Calculus – by Mrs. Lovenduski

[Ed. Note – the links are provided by me to help those who haven’t taken Mrs. Lovenduski’s class!]

In Calculus, we start by hearing out and thinking upon Zeno’s Paradoxes.  Inherent within his paradoxes is the questions ” Is motion possible?  and, if so, how?”  In working with these questions we also wrestle with; how does one gauge or calculate out motion and what is instantaneous velocity?

We begin our journey to reconcile the Dichotomy paradox with an introduction to sequences, summations and series. We develop the Power Series Formula and then expand that concept to arrive at the Infinite Power Series Formula.  The fundamental concept of limits is discovered and wrestled with.  We use the Infinite Power Series Formula to disprove Zeno’s Dichotomy Paradox and reconcile that motion is not an illusion and that instantaneous velocity must be possible.  Following in the footsteps of Newton, we commence our journey to calculate instantaneous velocity.

We work through average velocity and develop a way to calculate instantaneous velocity, which leads us to the difference quotient.  Through many lengthy calculations, the students begin to yearn for a short-cut and they soon discover a pattern that follows their work.  We now have the tool of being able to apply the derivative using the power rule, the addition rule and the constant rule.

Next, we follow in the footsteps of Leibniz and look at how one could calculate the area under any given curve.  Starting with the Left-hand Sum and Right-hand Sum calculations, the students quickly see the need for limits and again search for a short-cut through their lengthy calculations. Quickly, they find the pattern and we now have the ability to apply the anti-derivative and to integrate, landing on the Fundamental Theorem of Calculus.

This block pushes the students to explore questions through different vantage points and to find the connections that will lead them to the universal concepts underlying the answers.