2 - Equations, functions, vectors

Abstract

This chapter provides an overview of equations, functions, and vectors. An equation with at least one variable is a form of statement that becomes a true or false statement if values of the domain of definition or range of variables and the variables are coordinated. The chapter presents algebraic equations in one variable, linear equations, quadratic equations, cubic equations, definitive equation of the nth degree, transcendental equations, exponential equations, and logarithmic equations. It also discusses approximation methods for determining the roots of an equation, such as Newton's method of approximation and iterative methods, graphical solution of equations, and systems of equations. If the mapping of the set X onto the set Y is unique, that is, if exactly one y Є Y is made to correspond to each x ∈X, the mapping is called a function. It represents the set f of ordered pairs (x, y), which is a subset of the Cartesian product R × R. The chapter highlights real functions, rational functions, irrational functions, rational integral functions, rational fractional functions, algebraic function, transcendental function, identical functions, even and odd functions, monotonic functions, continuous functions, periodic functions, and inverse functions. It discusses approximate representation of functions by interpolation formulas and graphical representation of functions. The essential properties of a vector are a numerical value or magnitude (a length), a direction, and a directional sense. It is graphically represented by an arrow of which direction and sense correspond to those of the vector and length is proportional to a numerical value. The chapter describes representation of vectors in terms of their components, addition and subtraction of vectors, multiplication of vectors, and geometrical applications of vector calculus.