Using Computer Algebra To Extract Meaning from Parameters.
Carl Leinbach · 1999
Computer Algebra Systems (CAS) can be used to help understand the behavior of cubics and other families of curves. The analysis of the parameters associated with a family of curves can give insight into the behavior of the family and can be used within the context of a mathematical model to determine a policy for the implementation of the model in a real-world setting. Examples of this use are included from a calculus-based analysis of a mathematical model and the behavior of a model described by a differential equation. The case is made that the ability, afforded by the use of CAS, to keep the parameter in symbolic form enables an analysis that goes beyond the standard analysis that may be done in the standard situation using numerical coefficients. (MM) Reproductions supplied by EDRS are the best that can be made from the original document. PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIAL HAS BEEN GRANTED BY TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) Using Computer Algebra to Extract Meaning from Parameters John Moores University Liverpool, UK L3 3AF C.Leinbach @ 1 ivjm.ac.uk Introduction Carl Leinbach on leave from U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) Aq.hia document has been reproduced as received from the person or organization originating it. 0 Minor changes have been made to improve reproduction quality. Points of view or opinions stated in this document do not necessarily represent official OERI position or policy. Gettysburg College Gettysburg, PA USA [email protected] Understanding the meaning of a mathematical equation, model, or result comes from a variety of sources. It may come from the ability to estimate, a graphical intuition, or an understanding of the phenomena underlying the result. Ultimately, however, the most valuable insights are gained by understanding the parameters that are used to define the mathematical objects being considered. For example, the behavior of a cubic, ax3 + bx2 + cx + d, depends on the size, sign, and relationships that exist among the coefficients, a, b, c, and d. The fact that the polynomial is a cubic determines the general shape of the curve, but it is the parameters that determine the center of symmetry, the location of the maximum and minimum points, if any, and the distance between these points. In effect, the degree of a polynomial determines the general shape, and the parameters define the quality of the polynomial's behavior. While the analysis of parameters associated with a family of curves can give significant insights into the behavior of the family, there are other uses for this type of analysis. For example, parameters can be used within the context of a mathematical model to determine a policy for the implementation of the model in a real world setting. Examples of this use will be taken from a calculus based analysis of a mathematical model, and the discussion of the behavior of a model described by a differential equation. In each case we will show that the ability to keep the parameter in its symbolic form enables an analysis that goes beyond the standard analysis that may be done in the standard situation using numerical coefficients. Constructing Cubic Curves The general definition of a function determines the general shape of a process, but analysis of the parameters associated with the function detennine the quality of the process. For example, using our knowledge of trigonometry we can describe the process associated with the function, f(x)=ax + bsin(ax + p) as an oscillating curve that lies along a ray passing through the origin. The paranieter, a, determines the angle the curve makes with the x-axis, while b, a, and p determine the height, frequency and phase shift of the oscillation. The latter characteristics are important if one is describing a radio wave being broadcast to a destination. In a similar way the parameters associated with a polynomial function determine the behavior of the polynomial. We will first consider a cubic polynomial. 2 BEST COPY AVAILABLE Begin the analysis by considering a very straight forward cubic function of the form f(x) = ax3 13x We can learn a lot about this curve by doing some elementary reasoning and exploration r 1. The roots are located at x = 0, A , A a a 2. The graph is symmetric about the origin, i.e. flx) .f(x) 1 3. The graph has a and at x = ± --(1 or TI-1 if and only if 3a V3a a and 13 have the same sign. The last fact is easily determined using calculus techniques if the students have had an introduction to differential calculus. However, if they have not some elementary graphical analysis and algebraic manipulations using a CAS have the potential to make the exploration even more exciting. For example, one approach is to begin with a graphic justification of the tangent line as a limiting position for secant lines through a fixed point and a sequence of points becoming ever closer to the given point as is illustrated in the following screen taken from a TI-89 session. Each student can be given different points on the graph and a different sequence of points (values of h) each of which terminate at a point 'close' to the fixed point. The graphical evidence is convincing but not a proof. The CAS allows for a more convincing argument after the graphical case has been considered. It also leads us to the conclusion given in (3) above. rl. Tools F2. Algebra F3. Ca lc Fh. Other FT Pr nMO F6. Clean UP Define f(x) =a x3 B x Done f(x + h) f(x) 3-ax2+3.hax +h2.a-B (f(x+h)-f(x))/h MAIN RAD AUTO FUNC 2130 ri. Tools F2. Algebra F3. Cale Fh. Other FT rol0 F6. Clean Up MO 3. a x2 zeros(3. a x 2 B, {when(-,r378) -&l ,Trx wherl zeros ( 3*a*x^2-B, x) MAIN RAD AUTO FUNC ROO The screen on the left allows the student to easily manipulate the expression for the difference quotient. The obvious question is what happens as h becomes smaller and smaller in magnitude. This leads to the idea of a limit. The student now knows how to find the slope, and as a result the equation, of the tangent line to the graph of f(x) at any point on its graph. Now, the next question is what makes peaks and valleys interesting? Of course, these are points where the tangent line is horizontal! So the student solves for the points where the slope of the tangent is zero, and the students have shown statement number 3. In the process they see that the location of the peak and valley are related to Leinbach: The Role of Parameters