precalculus with geometry & trigonometry avinash sathaye [pdf]
Precalculus with Geometry and Trigonometry by Avinash Sathaye
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About this book :-
Precalculus with Geometry and Trigonometry written by
Avinash Sathaye .
This book on Precalculus with Geometry and Trigonometry should be treated as simply an enhanced version of our book on College Algebra. Most of the topics that appear here have already been discussed in the Algebra book and often the text here is a verbatim copy of the text in the other book.
We expect the student to already have a strong Algebraic background and thus the algebraic techniques presented here are more a refresher course than a first introduction. We also expect the student to be heading for higher level mathematics courses and try to supply the necessary connections and motivations for future use.
Book Detail :-
Title: Precalculus with Geometry and Trigonometry
Author(s): Avinash Sathaye
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About Author :-
Avinash Sathaye Professor of Mathematics, Department of Mathematics, University of Kentucky, Lexington, Kentucky, United States.
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Book Contents :-
Precalculus with Geometry and Trigonometry written by
cover the following topics.
Review of Basic Tools.
Underlying field of numbers, Working with Complex Numbers, Indeterminates, variables, parameters, Basics of Polynomials, Rational functions, Working with polynomials, Examples of polynomial operations, Example 1. Polynomial operations, Example 2. Collecting coefficients, Example 3. Using algebra for arithmetic, Example 4. The Binomial Theorem, Example 5. Substituting in a polynomial, Example 6. Completing the square
Solving linear equations.
What is a solution?, One linear equation in one variable, Several linear equations in one variable, Two or more equations in two variables, Several equations in several variables, Solving linear equations efficiently, Example 1. Manipulation of equations, Example 2. Cramer’s Rule, Example 3. Exceptions to Cramer’s Rule, Example 4. Cramer’s Rule with many variables
The division algorithm and applications
Division algorithm in integers, Example 1. GCD calculation in Integers, Aryabhat¯.a algorithm: Efficient Euclidean algorithm, Example 2. Kuttaka or Chinese Remainder Problem, More Kuttaka problems, Example 4. Disguised problems, Division algorithm in polynomials, Repeated Division, The GCD and LCM of two polynomials, Example 5. Aryabhat¯.a algorithm for polynomials, Example 6. Efficient division by a linear polynomial, Division by a quadratic polynomial
Introduction to analytic geometry.
Coordinate systems, Geometry: Distance formulas, Connection with complex numbers, Change of coordinates on a line, Change of coordinates in the plane, General change of coordinates, Description of Isometies, Complex numbers and plane transformations, Examples of complex transformations, Examples of changes of coordinates
Equations of lines in the plane.
Parametric equations of lines, Examples. Parametric equations of lines, Meaning of the parameter t:, Examples. Special points on parametric lines, Comparison with the usual equation of a line, Examples of equations of lines, Example 1. Points equidistant from two given points, Example 2. Right angle triangles
Special study of Linear and Quadratic Polynomials.
Linear Polynomials, Factored Quadratic Polynomial, Interval notation. Intervals on real line, The General Quadratic Polynomial, Examples of Quadratic polynomials
Plane algebraic curves, What is a function?, Modeling a function, Inverse Functions
Circle Basics, Parametric form of a circle, Application to Pythagorean Triples, Pythagorean Triples. Generation of, Examples of equations of a circle, Example 1. Intersection of two circles, Example 1a. Complex intersection of two circles, Example 2. Line joining through the intersection of two circles, Example 3. Circle through three given points, Example 4. Exceptions to a circle through three points, Example 5. Smallest circle with a given center meeting a given line, Example 6. Circle with a given center and tangent to a given line, Example 7. The distance between a point and a line, Example 8. Half plane defined by a line
Trigonometric parameterization of a circle, Definition. Trigonometric Functions, Basic Formulas for the Trigonometric Functions, Connection with the usual Trigonometric Functions, Important formulas, Using trigonometry, Proofs I, Proofs II
Looking closely at a function
Introductory examples, Parabola. Analysis near its points, Circle. Analysis near its points, Analyzing a general curve y = f(x) near a point (a, f(a)), The slope of the tangent, calculation of the derivative, Derivatives of more complicated functions, General power and chain rules, Using the derivatives for approximation, Linear Approximation. Examples
Newton’s Method, Limitations of the Newton’s Method
An analysis of √2 as a real number, Idea of a Real Number, Summation of series, A basic formula, Using the basic formula, On the exponential and logarithmic functions, Infinite series and their use, Inverse functions by series, Decimal expansion of a Rational Number, Matrices and determinants: a quick introduction
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