Classroom Resources Index – Calculus and Differential Equations

Classroom-Ready Resources and Teaching Suggestions

Browse index of informative background articles for calculus and differential equations.
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Calculus 1

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 1.

• Beyond Riemann Sums: Fermat's Method of Integration – A Mini-Primary Source Project for First-Year Calculus Students, by Dominic Klyve (can also be applicable to Calculus 2)
• Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students, by Kenneth M Monks
• The Derivatives of the Sine and Cosine Functions: A Mini-Primary Source Project for Calculus 1 Students, by Dominic Klyve
• Three Hundred Years of Helping Others: Maria Gaetana Agnesi on the Product Rule – A Mini-Primary Source Project for Calculus 1 Students, by Kenneth M Monks

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.

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.

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.

• Bhāskara’s Approximation to and Mādhava’s Series for Sine: A Mini-Primary Source Project for Calculus 2 Students, by Kenneth M Monks
• Euler’s Calculation of the Sum of the Reciprocals of the Squares: A Mini-Primary Source Project for Calculus 2 Students, by Kenneth M Monks
• Gaussian Guesswork: Three Mini-Primary Source Projects for Calculus 2 Students, by Janet Heine Barnett
• How to Calculate \(\pi\): Buffon's Needle – A Mini-Primary Source Project on Geometric Probability for Calculus 2 Students, Pre-service Teachers and Others, by Dominic Klyve
• How to Calculate \(\pi\): Machin's Inverse Tangents, A Mini-Primary Source Project for Calculus 2 Students, by Dominic Klyve
• Investigations Into d'Alembert's Definition of Limit: A Mini-Primary Source Project for Students of Real Analysis and Calculus 2, by Dave Ruch

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.

• Braess’ Paradox in City Planning: A Mini-Primary Source Project for Multivariable Calculus Students, by Kenneth M Monks
• Fourier's Heat Equation and the Birth of Modern Climate Science: A Mini-Primary Source Project for Differential Equations and Multivariable Calculus Students, by Kenneth M Monks

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.

• Fourier's Heat Equation and the Birth of Modern Climate Science: A Mini-Primary Source Project for Differential Equations and Multivariable Calculus Students, by Kenneth M Monks
• Solving Linear Higher Order Differential Equations with Euler and Johann Bernoulli: A Mini-Primary Source Project for Differential Equations Students, by Adam E. Parker
• Solving First-Order Linear Differential Equations: Three Mini-Primary Source Projects for Differential Equations Students, by Adam E. Parker
• Wronskians and Linear Independence: A Theorem Misunderstood by Many – A Mini-Primary Source Project for Students of Differential Equations, Linear Algebra and Others, by Adam Parker

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.
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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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