Worldwide Pre-Calculus

Kenneth Kuttler




Worldwide Pre-Calculus

Worldwide Pre-Calculus
Kenneth Kuttler – Brigham Young University

ISBN-10: 0-9842071-0-4
ISBN-13: 978-0-9842071-0-7
259 Pages
©2011 Worldwide Center of Mathematics, LLC

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When students have taken a good course in college algebra and one in trigonometry, they should be ready for calculus. However, they often either forget this material or some important topics were omitted. This is why there is a need for pre-calculus courses.

This book emphasizes the topics, which are often a source of difficulty for calculus students: functions, trigonometry identities, binomial theorem, and basic area formulas, for example. However, mathematics does not begin with calculus. There are many fascinating and useful concepts, which can be studied with no exposure to calculus. The book includes some of these, such as applications to finance, probability, partial fractions, and the abstract concept of a field. In addition, rigorous explanations of standard topics from algebra and trigonometry are included: the binomial theorem, the length of a circular arc, the definitions of the exponential and log functions, the theory of partial fractions, for example.


Introduction Video


Coming Soon




Chapter 1. Numbers
1.1 The Number Line And Field Axioms
1.2 Exercises
1.3 Order
1.4 Set Notation
1.5 Exercises
1.6 Order, The Short List
1.7 The Absolute Value
1.8 Exercises
1.9 Well Ordering And Archimedean Property
1.10 Exercises
1.11 Division Of Numbers
1.11.1 Review Of The Standard Algorithm
1.11.2 General Theory
1.12 Exercises
1.13 Rational Exponents
1.14 Completeness of R
1.15 Existence Of Roots
1.16 Exercises
1.17 Counting
1.17.1 Combinations
1.17.2 The Binomial Theorem
1.18 Exercises
1.19 Counting And Basic Probability
1.20 Exercises

Chapter 2. Functions      
2.1 Generalities
2.2 Real Functions
2.3 Cartesian Coordinates And Graphs
2.4 Exercises
2.5 Quadratic Functions
2.5.1 Maximizing And Minimizing
2.5.2 Solving Quadratic Equations
2.6 Exercises
2.7 Asymptotes
2.8 Exercises

Chapter 3. Division      
3.1 Division And Integers
3.2 Exercises
3.3 Rational Root Theorem
3.4 Exercises
3.5 Division And Polynomials
3.6 The Standard Algorithm
3.7 The Theory Of Division By Polynomials
3.8 Factoring Polynomials
3.9 Exercises
3.10 Technique Of Partial Fractions
3.11 Exercises
3.12 Theory Of Partial Fractions
3.12.1 Polynomials With Coefficients In A Field
3.12.2 Real Polynomials
3.13 Field Extensions
3.14 Exercises

Chapter 4. Sequences And Series
4.1 Basic Concepts
4.2 Finding A Formula
4.3 Exercises
4.4 Arithmetic And Geometric Sequences
4.4.1 The nth Term
4.4.2 The Sum
4.4.3 Compound Interest
4.4.4 Annuities
4.5 Exercises
4.6 The Limit Of A Sequence
4.6.1 Sequences And Completeness
4.6.2 Decimals
4.6.3 Infinite Series
4.7 Exercises

Chapter 5. Basic Geometry And Trigonometry
5.1 Similar Triangles And Pythagorean Theorem
5.2 Exercises
5.3 Distance Formula And Trigonometric Functions
5.4 Exercises
5.5 The Circular Arc Subtended By An Angle
5.6 The Length Of A Circular Arc
5.7 An Important Inequality
5.8 Exercises
5.9 The Trigonometric Functions
5.9.1 A Fundamental Identity
5.9.2 Reference Angles And Other Identities
5.10 Exercises
5.11 Some Basic Area Formulas
5.11.1 Areas Of Triangles And Parallelograms
5.11.2 The Area Of A Circular Sector
5.12 Exercises

Chapter 6. Exponential Functions And Logarithms
6.1 The Exponential Function
6.2 The Existence Of The Exponential Function
6.3 The Natural Logarithm
6.4 Another Approach
6.5 Raising A Positive Number To A Real Exponent
6.6 Applications
6.6.1 Interest Compounded Continuously
6.6.2 Exponential Growth And Decay
6.7 Logarithms
6.8 Exercises

Chapter 7. Parabolas, Ellipses, and Hyperbolas 
7.1 The Parabola
7.2 The Ellipse
7.3 The Hyperbola
7.4 Exercises

Chapter 8. Polar Coordinates And Graphs
8.1 Exercises

Chapter 9. The Complex Numbers
9.1 Polar Form Of Complex Numbers
9.2 Roots Of Complex Numbers
9.3 The Quadratic Formula
9.4 Exercises

Chapter 10. Systems Of Equations
10.1 Systems Of Equations, Geometric Interpretations
10.2 Systems Of Equations, Algebraic Procedures
10.2.1 Elementary Operations
10.2.2 Gauss Elimination
10.3 Exercises

Chapter 11. Vectors
11.1 Rn
11.2 Algebra in Rn
11.3 Geometric Meaning Of Vectors
11.4 Geometric Meaning Of Vector Addition
11.5 Distance Between Points In Rn Length Of A Vector
11.6 Meaning Of Scalar Multiplication
11.7 Lines
11.8 Exercises
11.9 Vectors And Physics
11.10 Exercises

Chapter 12. Vector Products
12.1 The Dot Product
12.2 The Significance Of The Dot Product
12.2.1 The Angle Between Two Vectors
12.2.2 Work And Projections
12.3 Exercises
12.4 The Cross Product
12.4.1 The Box Product
12.4.2 A Proof Of The Distributive Law
12.4.3 Torque
12.5 Exercises




- Each instructor gets a PDF copy of the student version for presentation purposes.

- Each instructor gets a PDF copy of the instructor's version which has a solutions manual at the end with clickable links to the corresponding exercises in the text.

- The text emphasizes the techniques and basic concepts, including some which are very elementary. Theoretical explanations are also given in a separate section for those who may be interested.

- Solutions to selected exercises are presented in the back of the student version. Detailed explanations are given for the most difficult exercises.

- Alternative explanations are often given for the most significant concepts and there are many illustrations to aid in understanding the concepts.

- Some topics are developed further in the exercises.

- Explanations are given for the need for precise denitions.

- Warnings are given about common fallacies.

- Emphasis is placed on writing understandable explanations.

- The content is not software intensive or specic. There are many links to videos which contain explanations about the topics discussed, but the book can be used either with or without technology.

- The book is comparatively short, allowing it to be read more easily.




Additional Exercises
Upon request, additional exercises with accompanying solution manuals are available. These can be used to provide students additional exercises to those already in the book, or as a source for exam problems. Alternatively, the solutions to these exercises could be given to students to provide an extensive list of supplementary examples. For example, an instructor could assign a set of exercises from one version of these files and hand out the solutions to another version in case the students are having difficulty. It is also possible to provide a complete solutions manual based on one version for these exercises without giving away all the answers to another version. The option exists to have a fresh set of exercises for the book each time it is used. Those who have scientific notebook 5.5 or scientific workplace 5.5, are welcome to the three files used to generated these random exercises and solutions, along with needed wmf files. These can be modified as desired.

Video Examples (Free)
Links to video example by the author cover each topic in the textbooks.

Faculty Solution Manual (Free)
Faculty can request a free faculty solution manual by contacting

WebWork Homework and Testing
Coming soon.




Kenneth Kuttler
Kenneth Kuttler has taught calculus and courses based on calculus for over thirty years. The first half of his career was spent at Michigan Tech. University, where he taught mainly calculus and differential equations to engineering students with an occasional course on vector analysis, linear algebra, advanced calculus or topology. He then went to BYU where he taught engineering math, advanced calculus, real analysis and linear algebra. Kuttler's main interest is in mathematical analysis, especially the mathematical questions encountered in the study of nonlinear problems from physics and engineering. He has also written several books, two graduate level analysis books and a calculus book. Recently, he has become interested in stochastic integration and stochastic evolution equations.Vermont.