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Foundations of mathematics : algebra, geometry, trigonometry, calculus

Author: Philip Brown
Publisher: Dulles, Virginia : Mercury Learning and Information, [2016] ©2016
Series: Essentials of mathematics.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"Foundations of Mathematics offers the university student or interested reader a unique reference book by covering the basics of algebra, trigonometry, geometry, and calculus. There are many instances in the book to demonstrate the interplay and interconnectedness of these topics. It presents definitions and examples throughout for clear, easy learning. Numerous exercises are included at the ends of the chapters,  Read more...
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Document Type: Book
All Authors / Contributors: Philip Brown
ISBN: 9781942270751 1942270755
OCLC Number: 922914280
Notes: Includes index.
Description: xvii, 381 pages : illustrations ; 28 cm.
Contents: 1.1. Introduction --
1.2. Numbers --
1.3. Fractions and Inequalities --
1.4. Notation for Sets --
1.5. Intervals of the Real Line and Set Intersections --
1.6. Absolute Value of a Real Number --
1.7. Algebraic Operations --
1.7.1. Addition and Subtraction --
1.7.2. Multiplication --
1.7.3. Division --
1.7.4. Recurring Decimals and Irrational Numbers --
1.8. Algebraic System on the Set of Real Numbers --
1.8.1. Foil --
1.9. Natural Numbers as Exponents --
1.10. Order of Operations --
1.11. Laws of Division --
1.12. Decimal Notation --
1.12.1. Decimal Representation of Fractions --
1.12.2. Scientific Notation and Precision --
1.12.3. Decimal Representation of Irrational Numbers --
1.12.4. Conversion of Decimals into Fractions --
1.12.5. Rounding Off Decimals --
1.13. Divisibility of Natural Numbers --
1.13.1. Prime Decomposition of a Natural Number --
1.13.2. Finding the Prime Factors of a Natural Number --
1.13.3. Testing for Divisibility by Small Prime Numbers --
1.13.4. Adding Fractions Using Their Lowest Common Denominator --
1.14. Laws for Exponents --
1.15. Radicals --
Exercises --
2.1. Introduction --
2.2. Working in a Coordinate System --
2.3. Linear Equations and Straight Lines --
2.3.1. Graph of a Linear Equation --
2.3.2. Linear Equation of a Line --
2.3.3. Linear Relationships in Statistical Analysis --
2.3.4. Parallel and Perpendicular Lines --
2.3.5. Distance between Points on a Line --
2.3.6. Equation of a Perpendicular Bisector --
2.4. Circles in the Cartesian Plane --
2.5. Conic Sections in the Cartesian Plane --
2.6. Vector Algebra --
2.6.1. Addition and Scalar Multiplication of Vectors --
2.6.2. Subtraction of Vectors --
2.6.3. Standard Basis Vectors --
2.6.4. Dot Product of Vectors --
2.6.5. Triangle Inequality and the Parallelogram Law --
Exercises --
3.1. Introduction --
3.2. Solving Linear Equations --
3.3. Solving Quadratic Equations by Completing the Square --
3.4. Polynomials --
3.4.1. Addition and Subtraction of Polynomials --
3.4.2. Multiplication of Polynomials --
3.4.3. Polynomials in More Than One Variable --
3.4.4. Long Division of Polynomials --
3.4.5. Remainder Theorem and the Factor Theorem --
3.5. Properties of Quadratic Polynomials --
3.5.1. Graphs of Quadratic Polynomials --
3.5.2. Nature of the Roots --
3.5.3. Factorizing Quadratic Polynomials --
3.5.4. Solving Quadratic Equations by Factorizing --
3.6. Complex Numbers as Matrices --
3.6.1. Algebra of 2 x 2 Matrices --
3.6.2. Complex Numbers as 2 x 2 Matrices --
3.7. Roots of Polynomials --
3.7.1. Factorization Theorems --
3.7.2. Method For Finding the Integer and Rational Roots of a Polynomial --
3.8. Graphs of Polynomials --
3.9. Solving Cubic, Quartic, and Quintic Equations --
3.9.1. Solving Cubic Equations --
3.9.2. Solving Quartic Equations --
3.9.3. Solving Quintic Equations --
Exercises --
4.1. Introduction --
4.2. Angles in the Cartesian Plane --
4.3. Trigonometric Ratios --
4.4. Special Angles --
4.5. Negative Angles and Periodicity --
4.6. Reciprocal Trigonometric Ratios --
4.7. Cofunction Identities --
4.8. Trigonometric Graphs --
4.8.1. Generation of a Sine Curve --
4.8.2. Sine and Cosine Graphs --
4.8.3. Scaling and Shifting of the Sine and Cosine Graphs --
4.8.4. Tangent and Cotangent Graphs --
4.8.5. Cosecant and Secant Graphs --
4.9. Pythagorean Identities --
4.10. Solving Basic Trigonometric Equations --
4.11. Addition Identities --
4.12. Double-Angle and Half-Angle Identities --
4.13. Solving Triangles --
4.13.1. Right Triangles --
4.13.2. Area Formula and the Sine Rule --
4.13.3. Cosine Rule --
4.14. Vectors and Trigonometry --
4.14.1. Components of a Vector --
4.14.2. Geometric Interpretation of the Dot Product --
4.15. More Identities --
Exercises --
5.1. Introduction --
5.2. Relations and Functions --
5.3. Visualizing Functions --
5.3.1. Graphs of Equations --
5.3.2. Vertical Line Test --
5.4. Absolute Value Function --
5.5. Exponential Functions --
5.5.1. Fractional Exponents --
5.5.2. Irrational Exponents --
5.5.3. Graphs of Exponential Functions --
5.6. Rational Functions --
5.7. Root Functions --
5.8. Piecewise Defined Functions --
5.9. Symmetry of Functions --
5.10. Operations on Functions --
5.10.1. Algebra of Functions --
5.10.2. Compositions of Functions --
5.11. Transformations of Functions --
5.11.1. Vertical and Horizontal Shifts --
5.11.2. Vertical and Horizontal Scaling --
5.11.3. Reflections Across the Axes --
5.12. Vector-Valued Functions --
5.12.1. Vector-Valued Function for a Circle --
5.12.2. Vector-Valued Function for a Line --
5.12.3. Exploring Vector-Valued Functions --
5.13. Inverse Functions --
5.13.1. Inverse of a Point --
5.13.2. Logarithmic Functions --
5.13.3. Inversion of One-to-One Functions --
5.13.4. Increasing and Decreasing Functions --
5.13.5. Inverse Trigonometric Functions --
Exercises --
6.1. Introduction --
6.2. Algebra of Rational Expressions --
6.2.1. Multiplying and Dividing Rational Expressions --
6.2.2. Adding and Subtracting Rational Expressions --
6.3. Algebra with Rational Exponents --
6.4. Solving Equations --
6.5. Partial Fractions --
6.6. Inequalities --
6.6.1. Simplifying Inequalities --
6.6.2. Solving Inequalities --
6.6.3. Two-Variable Inequalities --
Exercises --
7.1. Introduction --
7.2. Method of Exhaustion --
7.3. Sequences --
7.4. Limits of a Function --
7.5. Continuity --
7.5.1. Definition of Continuity at a Point --
7.5.2. Discontinuity at a Point --
7.5.3. Continuity on an Interval --
7.5.4. Continuous Functions --
7.5.5. More Continuous Functions --
7.6. Computing Limits --
7.6.1. Limits in the Domain of a Continuous Function --
7.6.2. Limits Involving Piecewise-Defined Functions --
7.6.3. Computing Limits by Simplification --
7.7. Applications of Continuity --
7.7.1. Intermediate Value Theorem --
7.8. Horizontal Asymptotes --
7.9. Vertical Asymptotes of Rational Functions --
7.10. Squeeze Theorem and Rules for Limits --
7.10.1. Squeeze Theorem --
7.10.2. Rules for Limits --
Exercises --
8.1. Introduction --
8.2. Definition of the Derivative --
8.2.1. Graphs Tangential to the x-axis at the Origin --
8.2.2. Tangent Line to a Graph at the Origin --
8.2.3. Formula for the Derivative --
8.2.4. Definition of the Derivative (General Case) --
8.3. Derivative Functions --
8.3.1. Power Rule for Natural Numbers --
8.3.2. Leibniz Notation --
8.3.3. Sum, Product, and Quotient Rules --
8.4. Tangent Line Problems --
8.5. Power Rule for Rational Exponents --
8.6. Derivatives of Trigonometric Functions --
8.7. Some Basic Applications of Calculus --
8.8. Chain Rule --
8.9. Calculus of Vector-Valued Functions --
8.10. Calculus of Exponential and Logarithmic Functions --
8.10.1. Formula fore --
8.10.2. Derivatives of Exponential Functions --
8.10.3. Derivatives of Logarithmic Functions --
8.10.4. Proof of the Power Rule --
8.11. Derivatives of the Inverse Trigonometric Functions --
Exercises --
9.1. Introduction --
9.2. Euclid's Elements --
9.3. Terminology --
9.3.1. Lines, Angles, and Polygons --
9.3.2. Circles --
9.3.3. Other Important Terms --
9.4. Basic Problem Solving in Geometry --
9.5. Elementary Theorems Relating to Lines and Polygons --
9.5.1. Theorems about Angles --
9.5.2. Theorems about Triangles --
9.5.3. Parallelograms and Parallel Lines --
9.5.4. Concurrency, Proportionality, and Similarity --
9.6. Elementary Theorems Relating to Circles --
9.6.1. Chords and Subtended Angles --
9.6.2. Cyclic Quadrilaterals --
9.6.3. Tangent Lines and Secant Lines --
9.7. Examples and Applications --
Exercises --
Appendix: Geometry Theorems --
10.1. Introduction --
10.2. Planes and Spheres --
10.2.1. Planes in Space --
10.2.2. Spheres --
10.2.3. Spherical Triangles --
10.3. Vectors in Space --
10.3.1. Cross Product of Two Vectors --
10.3.2. Parallelepipeds and Cross Product Identities --
10.3.2. Angle Between Two Planes --
10.4. Solving Spherical Triangles --
10.5. Solving Right Spherical Triangles (I) --
10.6. Solving Right Spherical Triangles (II) --
10.6.1. Rules for Quadrants --
10.6.2. Napier's Rules --
Exercises.
Series Title: Essentials of mathematics.
Responsibility: Philip Brown, Texas A & M University at Galveston.

Abstract:

Foundations of Mathematics offers the university student an unique reference book by covering the basics of algebra, trigonometry, geometry, and calculus. There are many instances in the book to  Read more...

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