Differential Equations:
A Motivating Theme for Sensible Calculus *
by Martin E. Flashman 
Department of Mathematics 
Humboldt State University 
Arcata, California 95521

 PREFACE:  For a calculus course to make sense as a whole, it is helpful to have themes that run through the course. These can provide thread for the instructor and the students to sew together what otherwise might seem to be an unconnected collection of topics.  I have suggested in [1] three themes for a sensible calculus course:  approximations, modelling, and differential equations.  In these remarks I will focus on some ways that differential equations can unify the first year calculus course.

OVERVIEW:  Because differential equations is critical to many scientific investigations, an initial calculus course should provide the student with basic literacy in this subject.  What should a student know about differential equations after this first calculus course?  Students who complete their formal mathematics study with calculus should be able to read with some understanding materials that use differential equations in other disciplines.  These students should also be aware of how modelling uses differential equations in many situations.  The students who will study differential equations further should understand what issues, concepts, and techniques might be explored in greater depth.

OBJECTIVES:  A student completing a first year calculus course should have experience and some understanding of the following aspects of mathematics related to differential equations:

    1.  How the modelling process in scientific and engineering studies uses differential equations to formulate problem assumptions .  What an acceptable solution is in a modelling situation.  [This may involve both first and second order differential  equations with one or more dependent variables.]
    2.  What it means to solve a differential equation.  [This may involve discussion of existence and uniqueness of solutions;  closed form vs. approximate solutions;  and qualitative , numerical and graphic features of solutions.]
    3.  How to solve some differential equations.  [This should include numerical and graphical methods along with the traditional closed form methods of integration and separation of variables.  For example , Euler's method and drawing tangent (direction) fields with integral curves can be introduced in the first semester.]
    4.  How to interpret a solution to a differential equation in a modelling situation.  [This may involve refinement of a model assumption and exploration of the sensitivity of the solution to changes in parameters and assumptions.]

RECOMMENDATIONS: Here are a seven specific changes that could be made in the first calculus course that might help achieve some of the objectives discussed.

     1.  Introduce Euler's method for numerical solution to differential equations before any discussion of the definite integral.  Euler's method is in fact a Riemann sum using a partition of equal subintervals and the left hand endpoints.  This treatment should be consistent with the interpretation of the derivative as a rate of change so that a solution (by Euler's method) can be interpreted as finding a net accumulated change in the dependent variable.

    2.  Introduce tangent fields and integral curves as methods to visualize the solution of differential equations before any discussion of the definite integral.  Tangent fields connect the work done on graphing and the tangent interpretation of the derivative to differential equations.  This can be connected as well to more analytic discussion of the qualitative features of a solution and to Euler's method.  For a more extensive discussion of the use of tangent fields see [2].

    3.  Place less emphasis on the definite integral as a tool for measuring area.  A dual interpretation of the integral as area and as the accumulation of change may help students to distinguish the mathematical object from its interpretation.  Students seem to do this now in distinguishing the derivative from its interpretations as a rate of change and the slope of a tangent line.

    4.  Give the fundamental theorem of calculus a discussion that includes differential equations in both its interpretation (existence and uniqueness of solutions) and in its proof (estimating the net change of a variable by accumulating approximate changes via Euler's method).

    5.  Use Taylor's theory to suggest how solutions to differential equations may be estimated by polynomial functions . This reinforces the important issues of existence and uniqueness along with approximate vs. closed form solutions.  The Taylor polynomials might also be recognized as solutions to higher order differential equations with initial conditions.

    6.  Use tangent fields to help visualize solutions to indefinite integrals.  Discussion of methods of integration can become extremely formal. Using tangent fields can help recover some of the meaning that the formalities hide.

    7.  Use differential equation models to motivate the discussion of (transcendental) functions.  The importance of many functions to contemporary problems is as the solutions to modelling problems.  These functions provide an opportunity for a student to experience a variety of modelling techniques where solutions can been developed thoroughly with a variety of approaches at an elementary level.

CONCLUSION:  The theme of differential equations can come into play at almost every turn of an initial calculus course.  The objectives and recommendations I have presented here should help in reorganizing and assessing how to reform the calculus curriculum for a more sensible approach.


[1] Flashman, Martin E. 1990. A Sensible Calculus. The UMAP Journal 11 : 93-96.

[2] Flashman, Martin E.  1989.  Using Computers to Make Integration More Visual with Tangent Fields.  In Proceedings of the Second Annual Conference on Technology in Collegiate Mathematics, edited by F.  Demana, B.  Waits,and J.  Harvey.  Addison-Wesley, 1991: 168-170.

*This paper was written to appear in "Calculus for All Users," the Report of A Conference on Calculus and Its Applications held at the University of Texas, San Antonio, (A NSF Calculus Reform Conference), October 5-8, 1990.

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