Hardcover ISBN:  9780883857885 
Product Code:  CLRM/49 
List Price:  $59.00 
MAA Member Price:  $44.25 
AMS Member Price:  $44.25 
eBook ISBN:  9781614441205 
Product Code:  CLRM/49.E 
List Price:  $55.00 
MAA Member Price:  $41.25 
AMS Member Price:  $41.25 
Hardcover ISBN:  9780883857885 
eBook: ISBN:  9781614441205 
Product Code:  CLRM/49.B 
List Price:  $114.00 $86.50 
MAA Member Price:  $85.50 $64.88 
AMS Member Price:  $85.50 $64.88 
Hardcover ISBN:  9780883857885 
Product Code:  CLRM/49 
List Price:  $59.00 
MAA Member Price:  $44.25 
AMS Member Price:  $44.25 
eBook ISBN:  9781614441205 
Product Code:  CLRM/49.E 
List Price:  $55.00 
MAA Member Price:  $41.25 
AMS Member Price:  $41.25 
Hardcover ISBN:  9780883857885 
eBook ISBN:  9781614441205 
Product Code:  CLRM/49.B 
List Price:  $114.00 $86.50 
MAA Member Price:  $85.50 $64.88 
AMS Member Price:  $85.50 $64.88 

Book DetailsClassroom Resource MaterialsVolume: 49; 2015; 169 pp
A 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 firstyear 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 CauchySchwarz inequality, the arithmetic meangeometric mean inequality, and the EulerMascheroni 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 the author hopes that they will be more accessible to teachers of calculus, both for use in the classroom and as supplementary explorations for students.

Table of Contents

Cover

Half title

Copyright

Title

Series

Dedication

Preface

Contents

Part I Limits and Differentiation

1 The limit of (sin t)/t

2 Approximating π with the limit of (sint)/t

3 Visualizing the derivative

4 The product rule

5 The quotient rule

6 The chain rule

7 The derivative of the sine

8 The derivative of the arctangent

9 The derivative of the arcsine

10 Means and the mean value theorem

11 Tangent line inequalities

12 A geometric illustration of the limit for e

13 Which is larger, eπ or πe? ab or ba?

14 Derivatives of area and volume

15 Means and optimization

Part II Integration

16 Combinatorial identities for Riemann sums

17 Summation by parts

18 Integration by parts

19 The world's sneakiest substitution

20 Symmetry and integration

21 Napier's inequality and the limit for e

22 The nth root of n! and another limit for e

23 Does shell volume equal disk volume?

24 Solids of revolution and the CauchySchwarz inequality

25 The midpoint rule is better than the trapezoidal rule

26 Can the midpoint rule be improved?

27 Why is Simpson's rule exact for cubics?

28 Approximating π with integration

29 The HermiteHadamard inequality

30 Polar area and Cartesian area

31 Polar area as a source of antiderivatives

32 The prismoidal formula

Part III Infinite Series

33 The geometry of geometric series

34 Geometric differentiation of geometric series

35 Illustrating a telescoping series

36 Illustrating applications of the monotone sequence theorem

37 The harmonic series and the EulerMascheroni constant

38 The alternating harmonic series

39 The alternating series test

40 Approximating π with Maclaurin series

Part IV Additional Topics

41 The hyperbolic functions I: Definitions

42 The hyperbolic functions II: Are they circular?

43 The conic sections

44 The conic sections revisited

45 The AMGM inequality for n positive numbers

Part V Appendix: Some Precalculus Topics

46 Are all parabolas similar?

47 Basic trigonometric identities

48 The addition formulas for the sine and cosine

49 The double angle formulas

50 Completing the square

Solutions to the Exercises

References

Index

About the Author


Additional Material

Reviews

Visualizing mathematical ideas usually reduces the complexity of topics and therefore has educational value. This plays an essential role in courses such as calculus, which are both fundamental and scheduled for firstyear students. ... The book under review is an interesting and pretty collection of proofs of material from the firstyear course, all based on visualizing ideas. ... This is not a standard textbook, but it is a very useful complement for both students and instructors in a firstyear calculus course.
Mehdi Hassani, MAA Reviews 
... This unique, studentfriendly text should be required reading for anyone enrolling in a firstyear calculus course, especially for those who are math challenged. ...
D. J. Gougeon, CHOICE Connect 
... Even the most experienced calculus instructor will likely find something new and useful in this slim volume.
CMS Notices


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 Book Details
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 Reviews
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A 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 firstyear 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 CauchySchwarz inequality, the arithmetic meangeometric mean inequality, and the EulerMascheroni 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 the author hopes that they will be more accessible to teachers of calculus, both for use in the classroom and as supplementary explorations for students.

Cover

Half title

Copyright

Title

Series

Dedication

Preface

Contents

Part I Limits and Differentiation

1 The limit of (sin t)/t

2 Approximating π with the limit of (sint)/t

3 Visualizing the derivative

4 The product rule

5 The quotient rule

6 The chain rule

7 The derivative of the sine

8 The derivative of the arctangent

9 The derivative of the arcsine

10 Means and the mean value theorem

11 Tangent line inequalities

12 A geometric illustration of the limit for e

13 Which is larger, eπ or πe? ab or ba?

14 Derivatives of area and volume

15 Means and optimization

Part II Integration

16 Combinatorial identities for Riemann sums

17 Summation by parts

18 Integration by parts

19 The world's sneakiest substitution

20 Symmetry and integration

21 Napier's inequality and the limit for e

22 The nth root of n! and another limit for e

23 Does shell volume equal disk volume?

24 Solids of revolution and the CauchySchwarz inequality

25 The midpoint rule is better than the trapezoidal rule

26 Can the midpoint rule be improved?

27 Why is Simpson's rule exact for cubics?

28 Approximating π with integration

29 The HermiteHadamard inequality

30 Polar area and Cartesian area

31 Polar area as a source of antiderivatives

32 The prismoidal formula

Part III Infinite Series

33 The geometry of geometric series

34 Geometric differentiation of geometric series

35 Illustrating a telescoping series

36 Illustrating applications of the monotone sequence theorem

37 The harmonic series and the EulerMascheroni constant

38 The alternating harmonic series

39 The alternating series test

40 Approximating π with Maclaurin series

Part IV Additional Topics

41 The hyperbolic functions I: Definitions

42 The hyperbolic functions II: Are they circular?

43 The conic sections

44 The conic sections revisited

45 The AMGM inequality for n positive numbers

Part V Appendix: Some Precalculus Topics

46 Are all parabolas similar?

47 Basic trigonometric identities

48 The addition formulas for the sine and cosine

49 The double angle formulas

50 Completing the square

Solutions to the Exercises

References

Index

About the Author

Visualizing mathematical ideas usually reduces the complexity of topics and therefore has educational value. This plays an essential role in courses such as calculus, which are both fundamental and scheduled for firstyear students. ... The book under review is an interesting and pretty collection of proofs of material from the firstyear course, all based on visualizing ideas. ... This is not a standard textbook, but it is a very useful complement for both students and instructors in a firstyear calculus course.
Mehdi Hassani, MAA Reviews 
... This unique, studentfriendly text should be required reading for anyone enrolling in a firstyear calculus course, especially for those who are math challenged. ...
D. J. Gougeon, CHOICE Connect 
... Even the most experienced calculus instructor will likely find something new and useful in this slim volume.
CMS Notices