This document discusses functions, their derivatives, and rates of change. It introduces the concept of the derivative as the instantaneous rate of change of a function. It then asks the reader to sketch and describe different types of functions without a calculator. It provides a sample graph showing sections that are increasing, decreasing, or neither, and asks the reader to identify these sections and determine where the rate of change is more or less quick.

1. Warm Up Sketch a graph of each of the following functions without a calculator and describe the type of function.

2. The Derivative We have been discussing instantaneous rate of change for the last two days. From now on, we are going to refer to this rate as the derivative . Today, we are going to investigate the derivative a little bit more… Since the derivative is a rate of change , we need to discuss how it is that we say functions are changing : A function can be either increasing , decreasing or neither . In the following graph, all three of these things occur: Where is the graph increasing? Where is the graph decreasing? Neither?

3. Of the two sections of the graph that are increasing, which increases more quickly? Of the two sections of the graph that are decreasing, which decreases more quickly? How can we define more quickly ?