Learning Resources

Outcomes

In this lesson you will

• Observe that the slopes of secant lines close to a point can be used to estimate the slope of the tangent to a curve at that point.
• Observe the relationship between the slope of the tangent to a curve at a point and the rate of change of the curve at that point.

By the end of this section you should be able to:

• Demonstrate an understanding that the slope of a line tangent to a curve is the instantaneous rate of change of the curve at the point of tangency.
• Approximate and interpret slopes of tangents to curves at various points on the curves, with and without technology.
• Describe and apply rates of change by analyzing graphs, equations, and descriptions of linear and quadratic functions.
• Demonstrate an understanding that slope depicts rate of change
• Solve problems involving instantaneous rates of change.

Introduction

In this section you will focus on the meaning of instantaneous change and the approximation of these rates. 

You will use technology to help you visualize how the slope of the tangent to a curve at a point is approximated more and more closely by the slopes of a sequence of secant lines connecting nearby points on the curve. Technology will help you see why it makes sense to view this slope of a tangent at a point as a description of instantaneous rate of change at that point. 

You will approximate rates of change in a variety of situations involving constant and variable rates of change by looking at the graphs of functions or working with the descriptions of those functions.

In this section, you will also become familiar with conventional notation used to describe rate of change.

This section should take you about 4 hours to complete.

Prerequisites

To be successful in this lesson, it would be helpful to know the following:

• The ability to construct a graph from a table of values.
• The concept of rate of change.