T

Learning:
The Journey of a Lifetime
A Cloud Chamber of the Mind

November 2005 Mathematics Notebook
Introduction      
Goals
Previous
 
Journal Page for This Day
 
     

An Example of a "Learning Process" Journal
November 11, 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: P1. Graph of an Equation

Introduction: Ideas in mathematics (and in calculus) can be represented visually, analytically (algebraically), numerically (tables) and linguistically. The idea is the same, it is the representation that is different, and each representation has strengths and weaknesses. The analytic approach is most precise whereas the visual approach is often more intuitively meaningful. Yet we use language to interpret and describe both approaches. Language is the mortar that holds different types of bricks together and allows these bricks to join to one another. A visual diagram like a concept map provides another way in which an image may supplement language and vice-versa.

Keywords: graph, x-intercept, y-intercept, symmetric with respect to the x-axis, symmetric with respect to the y-axis, symmetric with respect to the origin, point of intersection of graphs of two equations.

History: Rene Descartes [1596 - 1650] created analytic geometry.

Description: Intuitively, I understand the relationship between a graph and the equation of a function. The graph is a plot of all the points that satisfy the equation. When constructing a graph one needs to pay particular attention to the "viewing window" of the graph. This involves a consideration of both the scale of the axes as well as the minimum and maximum values for each axis. A graph may "appear" quite different depending on the values chosen for the axes.

Visually, the idea of symmetry is straight forward. Describing symmetry analytically adds a sense of precision to the idea.

Definition: Thus a graph is symmetric wrt the y-axis if replacing x with -x gives an equivalent equation, symmetric wrt the x-axis if replacing y with -y gives an equivalent equation, and symmetric wrt the origin if replacing x with -x and y with -y gives an equivalent equation.

Example: The equation y = x^2 is symmetric wrt the y-axis since y = x^2 is equivalent to y = (-x)^2. Similarly, the equation y^2 = x is symmetric wrt the x-axis since y^2 = x is equivalent to (-y)^2 = x. Finally, y = x is symmetric wrt the origin since y = x is equivalent to (-y) = (-x).

Two common tests for symmetry are:
(1) If all the exponents of x are even, then the graph is symmetric wrt the x-axis.
(2) If all the exponents of x are odd, then the graph is symmetic wrt the origin.

Special Note: We have not discussed other axes of symmetry such as being symmetric about an arbitrary line or point.

pdf Files: none

Mathematica: I should be able to obtain the graph of any equation or function using Mathematica. I should be able to specify the color of the line as well as the viewing window. Finally, I should be able to superimpose the graphs of more than one equation/function on the same axis.

Mathematica

web

Problems:

Page 9, #9. Find any intercepts for x^2y - x^2 + 4y = 0.

One approach is to rearrange the equation to be of the form y = f(x). Then I can use Mathematica to create the graph. This means that I must be comfortable with algebraic manipulation.

Another approach is to use Mathematica directly, but in this case I must use the ImplicitPlot function.

Interesting! The first approach works fine but the ImplicitPlot approach gives a totally different result. My first reaction is to assume that I do not properly understand the ImplicitPlot routine.

Page 9, #13. Check for symmetry wrt each axis and to the origin. y^2 = x^3 - 4x

Since y can be replaced by -y with no effect, the graph is symmetric about the x-axis.

Here is the graph of this function, created in Mathematica (web):

Page 9, #17. Check for symmetry wrt each axis and to the origin. y = x/(x^2 + 1).

For this graph, the origin is an axis of symmetry since replacing both x and y with -x and -y leaves the equation unchanged.

Here is the graph of this function, created in Mathematica (web)

Special Note: In Mathematica, select a graph and then hold down the Ctrl key while moving the cursor to see the coordinates displayed in the bottom left corner of the window.

Page 9, #34. Graph y = x Sqrt(25 - x^2). Identify intercepts and test for symmetry.

The graph is symmetric about the origin, and thus goes through (0,0).

Here is the graph of this function, created in Mathematica (web)

Page 10, #55. Modeling data.

Page 10, #56. Modeling data.

Summary:

Obtaining the graph of a function, equation or for data is a valuable component of understanding the nature of the relationship. Being able to use software packages such as Mathematica can give one incredible power for exploring ideas related to functions.

Reflection:

Preparing these notes involves three different domains of knowledge:

  • the mathematics
  • use of Mathematica software
  • combining Mathematica with Dreamweaver web authoring.

By far the most difficult of the three is Mathematica. But I believe that this will pay off in the long run as I become more familiar with it. The difficulty is finding a balance between Learning Mathematica and simply Learning specific routines for specific problems.

As illustrated above, it is possible to click on a graph in Mathematica to select it, then do a copy and then open up a graphics software program such as ImageForge to paste it and then save it as a png file. (Mathematica does not have an option for saving the graph as a jpg or gif or png.)

Links:

Descartes

Mathematica

Topic: P2. Linear Models

Introduction: The idea of a linear model (an analytic/algebraic idea) is equivalent to the geometric idea of a straight line.

Keywords: slope of a line, point-slope equation of a line, slope-intercept equation of a line, ratio, average rate of change, general form of equation of a line, parallel lines, perpendicular lines.

History: none

Description: The important property of a straight line is its slope. The slope of a line is determined by any two points on the line. Once one knows the slope, there are two main forms for the equation of the line, one where one also knows a point that the line passes through, and the other when one knows the value of the y-intercept (which is a special point that the line passes through).

Definition: The slope of a nonvertical line passing through the points and is

A positive slope rises to the right, a negative slope decreases to the right. The greater the absolute value of m, the steeper the line. The letter m to represent the slope comes from the French word monter, meaning to mount, to climb, to rise.

If the x- and y-axis both have the same unit of measure, then the slope has no units and is a ratio. If they have different units, then the slope is the rate of change.

Nonvertical lines with the same slope are parallel, and nonvertical lines whose slopes are negative reciprocals are perpendicular.

Definition: The point-slope equation of a line with slope m passing through the point is

Definition: The slope-intercept equation of a line with slope m and y-intercept (0, b) is

y = mx + b

Definition: The general form of the equation of a line is

Ax + By + C = 0

pdf Files: none

Mathematica: none

Problems: Any situation involving two variables where one is able to plot two points gives rise to a straight line showing the relationship between the variables.

Summary: This section gives the formula for determining the slope of a straight line plus a couple of formulas for straight lines in terms of the slope and a point.

Reflection: The ideas in this section are simple and straight forward, and the problems are not necessary as they only involve simple algebraic or numerical operations.

Links: none