Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.
Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.
Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts
- Students can connect concepts to their visual representation with and without technology.
How do you introduce the concept of a derivative?
We start with visuals of the derivative (from a Getting Started with Calculus activity called Derivative Trace).
Tangent Line Demonstration from Calculus Nspired has some similar ideas.
We then look at the Derivative Grapher to connect the slope of the tangent line to the graph of the derivative, changing the original function as requested by the students.
We begin to develop some common language around derivatives before we formalize what is a derivative, before we formalize the definition of a derivative.