Moving from Average Rate of Change (AROC) to Instantaneous

Rate of Change (IROC)Today you will use the average

rate of change to find the instantaneous rate of change

and see how the slope of secant lines can be used to approximate the slope of

tangent lines.

Moving from AROC to IROC

• You have done several problems where you investigated the average rate of change for functions over different intervals. In this lesson, you will learn how to find the instantaneous rate of change at a specific point.

Average Rate of Change for Linear Functions• You investigated the average rates of change for

several linear functions over various intervals. You found that the average rate of change for y=3x-5 was 3, the average rate of change for y=2x -1 was 2, and the average rate of change of y=6-2x was -2.

• What is true about the average rate of change of linear functions?

• Has this been proven or are you working from patterns?

Proving that the average rate of change for linear functions is the slope.

Let f(x)=mx+b. We want to find the average rate of change of f(x) over the interval [c,d].

• Explain why we want to compute the ratio

• Compute the ratio and show that it equals m, the slope.

• Explain why what you have done shows that the average rate of change of every linear function is constant over every interval.

( ) ( )f d f cd c

Making Connections

• You have seen how slope, the average rate of change, and the instantaneous rate of change are all related. Now let’s look specifically at the lines that we have been using to calculate these two types of rates of change. We begin by developing a general idea of what we mean by secant and tangent lines as they relate to graphs.

Let’s look at the secant line and the tangent line using a dynamic tool.

Internet Applet

Consider the function f(x)=2x2+5

• Write an expression for the average rate of change from x = 1 to x = 1+h

The average rate of change function approaches a number as h→0. What is the number?

Geometrically, what is the significance of this number?

2.5,2.5 0, 15

Average vs. Instantaneous Rate of Change

Example:

2( ) 3 2f x x

Try Another One

2( ) 4f x x x

The Derivative• This is one of the central ideas of all of calculus. The

concept of an “instantaneous rate of change” of a function is an important idea in calculus. Its formal name is the derivative. This idea is useful in finding the slope of a tangent line of a graph and finding the velocity from a distance function. On Monday, we will formally define the derivative and learn about various notations used to express it.

AssignmentA 1.17 Sect III