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Glossary |
Besides being a beautiful and fascinating study in its own right, mathematics has grown in importance as a fundamental tool of modern science. In this introductory chapter, we will discuss the role of mathematics in science, in order to motivate the fundamental objectives of this text.
The scientific method is based on the following process:
Derive mathematical equations based on this logical system; in our previous example, we assume that "gravity" causes an object which is dropped to accelerate at a constant rate of approximately 9.8 m/sec2, or mathematically, a = 9.8, where a = the acceleration in m/sec2.
A collection of equations such as:
| a = 9.8 d = 4.9t2 |
where | a = the acceleration (in m/sec2) d = the distance (in meters) fallen in t seconds t = the number of seconds the object has fallen |
from our example is called a mathematical model. Such a model consists of equations which constrain the values that the variables may take and/or describe relationships between the variables, such as the two equations on the left. The first equation, a = 9.8, says that the acceleration is constant, that is, its value does not change. The second equation describes how the distance, d, is related to the time, t (i.e., it is a constant multiple of the time squared). Mathematically, we say that d is a function of t. Notice how a model also includes an equation which gives an interpretation of each of its variables; this connects the mathematical equations to their applied setting.
The scientific method relies on mathematical models to make predictions that can be tested to see how well our conceptual model of the world agrees with our experience. This leads to the most crucial step in the scientific method:
It is this step that allows us to weed out bad models, and identify errors in our beliefs about the world. However, to make predictions from a mathematical model, one must be able to:
For example, Galileo could have tested the previous model by dropping a ball from the top of the tower of Pisa. Since it is approximately 58 m. tall (not taking into account its lean), he would solve the equation 58 = 4.9t2 for t (to obtain t = 3 sec., approximately), and see if it in fact took that long to drop. Note: Since Galileo did not know the gravitational constant 9.8 m/sec2, he only used the general model: a = k, d = kt2/2; as a sign of his genius, he designed his original experiment to verify the model even without knowing the value of the constant k!
We use the scientific method to refine our understanding of the world around us. If the model yields inaccurate predictions, we will adjust the model until it is more accurate. Given a complete enough understanding, we can make fairly accurate predictions ("Since my grandmother and father both developed leukemia, how likely am I to contract it?") and to attempt to change our world for the better ("If I develop leukemia, what treatment can I take?"). Important: This technique of experimentation and verification is a very powerful technique; we will learn to see how it can be applied even more broadly to check our mathematical calculations, to root out other types of errors we may make, and to uncover other misconceptions we may hold.
Notice that, to be able to understand and use the scientific method, you must be able to:
In previous mathematical studies, you have probably spent some time with item 1, and a lot more time with item 3. In Algebra, you probably studied how to manipulate mathematical expressions and equations in order to "simplify" them. This may have been done as a largely mechanical task.
However, items 2 and 4 are probably the more challenging and important skills. These require experience, judgment, and insight:
This means that the primary goal of this text will be to introduce you to the function concept, in general, and a large number of important examples of functions, such polynomial (e.g., ), rational (e.g., ), radical (e.g., ), exponential (e.g., ), logarithmic (e.g., ), and trigonometric (e.g., ) functions. Note: If these functions do not appear in your browser, make sure to install the Java-based MathViewer. When working with functions, we will learn to view them from three different, but related, perspectives: algebraic, numerical, and graphical. Thus, another important goal of the course is to learn general principles that relate the formula of a function to its numerical values and to its graph. All of the functions we will discuss will have their own special properties and unusual features, so we will also stress the algebraic and graphical features of each of these important functions.
This text is designed to be a preparation for Calculus, specifically, and for work in scientific disciplines that rely on mathematics, more generally. If you do not intend to go on in science or Calculus, then this course may not be the right one for you. However, the general abilities, to think abstractly, plan, implement, and verify a problem-solving strategy, that you should gain while working through this text, is valuable for people in every field of endeavor.
Also, this text will challenge you to learn to become more flexible in your thinking. The most fundamental problem-solving skill, that undergirds all of mathematics, is the ability to view the same situation from many different points of view. For example, the process of mathematical modeling described earlier is a way of taking a "real" world situation and creating an alternative viewpoint, using mathematical variables and equations. Eventually, if we achieve a good enough point of view, the solution to our original problem becomes "obvious". Even if this does not seem clear to you now, we will illustrate this principle throughout the text.
This collection of Web pages are designed to form an interactive text. We will begin with a general discussion of the function concept. We will then discuss transcendental functions (i.e., exponential, logarithmic, and trigonometric functions), and conclude by examining algebraic functions (i.e., polynomial, rational, and radical functions). With each type of function, we will:
More than learning any specific facts, an important goal of this text is to help you become independent learners, careful thinkers, and confident problem-solvers. Along the way, we will try to highlight and bring together a lot of separate facts that you have seen in Algebra classes to help you see the "big picture", and solidify your algebra skills. We will see that the function concept provides a powerful tool for unifying many of the ideas and techniques of Algebra.
To become independent learners and problem-solvers, we all need sufficient guidance and help along the way. This text is "interactive", in several ways. As an Web-based text, it uses hyperlinks (like this) to allow you to navigate between different parts of the text. Each hyperlink is associated with a specific location in a document. This text uses hyperlinks in three major ways:
Besides hyperlinks, this text also incorporates a Java program, called XFunctions, as a graphing tool.
Note: Historical references are provided courtesy of the MacTutor History of Mathematics archive, The WordMan, and Jeff Miller, a teacher at Gulf High School in New Port Richey, Florida.
Go to Functions and Graphing
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Glossary |