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November 12, 2005

Advance Organizer for Calculus (based on the Larson textbook)

Overview

The introductory chapter from Larson provides a review of the prerequisites for the Learning of calculus. The basic ideas are those of the graph of an equation, linear models, functions and fitting models to data.

Topic: P3. Functions and Their Graphs

Introduction: The two most important ideas for having a deep understanding of calculus are function and limit. This section focuses on the concept of function and then describes many different types of function.

Keywords: function, independent variable, dependent variable, domain, range, implicit form, explicit form, vertical line test, transformations of graphs, elementary functions, algebraic functions, polynomial functions, rational functions, trigonometric functions, exponential functions, logarithmic functions, degree of a polynomial function, transcendental functions, composite functions, odd function, even function.

History: Leonhard Euler [1707 - 1783] was one of first to apply calculus to real-life problems in physics.

Description: This description of the concept of function begins with a general definition, then discusses notational conventions, evaluation of a function, the graph of a function, and various transformations of functions. This is followed by an introduction into many different kinds of functions.

Definition: Let X and Y be sets of real numbers. A real-valued function f of a real variable x from X to Y is a correspondence that assigns to each number x in X a unique number y in Y. The domain of f is the set X. The number y is the image of x under f and is denoted by f(x). The range of f is the subset of Y consisting of all images of numbers in X.

Functions can be specified in a variety of ways: language (giving a rule for the correspondence), set theory notation, an equation (analytic approach) or a set of equations (piecewise definition), a table (numeric approach) or a diagram (visual approach - e.g. a Venn diagram). The set theoretic approach is used in analysis, but in basic and applied settings one usually uses equations.

If the equation is of the form y = f(x), then this is called an explicit form. This form is relatively easy to evaluate and graph. If the equation is of the form f(x,y), it is called an implicit form and is usually much more difficult to evaluate or graph.

History: The word function was first used by Leibniz in 1694 as a term to denote any quantity connected with a curve. In 1734 Euler introduced the notation y = f(x). See References for further information on the development of the function concept.

Since a function is defined as a unique mapping of x into y, a vertical line can intersect the graph of a function only once. This is called the vertical line test for a function. Many equations give rise to graphs that fail this test for a function. The equation and the graph are still meaningful, but it is not proper to refer to these as functions.

It is possible to "shift" a graph either horizontally or vertically and to provide a mirror image of the original by means of a simple transformation.

Definition: Basic types of transformation (c > 0)

Original graph y = f(x)
Horizontal shift c units to the right y = f(x - c)
Horizontal shift c units to the left y = f(x + c)
Vertical shift c units down y = f(x) - c
Vertical shift c units up y = f(x) + c
Reflection about the x-axis y = -f(x)
Reflection about the y-axis y = f(-x)
Reflection about the origin y = -f(-x)

There are many types of functions.

Definition: A polynomial function is of the form .

Definition: A radical function is of the form .

Definition: A rational function is of the form .

Definition: A composite function is of the form and is usually written or

Definition: A function is even if f(-x) = f(x).

Definition: A function is odd if f(-x) = -f(x).

pdf Files:

Mathematica: none

Problems: none

Summary: The general idea of function is straight forward. Most introductory calculus material focuses on the elementary functions. The idea of a composite function is simply the idea of recursion!

Reflection: The identification of the various types of elementary functions is worthwhile. This leaves open the idea that there are many other types of functions. This list is likely endless as one can create a new category by simply identifying a new property.

References: none

Links:

Function

Mathematics Classifications

Topic:P4. Fitting Models to Data

Introduction: This is a relatively short section that extends the ideas first presented in P1 on fitting regression models to data.

Keywords: fitting a linear model to data, fitting a quadratic model to data, fitting a trigonometric model to data.

History: none

Description: The idea of regression analysis is one I am familiar with. The material in this chapter involves the use of software to conduct various regression analyses. All of the examples in the text are taken from either physics or economics.

Quotation: "Learning to apply mathematical skills is very different from learning mathematics itself." Edwards, D. & Hamson, M. (1990). Guide to Mathematical Modelling. Boca Raton: CRC Press.

Comment: While I agree with this quote, I also see that having a strong background in the mathematics of a topic helps one to appreciate and interpret the results of an analysis. It turns the black box of the software to at least a grey box, and ideally, a glass box.

pdf Files: see Links section below

Mathematica: see examples in section P1

Summary: The idea of seeing how close a particular mathematical expression can approximate a set of data points is straightforward. The real challenge is not in the mathematics but is in learning to use a particular software package. I am fairly familiar with SPSS and now know how to use Mathematica to perform the same analysis.

I am feeling very comfortable and confident with the material in this chaper (all four sections). It is time to move on to Chapter 1 Limits and Their Properties [p. 39 - 88].

Reflection: Now that I have two different tools for carrying out a regression analysis, I will use both of them to provide a check on the results.

I have created a well-structured set of notes for chapter P in the Larson texbook on calculus. I now want to back off from the mathematics for a bit and instead play with a software package called Stylus Studio which should help me create an XML set of files for saving and displaying (for both screen and printer) these notes.

Links: Google [mathematica "regression analysis"]

Google ["regression analysis" history]