Precalculus: A Study of Functions and Their Applications

Annotated Table of Contents

Precalculus: A Study of Functions and Their Applications
Swanson, Andersen, and Keeley

Chapter 1: An Introduction to Functions. Various representations of functions as well as the language and notation associated with functions are introduced. This chapter also illustrates how graphing calculators can be used and misused in the study of functions.

1.1 Functions
1.2 Graphical Representations of Functions
1.3 Calculator Graphics
1.4 Mathematical Modeling
1.5 Project: Crickets - Nature's Thermometer Chapter 2: Families of Functions. Linear, exponential, logarithmic, periodic, and power functions are introduced. Students are shown how to recognize these functions in their various representations. Students are also shown how to obtain a formula when given a linear, exponential, or power function either numerically or graphically. This lays the groundwork for these functions and for their use throughout the remainder of the book. Examples of realworld situations are included for each type of function. 2.1 Linear Functions
2.2 Exponential Functions
2.3 Logarithmic Functions
2.4 Periodic Functions
2.5 Power Functions
2.6 Project: Newton - A Real Swinger Chapter 3: New Functions from Old. The basic functions from chapter 2 are transformed to form new functions in a variety of ways. In particular, the relationship between a transformed function in its symbolic form is compared with its graphical form. Transformations include addition and multiplication as well as composition. The relationship between a function and its inverse is also explored. 3.1 Function Transformations: Changes in the Output
3.2 Function Transformations: Changes in the Input
3.3 Combining Functions
3.4 Composition of Functions
3.5 Inverse Functions
3.6 Project: Setting the Tone Chapter 4: Polynomial and Rational Functions. Polynomials, introduced as transformations of particular power functions, are important enough to study as independent objects. We look at their properties as well as how they can be combined through division to form rational functions. Applications are given throughout the chapter. 4.1 Quadratic Functions
4.2 Polynomial Functions
4.3 Power Functions with Negative Exponents
4.4 Rational Functions
4.5 Project: The Amazing GolfOMeter Chapter 5: Trigonometric Functions. The periodic functions of sine and cosine, introduced in chapter 2, are reviewed and other trigonometric functions are introduced in this chapter. These functions are introduced by using the unit circle definitions. The geometry of a circle, including arc length and area, are also explored. The transformations, introduced in chapter 3, are applied to the trigonometric functions. Trigonometric identities are introduced throughout the chapter and are the focus of section 5.4. 5.1 Two Ways of Defining Trigonometric Functions
5.2 Arc Length and Area
5.3 Transformations of Trigonometric Functions
5.4 Trigonometric Identities
5.5 Project: Looking Out to Sea Chapter 6: Applications of Trigonometric Functions. Students look at additional types of applications for trigonometric functions by exploring triangle applications as well as periodic and "periodiclike" applications. 6.1 Right Triangle Applications
6.2 Law of Sines and Law of Cosines
6.3 Modeling Behavior with Sums of Sine and Cosine
6.4 Other Applications for Trigonometric Functions
6.5 Project: Life in the Fast Lane Chapter 7: Solving Equations and Fitting Functions to Data. Different methods for solving equations are introduced. This structure gives students a review of the functions first introduced in chapter 2. The techniques of linear, exponential, and power regression are introduced as methods of fitting functions to data. 7.1 Introduction to Solving Equations
7.2 Solving Exponential Equations
7.3 Solving Trigonometric Equations
7.4 Regression and Correlation
7.5 Fitting Exponential and Power Functions to Data
7.6 Project: The Population Problem Chapter 8: Getting Ready for Calculus. This chapter serves as an introduction to calculus by exploring the concept of limit, the derivative, and the integral. The focus is on these mathematical concepts with various representations of functions. 8.1 Limits
8.2 Slopes of Secant Lines and the Derivative
8.3 Sequences and Series
8.4 Area and the Integral
8.5 Project: Zero to Sixty Chapter 9: Additional Topics. The text concludes with a look at parametric equations, vectors, and multivariable functions. A property of a conic section is the focus of the project in this chapter. 9.1 Parametric Equations
9.2 Vectors
9.3 Multivariable Functions
9.4 Project: Elliptipool