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Publisher:

Mathematical Association of America

Publication Date:

2009

Number of Pages:

507

Format:

Hardcover

Series:

Classroom Resource Materials

Price:

74.95

ISBN:

9780883857618

Category:

Anthology

[Reviewed by , on ]

Allen Stenger

01/26/2010

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 MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

**Preface**

**Introduction**

**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 lim

**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 0^{0}

On the Indeterminate Form 0^{0}

Variations on a Theme of Newton

A Useful Notation for Rules of Integration

Wavefronts, Box Diagrams, and the Product Rule: A Discovery Approach

(x^{n})' = nx^{n-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 x^{1/2}, 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**

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