Archive for the ‘The Derivative’ Category

Yes, I prefer introducing integral calculus prior to differential for various reasons, one of which is that I believe that the former should not be dependent on the latter (my reasoning for this approach is very sound, in my opinion). The two disciplines are each introduced on their own merits with the inverse relationship between the two being discovered soon thereafter. These are my notes introducing differential calculus.




Click on secant line and adjust slider.

Thanks for looking.



Courant, Richard., John, Fritz (1999).  Introduction to Calculus and Analytics: Classics of Mathematics. New York, NY:  Springer-Verlag Berlin Heidelberg.

When introducing my high school students to Calculus, I begin by asking them if they would like to see why the volume of a cone is what it is; they invariably respond “yes”.  This leads immediately into some diagrams and discussion on what they already know and information that is missing. I tell them up front that the sum of squares formula will be needed; this is derived through telescopic sums, a rich task in itself.

A diagram of a cone is then sectioned into cylindrical slabs and, through further discussion, it is decided that a reasonable estimate for volume of the entire cone could be determined if these slabs were very thin, there volumes calculated and then summed.  The basic idea behind integration has now been considered and off we go.

By the end of day 2, students have seen how the discrete representation (Riemann sums) gives way to its continuous counterpart (integration) through use of the power rule. We work through several examples with uniform subintervals and then develop the power rule for integration using non-uniform partitions on f(x)=x^n.  The FTC follows shortly thereafter; students are then given some time to practice finding area by setting up simple integrals and working through the process.

Simple differential equations soon follow, as shown below.

Differential Equations.png

Rates of change (and related rates) can very easily be folded into the fabric of our work from thoughtful consideration of the topic shown above.  This is a necessary outcome as the task of deriving integrals of logarithms, trig functions, etc. is too demanding at this level.  Derivatives of these functions are, however, well within the grasp of high school students. Once these are derived, the inverse nature of integrals and derivatives can be exploited to continue on our little journey.

I realize that this approach isn’t for everyone but it works for me and my students seem to enjoy their work as well.

That is all for now. Thanks for reading.



Four towns (A, B, C, D) are situated to form a square with side length of 1 unit. Determine the minimum length of roadway that will link these four towns together.

View the James Grime video first.

Minimum Length

Click on the following link for an interactive exploration: GeoGebra


The graph below shows the length of roadways linking these towns as a function.

Length as a Function of  “x”

Four Towns Minimum

Click on the link provided here to determine local minimum.


Thanks for reading.


Determine a position “c” in terms of “a” and “b” that will result in triangle ABC having maximum area. Prove that the areas bounded by f(x) and the line segments AC & BC at that position are equal.

Putnam Area

Click on the link provided here to explore the triangle area above.