Mathematical Time Capsules: Cusps: Horns and Beaks

This is the mathematical tale of a cusp in the shape of a bird's beak. Although precalculus and calculus courses must stress the idea of function over that of equation, they nevertheless include a number of important topics concerning polynomial equations in two variables, including implicit differentiation and the study of conic sections. Whereas polynomial functions of one variable have very simple graphs, the graphs of polynomial equations in x and y — even those of relatively low degree — can exhibit wonderfully exotic features.

The story of the bird's beak can be used to enrich a course in analytic geometry, precalculus or calculus. For students who know some calculus, it also provides insight into continuous nondifferentiable functions. There is also a connection to power series representations, although this will not be discussed in this chapter (Euler treats them in §5–9 of [1, 2]).

For further reading on these topics, see [3, 4].

Historical Background

In the 18th century, calculus and the related branches of mathematics gradually changed their perspective from the geometric to the algebraic. When Renée Descartes (1596–1650) and Pierre de Fermat (1601–1665) invented analytic geometry, for example, mathematicians were already familiar with a large assortment of curves, given by a variety of geometric constructions. Analytic geometry gave them a means of associating equations with these curves. With passing time, the study of equations took primacy, so that the graph came to be seen as an attribute of the equation.

Frontmatter: Read PDF	pp. i-vi
Preface: Read PDF	pp. vii-x
Contents: Read PDF	pp. xi-xviii
1 - The Sources of Algebra: Read PDF By Roger Cooke	pp. 1-6
2 - How to Measure the Earth: Read PDF By Lawrence D'Antonio	pp. 7-16
3 - Numerical solution of equations: Read PDF By Roger Cooke	pp. 17-22
4 - Completing the Square through the Millennia: Read PDF By Dick Jardine	pp. 23-28
5 - Adapting the Medieval “Rule of Double False Position” to the Modern Classroom: Read PDF By Randy K. Schwartz	pp. 29-38
6 - Complex Numbers, Cubic Equations, and Sixteenth-Century Italy: Read PDF By Daniel J. Curtin	pp. 39-44
7 - Shearing with Euclid: Read PDF By Davida Fischman and Shawnee McMurran	pp. 45-54
8 - The Mathematics of Measuring Time: Read PDF By Kim Plofker	pp. 55-62
9 - Clear Sailing with Trigonometry: Read PDF By Glen Van Brummelen	pp. 63-72
10 - Copernican Trigonometry: Read PDF By Victor J. Katz	pp. 73-88
11 - Cusps: Horns and Beaks: Read PDF By Robert E. Bradley	pp. 89-100
12 - The Latitude of Forms, Area, and Velocity: Read PDF By Daniel J. Curtin	pp. 101-106
13 - Descartes' Approach to Tangents: Read PDF By Daniel J. Curtin	pp. 107-110
14 - Integration à la Fermat: Read PDF By Amy Shell-Gellasch	pp. 111-116
15 - Sharing the Fun: Student Presentations: Read PDF By Amy Shell-Gellasch and Dick Jardine	pp. 117-122
16 - Digging up History on the Internet: Discovery Worksheets: Read PDF By Betty Mayfield	pp. 123-126
17 - Newton vs. Leibniz in One Hour!: Read PDF By Betty Mayfield	pp. 127-132
18 - Connections between Newton, Leibniz, and Calculus I: Read PDF By Andrew B. Perry	pp. 133-138
19 - A Different Sort of Calculus Debate: Read PDF By Vicky Williams Klima	pp. 139-150
20 - A ‘Symbolic’ History of the Derivative: Read PDF By Clemency Montelle	pp. 151-158
21 - Leibniz's Calculus (Real Retro Calc.): Read PDF By Robert Rogers	pp. 159-168
22 - An “Impossible” Problem, Courtesy of Leonhard Euler: Read PDF By Homer S. White	pp. 169-178
23 - Multiple Representations of Functions in the History of Mathematics: Read PDF By Robert Rogers	pp. 179-188
24 - The Unity of all Science: Karl Pearson, the Mean and the Standard Deviation: Read PDF By Joe Albree	pp. 189-198
25 - Finding the Greatest Common Divisor: Read PDF By J.J. Tattersall	pp. 199-202
26 - Two-Way Numbers and an Alternate Technique for Multiplying Two Numbers: Read PDF By J.J. Tattersall	pp. 203-208
27 - The Origins of Integrating Factors: Read PDF By Dick Jardine	pp. 209-214
28 - Euler's Method in Euler's Words: Read PDF By Dick Jardine	pp. 215-222
29 - Newton's Differential Equation ẏ/ẋ = 1 − 3x + y + xx + xy: Read PDF By Hüseyin Koçak	pp. 223-228
30 - Roots, Rocks, and Newton-Raphson Algorithms for Approximating √2 3000 Years Apart: Read PDF By Clemency Montelle	pp. 229-240
31 - Plimpton 322: The Pythagorean Theorem, More than a Thousand Years before Pythagoras: Read PDF By Daniel E. Otero	pp. 241-250
32 - Thomas Harriot's Pythagorean Triples: Could He List Them All?: Read PDF By Janet L. Beery	pp. 251-260
33 - Amo, Amas, Amat! What's the sum of that?: Read PDF By Clemency Montelle	pp. 261-268
34 - The Harmonic Series: A Primer: Read PDF By Adrian Rice	pp. 269-276
35 - Learning to Move with Dedekind: Read PDF By Fernando Q. Gouvêa	pp. 277-284
About the Editors: Read PDF	pp. 285-285

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