Sinopsis
One of the important themes in calculus is the analysis of relationships between physical or mathematical quantities. Such relationships can be described in terms of graphs, formulas, numerical data, or words. In this chapter we will develop the concept of a “function,” which is the basic idea that underlies almost all mathematical and physical relationships, regardless of the form in which they are expressed. We will study properties of some of the most basic functions that occur in calculus, including polynomials, trigonometric functions, inverse trigonometric functions, exponential functions, and logarithmic functions.
Many scientific laws and engineering principles describe how one quantity depends on another. This idea was formalized in 1673 by Gottfried Wilhelm Leibniz (see p. xx) who coined the term function to indicate the dependence of one quantity on another, as described in the following definition.
In the mid-eighteenth century the Swiss mathematician Leonhard Euler (pronounced “oiler”) conceived the idea of denoting functions by letters of the alphabet, thereby making it possible to refer to functions without stating specific formulas, graphs, or tables. To understand Euler’s idea, think of a function as a computer program that takes an input x, operates on it in some way, and produces exactly one output y. The computer program is an object in its own right, so we can give it a name, say f . Thus, the function f (the computer program) associates a unique output y with each input x (Figure 0.1.2). This suggests the following definition.
Content
- BEFORE CALCULUS
- LIMITS AND CONTINUITY
- THE DERIVATIVE
- TOPICS IN DIFFERENTIATION
- THE DERIVATIVE IN GRAPHING AND APPLICATIONS
- INTEGRATION
- APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING
- PRINCIPLES OF INTEGRAL EVALUATION
- MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS
- INFINITE SERIES
- PARAMETRIC AND POLAR CURVES; CONIC SECTIONS
- THREE-DIMENSIONAL SPACE; VECTORS
- VECTOR-VALUED FUNCTIONS
- PARTIAL DERIVATIVES
- MULTIPLE INTEGRALS
- TOPICS IN VECTOR CALCULUS
