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
- Differentiation is an operation you perform on a function. The result of the operation is another function with the same arguments as the input.
- Anti-differentiation is the inverse operation that swaps the roles of the inputs and outputs.
There are two other important operations that you perform on functions.
- Evaluation — given the inputs to the function, find the output. Plotting is a form of evaluation done for a range of inputs.
- Solving — given the output and perhaps some of the inputs, find the value of inputs that will produce the given output.
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.
- Cumulative/Density. Explain where the cumulative comes from (e.g., with income percentiles, the 99%) and that the cumulative is a non-decreasing function on 0 to 1. The density is the derivative of this with respect to the variable. What densities are good for: you can see where most of the stuff is, easily.
- Amount of substance/reaction rate
- Energy-Entropy/Temperature – temperature is the change in Entropy divided by the change in Energy
- Total cost/Marginal cost
- Assets/Cash Flow
- Total tax/Marginal tax
- Prices/Inflation.Watch out: they report some changes over a year (seasonally adjusted — the change from last year), but it would be best if they reported a rate and used a sine model to do the seasonal adjustment.
- Height/Growth — or mass or volume, etc.
- longitude problem and dead reckoning at sea.
- "Stock” / “Flow”
- “Amount” / “Density”
- How long has the bathtub been filling VERSUS how much water is in the tub.
- How long has the food been in the oven. Heat is flowing into the food. VERSUS How much heat is in the food.
- Changes in salary versus salary itself.
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?
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
- Rate of orange juice production. Construct the function Questions about flow
- Modeling instantaneous fun: speed, slope, curvature. SuperSlide project