As we have seen, there are a couple of viewpoints to functions that drive our investigations.

\(\blacktriangleright \) One is a point-wise view. What is the function value at each domain number?

\(\blacktriangleright \) The other is a behavioral view. How are the function values changing compared to how domain numbers are changing?

We use these two viewpoints together when analyzing individual functions. We use these viewpoints together when comparing two functions.

When comparing functions, it is not enough to compare their values at each domain number. There are too many. We need to move faster. We need to predict what is going to happen. That is where rates-of-change come in.

We would like rates-of-change on intervals. We would also like a pointwise viewpoint of rates-of-change.

A rate-of-change at a point seems like a contradiction, but fits in with the story of tangent lines very well.

Learning Outcomes

In this section, students will

• investigate rate of change.