Derivatives and integrals are traditionally introduced as very different things — slopes versus areas — with the relationship between them being introduced as if it were a surprise.

We advocate a different approach

There are two other important operations that you perform on functions.

Much of what you will do with functions involves picking one of these four operations. The student needs to know which operation is appropriate for a given task or setting. Rather than introducing derivatives as “slopes'' without saying why you would be interested in a slope, give some contexts.

Important settings for integration and differentiation involve knowing a quantity and wanting to know a related quantity.

Some examples of Function/Derivative pairs.










The Cognitive Problem of Reading Graphs

You can read off the first and second derivatives easily from a graph of \( f(x) \). Drawing graphs of derivatives is an exercise in translation, taking a quantity that you perceive as a slope or as a curvature and representing that quantity in a different mode: as a height.

It's much easier for students to answer questions about derivatives from a graph if they are asked about easily perceived quantities: Is it positive, negative, or zero? Is it big or small in magnitude?

Higher-Order Derivatives

An effective metaphor for differentiation and integration is genealogical. Let \( f(x) \) be you. Then \( F(x) \) is your child and \( f'(x) \) is your mother. Integrating is like going forward a generation, from mother to child. Differentiating is like going backward a generation, from child to mother.

A mother can have many children; they are all similar in many ways. A child has only one mother. Similarly, a function has only one derivative, but it can have many different anti-derivatives that are all similar. (They differ only by a constant.)

The second derivative is like your mother's mother. There's a third derivative and so on, from generation to generation. Similarly, you can integrate a function to produce a child function.

Differentiation Modeling Settings

  1. Rate of orange juice production. Construct the function Questions about flow
  2. Modeling instantaneous fun: speed, slope, curvature. SuperSlide project