Nelsen R.B. Cameos for Calculus: Visualization in the First-Year Course

Mathematical Association of America, 2015. — 186 p. — (Classroom Resource Materials) — ISBN: 9780883857885, 088385788XA thespian or cinematographer might define a cameo as a brief appearance of a known figure, while a gemologist or lapidary might define it as a precious or semiprecious stone. This book presents fifty short enhancements or supplements (the Cameos) for the first-year calculus course in which a geometric figure briefly appears. Some of the Cameos illustrate mainstream topics such as the derivative, combinatorial formulas used to compute Riemann sums, or the geometry behind many geometric series. Other Cameos present topics accessible to students at the calculus level but not usually encountered in the course, such as the Cauchy-Schwarz inequality, the arithmetic mean-geometric mean inequality, and the Euler-Mascheroni constant. There are fifty Cameos in the book, grouped into five sections: Part I Limits and Differentiation; Part II Integration; Part III Infinite Series; Part IV Additional Topics, and Part V Appendix: Some Precalculus Topics. Many of the Cameos include exercises, so Solutions to all the Exercises follows Part V. The book concludes with References and an Index. Many of the Cameos are adapted from articles published in journals of the MAA, such as The American Mathematical Monthly, Mathematics Magazine, and The College Mathematics Journal. Some come from other mathematical journals, and some were created for this book. By gathering the Cameos into a book we hope that they will be more accessible to teachers of calculus, both for use in the classroom and as supplementary explorations for students.Limits and Differentiation
The limit of (sin t)/t
Approximating π with the limit of (sin t)/=t
Visualizing the derivative
The product rule
The quotient rule
The chain rule
The derivative of the sine
The derivative of the arctangent
The derivative of the arcsine
Means and the mean value theorem
Tangent line inequalities
A geometric illustration of the limit for e
Which is larger, eπ or πe? a^b or b^a?
Derivatives of area and volume
Means and optimization
Integration
Combinatorial identities for Riemann sums
Summation by parts
Integration by parts
The world’s sneakiest substitution
Symmetry and integration
Napier’s inequality and the limit for e
The nth root of n! and another limit for e
Does shell volume equal disk volume?
Solids of revolution and the Cauchy-Schwarz inequality
The midpoint rule is better than the trapezoidal rule
Can the midpoint rule be improved?
Why is Simpson’s rule exact for cubics?
Approximating with integration
The Hermite-Hadamard inequality
Polar area and Cartesian area
Polar area as a source of antiderivatives
The prismoidal formula
Infinite Series
The geometry of geometric series
Geometric differentiation of geometric series
Illustrating a telescoping series
Illustrating applications of the monotone sequence theorem
The harmonic series and the Euler-Mascheroni constant
The alternating harmonic series
The alternating series test
Approximating with Maclaurin series
Additional Topics
The hyperbolic functions I: Definitions
The hyperbolic functions II: Are they circular?
The conic sections
The conic sections revisited
The AM-GM inequality for n positive numbers
Appendix: Some Precalculus Topics
Are all parabolas similar?
Basic trigonometric identities
The addition formulas for the sine and cosine
The double angle formulas
Completing the square
Solutions to the ExercisesAbout the Author