We begin now an investigation of the second question raised in Chapter 1: How fast does y vary with t? We shall return to the first question (How does y vary with t?) from time to time as we encounter more kinds of functional relationships.

The development of this chapter follows this sequence of topics:

1. Average speed as a familiar example of rate of change – of how fast distance traveled varies with time

2. Average velocity as rate of change of displacement with respect to time

3. The intuitive idea of instantaneous velocity leading to the concept of limit

4. Computations of instantaneous velocity

5. Statements of theorems on limits, to simplify evaluations of limits

6. Proofs of some results on limits

7. Geometrical examples: slope of a line, average slope of a curve in an interval, slope of a curve at a point

8. Tangent to a curve defined in terms of slope

9. Unifying concept: the derivative as a rate of change, showing how fast one quantity varies with another

10. A specific formula for the derivative, and several general formulas As you proceed through the chapter, you may find it helpful to refer back to this outline.

Average speed and average velocity

Probably the most familiar example of a rate is found in the concept of average speed.