## Classroom-Ready Resources and Teaching Suggestions

Browse index of informative background articles for calculus and differential equations.

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**Calculus 1**

Archimedes’ Method for Computing Areas and Volumes, by Gabriela R. Sanchis

Archimedes’ use of the Law of the Lever to compute areas and volumes in *The Method,* with classroom-ready examples, exercises, and interactive applets.

Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students, by Kenneth M Monks

One of a collection of student-ready modules based on primary historical sources presented in the article A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources.

Historical Activities for the Calculus Classroom, by Gabriela R. Sanchis

History and mathematics of curve sketching, tangent lines, and optimization, explored using interactive applets.

Jan Hudde’s Second Letter: On Maxima and Minima, by Daniel J. Curtin

English translation of Hudde’s description of optimization via algebra and arithmetic progressions with an early appearance of the Quotient Rule; includes suggested exercises for its classroom use.

The Derivatives of the Sine and Cosine Functions: A Mini-Primary Source Project for Calculus 1 Students, by Dominic Klyve

One of a collection of student-ready modules based on primary historical sources presented in the article A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources.

**Calculus 2**

A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources

A collection of student-ready modules based on primary historical sources designed to teach standard topics from the curriculum of a wide range of undergraduate courses. Includes the following specific projects suitable for use in Calculus 2.

Kepler: The Volume of a Wine Barrel, by Roberto Cardil

In his analysis of volumes of wine barrels, Kepler used ideas that would become important in differential and integral calculus. This article provides visual imagery, much of it animated, to help instructors share Kepler's ideas with students.

**Multivariable Calculus **

A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources

A collection of student-ready modules based on primary historical sources designed to teach standard topics from the curriculum of a wide range of undergraduate courses. Includes the following specific projects suitable for use in Multivariable Calculus.

**Differential Equations**

A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical SourcesA collection of student-ready modules based on primary historical sources designed to teach standard topics from the curriculum of a wide range of undergraduate courses. Includes the following specific projects suitable for use in Differential Equations.

D'Alembert, Lagrange, and Reduction of Order, by Sarah Cummings and Adam Parker

Two historical approaches, one familiar and one unfamiliar, to enrich an ODE classroom.

Peano on Wronskians: A Translation, by Susannah M. Engdahl and Adam E. Parker

How a translation of Peano’s counterexample to the ‘theorem’ that a zero Wronskian implies linear dependence can help differential equations students; includes a series of suggested student tasks.

Things Certain and Uncertain, by Michael P. Saclolo and Erik R. Tou

The story of a mathematical problem on the mechanics of hot air balloon flight that Euler was working on the very day of his death, presented in its historical context and accompanied by a classroom capsule with suggestions for how the mathematics of balloon flight can be used in a contemporary differential equations or physics course.

## Informative Background Articles

Browse index of classroom-ready resources and teaching suggestions for calculus and differential equations.

Return to master index.

**General**

Math Origins: The Language of Change, by Erik Tou

An article from the Math Origins series in which the author explores how concepts, definitions, and theorems familiar to today’s students of mathematics developed over time.

**Calculus 1**

A Euclidean Approach to the FTC, by Andrew Leahy

The Fundamental Theorem of Calculus is presented in the version of Scottish mathematician James Gregory—without the use of limits.

Helping Ada Lovelace with her Homework: Classroom Exercises from a Victorian Calculus Course, by Adrian Rice

Highlights from Ada Lovelace’s correspondence course on calculus with Augustus De Morgan that shed light on common confusions that still arise today.

Teaching the Fundamental Theorem of Calculus: A Historical Reflection, by Jorge López Fernández and Omar Hernández Rodrígue

The authors argue that the teaching of elementary integration should better reflect its historical development.

The Four Curves of Alexis Clairaut, by Taner Kiral, Jonathan Murdock, and Colin B. P. McKinney

Translation of a paper on families of algebraic curves (along with a transcription of the French original) written when Clairaut was only twelve years old, with comments about what Clairaut’s paper might offer today’s students, based on the authors’ experience as a faculty-student research-and-translation team.

Thomas Simpson and Maxima and Minima, by Michel Helfgott

Simpson’s methods for finding maxima and minima are explored by using examples from his “Doctrine and Application of Fluxions.” Many of his techniques could be used in today’s classroom.

**Calculus 2**

What is 0^0?, by Michael Huber and V. Frederick Rickey

The expression 0^0 is usually called an indeterminate form. This article details the history of the meaning of this expression and concludes that, in some cases, we should evaluate it as 1.

**Multivariable Calculus**

Limit Points and Connected Sets in the Plane, by David R. Hill and David E. Zitarelli

A study of Mullikan's Nautilus, using movies to illustrate the important idea, that could be used as an instructive vehicle for introducing limits in the plane to students who have studied limits on the real line.

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