Warm-up/Activator: Rates of Change

What are 3 different rates of change that you are familiar with?

What are the Units they are written with?

When discussing graphing a line, rate of change is synonymous with ___________.

ESSENTIAL QUESTION: HOW DO YOU FIND THE

AVERAGE RATE OF CHANGE OF A FUNCTION?

Velocity

Average Rates of Change

If you travel 369 miles from Lancaster, PA to Sturbridge, MA and it takes you 6 hours and 20 minutes. What is your average speed?

Is it always the same speed?

Average Rates of Change

What is it?

x

y

ab

afbf

)()(

Example

If a free-falling object is dropped from a height of 100ft and air-resistance is neglected, the height, in feet, of the object at time ,t in seconds, is given by h = -16t2 + 100. Find the average speed of the object over the following intervals.a) [1,2]

b) [1,1.5]

c) [1,1.1]

ab

afbf

)()(

h = -16t2 + 100

What is happening to the average rates of change as we shrink the intervals?

[1,2] [1,1.5] [1,1.1]

Instantaneous Rate of Change & Velocity

Below is the limit definition of taking the derivative of a function.

Notice how this would give us the instantaneous velocity.

So, if we want to know the instantaneous velocity of an object, we must take the derivative of the position function to find the velocity function. By plugging an x-value into the velocity function, we find an instantaneous velocity.

x

xfxxf

x

yxx

)()(limlim

00

Instantaneous Rate of Change & Velocity

While there is much to learn about taking derivatives in a calculus course, taking the derivative of a polynomial function is fairly simple. For each term, take the exponent times the coefficient for the derivative coefficient. The derivative exponent is one less than the original exponent. The derivative of a constant term is zero.

For example, h = -16t2 + 100 then it’s derivative h’ = -32t1 + 0 = -32t

Find the velocity of the object at t = 1 (instantaneous velocity).

Example

At t = 0, a diver jumps from a diving board that is 32ft high. Because the diver’s initial velocity is 16 ft/s his position is h = -16t2 + 16t + 32

When does the diver hit the water? How fast was he going?

When does the diver reach his peak? How high did he go?