The modern rule of three suggests looking at mathematical concepts from each of the numerical, algebraic, and graphical perspectives.
The purely algebraic perspective is the most common in the introductory course:
Find the solution of the initial value problem P' = 0.015P - 0.209, P(0) = 10.
The geometric perspective is too rich to ignore:
Sketch graphs of solutions of P' = 0.015P - 0.209 for various positive initial values. Determine the regions where the solution curves are increasing, decreasing, concave up, and concave down.Does P' = 0.015P - 0.209 have any constant solutions? If so, what initial condition(s) lead to such solutions?
Sketch the direction field of P' = 0.015P - 0.209 in the first quadrant. Identify ranges of initial conditions that lead to qualitatively similar solution behavior.
Which of the curves shown in the figure (referring to a family of curves, some increasing and some decreasing, of varying concavity, including a horizontal line) might be solutions of P' = 0.015P - 0.209? Justify accepting or rejecting each curve.
Repeat the previous exercise using tables of solution values rather than graphs.
Similarly, the numerical view is too important to postpone until late in the course. And its exercise need not demand extensive computation.:
The previous exercises suggest that P' = 0.015P - 0.209, P(0) = 10, will have a decreasing solution. Use one step of Euler's method to estimate the time required for the solution to reach P= 0.Use Euler's method with
to construct a table of values of the solution of P' = 0.015P - 0.209, P(0) = 10 for
. Include in the table the values of
. Is the behavior you see in that table consistent with a direction field diagram for this equation? How could you use the entries in that table to help you draw an accurate direction field diagram? Are the entries in the table consistent with the exact solution?
Use Euler's method with
and
. How does the error in the estimated value of P(10) vary with
? What return in improved accuracy does Euler's method give for the increased work caused by halving the step size?
The graphical and numerical perspectives are also abundant sources of ``real'' mathematics, that material we would love to teach but for the stupor it induces in our students.
For example, a graphical examination of solution curves can
motivate a uniqueness theorem. After students have sketched solutions of
the logistic equation 
, ask whether the (stable) steady
state P = a/s is reached in finite or infinite time. The uniqueness
theorem shows that the approach is indeed asymptotic because two
solution curves can not cross.
Analysis of the error in initial value solvers like Euler's method requires the mean value theorem and Taylor polynomials. (Taylor polynomials also motivate higher order Runge-Kutta methods.) When students have explored the numerical relation between step size and error, they are prepared to approach error analysis as a way of understanding the potential pay back in improved accuracy from the added computational effort of reducing step size. Central ideas of analysis are brought into play because they help answer questions important to the students, not just because the professor likes the concepts.
Good exercises and projects can be formulated in both the graphical and numerical settings by letting students first discover for themselves the basic phenomena - the possibility of an asymptote among logistic equation solution graphs or the proportionality between numerical error and step size in Euler's method. Then build on that motivation to introduce the mathematical ideas that will permit students to answer the questions their experiences have raised.