's Calculus! videos are made to expand students' understanding and appreciation of mathematics.
These videos were produced under the auspices of the Mathematics Across the Curriculum project at Dartmouth College, funded by the National Science Foundation.
To The Limit
(Running time 20 minutes)
In this video, the first in the series, Josh Kombluth, the nationally acclaimed performance artist and star of the series, examines the limits of various functions f(x) as x approaches a point where the function is undefined. In addition, he takes a culinary look at functions. As you will see, he is unlike most instructors students are likely to encounter.
Mathematical Content: This video concerns the intuitive notion of limit. Josh presents a treatment of the limit of a function f(x) as x approaches a point where f(x) is undefined. He looks at this limit in spveral ways, with a visual examination of thp graph of the funntion, a computational analysis of the formula for f(x) (in this case, a quotient of polynomials), and a table of values. He also makes a connection to the puzzling claim that .99999 ... =1.
The Formal Limit
(Part 1, running time 25 minutes; Part 2, running time 27 minutes)
In this program Josh looks at epsilon-delta from a number of perspectives, including a magical blackboard discussion of the topic, and some imagined and some real-world demonstrations. It follows the action of an epsilon-delta game, discusses the squeeze theorem, and looks at a number of unlikely places you might encounter epsilon-delta in your social life.
Mathematical Content: Josh presents the formal, epsilon-delta definition of limit on the blackboard and uses it to give a formal proof of the "squeeze theorem." The quantifier structure of the definition ("for all epsilon ... there exits delta ... for all x") is also explicated by means of a board game (in which players alternate playing epsilon, delta and x) and through a character who utters English sentences having the same for all ... there exists ... for all ... quantifier structure.
Josh and the Mathemagician
(Part 1, running time 43 minutes; Part 2, running time 38 minutes)
In Part 1, Josh meets a mysterious woman dressed as a bag lady who can find where any function has its largest and smallest values. He undertakes a quest to find the secret to her arcane knowledge of max an min. In Part 2, Josh advances his status in the mysterious cult by learning to compute the derivatives of various and complicated functions. His brave efforts earn his induction into the cult.
Mathematical Content: Part 1 covers the definition of the derivative. Josh begins with the formula for the slope of a line, derives the (limit) definition of the derivative, and uses this definition to find the derivatives of the functions x squared, and then x to any (positive integer) power. Part 2 covers some rules for computing derivatives, including a proof of the product rule, and examples for use of the sum, product and chain rules.
In Search of the Derivative
(Part 1, running time 23 minutes; Part 2, running time 27 minutes)
In this applications-oriented show, Josh plays the student. Part I begins with Josh's visit to a mathematician, followed by an explosive stop in a chemist's lab. In Part 2 he continues his search for real-world applications, making field visits with a geologist who examines radioactive decay and an economist who looks at the problems associated with selecting insurance for a bike messenger.
Mathematical Content: In this video, various scientists show Josh how the derivative is used in their respective fields. In Part 1, a mathematician reviews the definition of the derivative and its use in analyzing shapes of curves, and a chemist explains that reaction rates of chemical reactions are derivatives. In Part 2, a geologist talks about exponential decay, and an economist explains marginal cost.