Pre-Calculus, Calculus, and Beyond

This is the last of three volumes that, together, give an exposition of the mathematics of grades 9–12 that is simultaneously mathematically correct and grade-level appropriate. The volumes are consistent with CCSSM (Common Core State Standards for Mathematics) and aim at presenting the mathematics of K–12 as a totally transparent subject.

This volume distinguishes itself from ...

(展开全部)

This is the last of three volumes that, together, give an exposition of the mathematics of grades 9–12 that is simultaneously mathematically correct and grade-level appropriate. The volumes are consistent with CCSSM (Common Core State Standards for Mathematics) and aim at presenting the mathematics of K–12 as a totally transparent subject.

This volume distinguishes itself from others of the same genre in getting the mathematics right. In trigonometry, this volume makes explicit the fact that the trigonometric functions cannot even be defined without the theory of similar triangles. It also provides details for extending the domain of definition of sine and cosine to all real numbers. It explains as well why radians should be used for angle measurements and gives a proof of the conversion formulas between degrees and radians.

In calculus, this volume pares the technicalities concerning limits down to the essential minimum to make the proofs of basic facts about differentiation and integration both correct and accessible to school teachers and educators; the exposition may also benefit beginning math majors who are learning to write proofs. An added bonus is a correct proof that one can get a repeating decimal equal to a given fraction by the “long division” of the numerator by the denominator. This proof attends to all three things all at once: what an infinite decimal is, why it is equal to the fraction, and how long division enters the picture.

This book should be useful for current and future teachers of K–12 mathematics, as well as for some high school students and for education professionals.

Cover1
Title page4
Contents8
Contents of the Companion Volumes and Structure of the Chapters10
Preface12
To the Instructor20
· · · · · · (更多)

Cover1
Title page4
Contents8
Contents of the Companion Volumes and Structure of the Chapters10
Preface12
To the Instructor20
To the Pre-Service Teacher34
Prerequisites38
Some Conventions40
Chapter 1. Trigonometry42
1.1. Sine and cosine43
1.2. The unit circle48
1.3. Basic facts73
1.4. The addition formulas82
1.5. Radians94
1.6. Multiplication of complex numbers111
1.7. Graphs of equations of degree 2, revisited120
1.8. Inverse trigonometric functions129
1.9. Epilogue139
Chapter 2. The Concept of Limit144
2.1. The real numbers and FASM144
2.2. The meaning of convergence159
2.3. Basic properties of convergent sequences175
2.4. First consequences of the least upper bound axiom187
2.5. The existence of positive 𝑛-th roots196
2.6. Fundamental theorem of similarity204
Chapter 3. The Decimal Expansion of a Number208
3.1. Decimals and infinite series208
3.2. Repeating decimals214
3.3. The decimal expansion of a real number223
3.4. The decimal expansion of a fraction231
3.5. More on infinite series241
Chapter 4. Length and Area250
Overview of Chapters 4 and 5250
4.1. Fundamental principles of geometric measurements252
4.2. Length257
4.3. Rectifiable curves265
4.4. Area of rectangles and the Pythagorean theorem270
4.5. Areas of triangles and polygons278
4.6. Areas of disks and circumferences of circles289
4.7. The general concept of area294
Chapter 5. 3-Dimensional Geometry and Volume306
5.1. Comments about three dimensions306
5.2. Cavalieri’s principle311
5.3. General remarks on volume313
5.4. Volume of a sphere319
5.5. Pedagogical comments323
Chapter 6. Derivatives and Integrals326
6.1. Continuity326
6.2. Basic theorems on continuous functions337
6.3. The derivative349
6.4. The mean value theorem355
6.5. Integrals of continuous functions369
6.6. The fundamental theorem of calculus381
6.7. Appendix. The trigonometric functions386
Chapter 7. Exponents and Logarithms, Revisited404
7.1. Logarithm as an integral405
7.2. The exponential function409
7.3. The laws of exponents413
7.4. Other exponential and logarithmic functions419
Appendix: Facts from the Companion Volumes424
Glossary of Symbols438
Bibliography442
Index446
· · · · · · (收起)