Learning Resources

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Outcomes

In this lesson you will

• observe the relationship between the slope of a tangent to a curve at a point and the rate of change of the curve at that point 
• 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

By the end of this section students will be able to:

• demonstrate an understanding that the slope of a line tangent to a curve at a point 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

Mathematical Modeling, Book 3 p.85 - 96

The average rate of change developed earlier will lead you to an expression of instantaneous rate of change in situations involving constant and variable rates of change. The emphasis is on graphing data and analyzing the average rate of change over smaller and smaller intervals of time to express the rate at a given instant in time. From these descriptions,  you will become familiar with the conventional notation use to describe the instantaneous rate of change.

This section should take 4 hours to complete.

Prerequisites

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

• calculate the slope of a line between two points.