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The Calculus Collection: A Resource for AP and Beyond

Caren L. Diefenderfer and Roger B. Nelsen, editors
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
[Reviewed by
Allen Stenger
, on

This is a collection of articles on calculus from the MAA journals, aimed at teachers of AP calculus. It is in a sense a continuation of the earlier two-volume set A Century of Calculus, containing mostly articles published since that collection was published in 1992. The flavor is subtly different: the earlier volumes focused on clever derivations, while the present volume deals mostly with better ways to explain mainstream topics. The present volume is aimed at high-school teachers, and they will find much here that is useful, with most of it is presented in a way that they will be able to understand without too much work. For calculus experts it is less interesting to browse through that the earlier volumes, because experts will already know most of the approaches here.

I browsed through the book and did not read every article. My favorites were: “A Tale of Two CDs” (technological progress and how it has affected everything in our lives except calculus teaching), “The Best Shape for a Tin Can” (why tin cans in grocery stores do not follow the model predicted by calculus, and developing a more realistic model), and “Gabriel’s Wedding Cake” (a squared-off version of Gabriel’s Horn that is manifested as a wedding cake with an infinite number of layers, and for which it is easier to estimate the volume and surface area).

The articles are very clear and sharp reproductions, and appear to have been printed photographically from the originals (except for articles from MAA FOCUS, which were reflowed to fit the page). The titles of all the articles were re-typeset, and unfortunately this introduced a lot of typographical errors, both on the article pages and in the table of contents (different errors in the two places). The articles were cut up and arranged into whole pages in the book, and in a few cases this caused some glitches. The article on p. 420 has lost its first sentence and starts abruptly with a series of formulas. On p. 69 is a review of a book titled “Calculus with Analytic Geometry,” but the author and publisher of the work were lost. The review spends most of its time discussing calculus books throughout history, and as no bibliographic data is shown we get the impression that we are reading a post-modern meta-review of all possible calculus books. (The review is of George F. Simmons’s 1985 volume from McGraw-Hill, and it gets only one paragraph.)

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.



Part 0. General and Historical Articles
Touring the Calculus Gallery
Calculus: A Modern Perspective
Two Historical Applications of Calculus
Ideas of Calculus in Islam and India
A Tale of Two CDs
The Changing Face of Calculus
Things I Have Learned at the AP Calculus Reading
Book Review: “Calculus with Analytic Geometry”
The All-Purpose Calculus Problem

Part 1. Functions, Graphs and Limits
Graphs of Rational Functions for Computer Assisted Calculus
Computer-Aided Delusions
An Overlooked Calculus Question
Introduction to Limits, or Why Can’t We Just Trust the Table?
A Circular Argument
A Geometric Proof of limd→0+{–d ln(d)} = 0

Part 2. Derivatives
The Changing Concept of Change: The Derivative from Fermat to Weierstrass
Derivatives Without Limits
Rethinking Rigor in Calculus: The Role of the Mean Value Theorem
Rolle over Lagrange — Another Shot at the Mean Value Theorem
An Elementary Proof of a Theorem in Calculus
A Simple Auxiliary Function for the Mean Value Theorem
A Note on the Derivative of a Composite Function
Do Dogs Know Calculus?
Do Dogs Know Related Rates Rather than Optimization?
Do Dogs Know Bifurcations?
The Lengthening Shadow: The Story of Related Rates
The Falling Ladder Paradox
Solving the Ladder Problem on the Back of an Envelope
How Not to Land at Lake Tahoe!
The Best Shape for a Tin Can
To Build a Better Box
The Curious 1/3
Hanging a Bird Feeder: Food for Thought
Honey, Where Should We Sit?
A Dozen Minima for a Parabola
Maximizing the Area of a Quadrilateral
A Generalization of the Minimum Area Problem
Constrained Optimization and Implicit Differentiation
For Every Answer There Are Two Questions
Old Calculus Chestnuts: Roast, or Light a Fire?
Cable-laying and Intuition
Descartes Tangent Lines
Can We Use the First Derivative to Determine Inflection Points?
Differentiate Early. Differentiate Often!
A Calculus Exercise For the Sums or Integer Powers
L’Hopital’s Rule Via Integration
Indeterminate Forms Revisited
The Indeterminate Form 00
On the Indeterminate Form 00
Variations on a Theme of Newton
A Useful Notation for Rules of Integration
Wavefronts, Box Diagrams, and the Product Rule: A Discovery Approach
(xn)' = nxn-1: Six Proofs
Sines and Cosines of the Times
The Spider’s Spacewalk Derivation of sin' and cos'
Differentiability of Exponential Functions
A Discover-e
An Exponential Rule
The Derivative of Arctan x
The Derivative of the Inverse Sine
Graphs and Derivatives of the Inverse Trig Functions
Logarithmic Differentiation: Two Wrongs Make a Right
Comparison of Two Elementary Approximation Methods

Part 3. Integrals
How Should We Introduce Integration?
Evaluating Integrals Using Self-Similarity
Self-Integrating Polynomials
Symmetry and Integration
Sums and Differences vs lntegrals and Derivatives
How Do You Slice the Bread?
Disks and Shells Revisited
Disks, Shells, and Integrals of Inverse Functions
Characterizing Power Functions by Volumes of Revolution
Gabriel’s Wedding Cake
Can You Paint a Can or Paint?
A Paradoxical Paint Pail
Dipsticks for Cylindrical Storage Tanks — Exact and Approximate
Finding Curves with Computable Arc Length
Arc Length and Pythagorean Triples
Maximizing the Arclength in the Cannonball Problem
An Example Demonstrating the Fundamental Theorem of Calculus
Barrow’s Fundamental Theorem
The Point-slope Formula leads to the Fundamental Theorem of Calculus
A Generalization of the Mean Value Theorem for Integrals
Proof Without Words: Look Ma, No Substitution!
Integration by Parts
Tabular Integration by Parts
More on Tabular Integration by Parts
A Quotient Rule Integration by Parts Formula
Partial Fraction Decomposition by Division
Partial Fractions by Substitution
Proof Without Words: A Partial Fraction Decomposition
Four Crotchets on Elementary Integration
An Application of Geography to Mathematics: History of the Integral of the Secant
How to Avoid the Inverse Secant (and Even the Secant Itself)
The Integral of x1/2, etc.
A Direct Proof of the Integral Formula for Arctangent
Riemann Sums and the Exponential Function
Proofs Without Words Under the Magie Curve
Math Without Words: Integrating the Natural Logarithm
Integrals of Products of Sine and Cosine with Different Arguments
Moments on a Rose Petal
A Calculation of ∫0 e-x2 dx
Calculus in the Operating Room
Physical Demonstrations in the Calculus Classroom
Who Needs the Sine Anyway?
Finding Bounds for Definite Integrals
Estimating Definite Integrals
Proof Without Words: The Trapezoidal Rule (for Increasing Functions)
Behold! The Midpoint Rule is Better Than the Trapezoidal Rule for Concave Functions
An Elementary Proof of Error Estimates for the Trapezoidal Rule
Pictures Suggest How to Improve Elementary Numeric Integration

Part 4. Polynomial Approximation and Series
The Geometric Series in Calculus
A Visual Approach to Geometric Series
The Telescoping Series in Perspective
Proof Without Words (Alternating series)
The Bernoullis and the Harmonic Series
On Rearrangements of the Alternating Harmonic Series
An Improved Remainder Estimate for Use with the Integral Test
A Differentiation Test for Absolute Convergence
Math Bite: Equality of Limits in Ratio and Root Tests
Another Proof of the Formula e = Σ (1/n!)
Taylor Polynomial Approximations in Polar Coordinates
The Taylor Polynomials of sin θ

Appendixes [sic]
I. Topic Outline for AP Calculus Courses
II. Suggested Uses for the Articles in a First-year Calculus Course


Author Index

About the Editors