Average Rate of Change and Equations…

Do Now: The height, in feet, of a ball thrown into the air is modeled by h(t) = - 16t2 – 96t , where t is measured in seconds. Graph the height of the ball seconds on the following grid. Use a scale on the x-axis of

3 boxes = 1 second. On the y -axis use 5 boxes = 40 feet.

• How far did the ball travel from t = 0 to t = 2 seconds?128 Ft

• What was the average distance traveled for the first two seconds?64 ft per second

• What is another name for this?average velocity

• Draw a line connecting (0, h(0)) and (2,h(2)). What is this line?secant line

• What does the average distance traveled (per second) represent in terms of this line?

slope• Write the equation of this line.

y – 0 = 64(x – 0) or y – 128 = 64(x – 2)

In general, __________________________________, can be found by using the formula:

The average rate of change of a function (slope of the secant)

AROCf b f a

b a

( ) ( )

Where [a,b] is a closed interval on which f is defined

The equation of ___________________________, can be found by:

1. Finding the slope of the secant (AROC)

2. using point/slope with either point

The secant line

• In general, as the interval get smaller (as b gets closer to a), what line are we approximating?

tangent line

• In terms of the original problem, what does this represent?

Instantaneous velocity

In general, how can we represent this?

ab

afbfab

)()(lim