For this activity, we will not begin by rattling off definitions, but work through an activity that will leave you with a general understanding of the concept of derivative. After completing the activity, definitions will be provided.

For this activity, you will need the geometric software program,
*Geometer's Sketchpad (GSP). *Before we can begin exploring,
you will need to download this
GSP file copyright by Scher, Steketee, Kunkel, & Lyublinskaya,
2005.

To begin, open the GSP file, Instantaneous Rate.gsp. You should have an opend sketchpad that looks like the following.

This activity will explore the angle of an automatic door (*d*)
as function of time (*t*). Before going further, press the"open
door" button to operate the door. Watch the movement of the
point t1 as the door opens and closes. You can also manipulate
the point t1 youself, by dragging it back and forth along the
*t*-axis.

As you observe the movement of the door, what happens to the values of t1 and d1? For what values of t1 is the door opening, or the angle incresing? For what values of t1 is the door closing, or the angle decresing? What is the time when the door is open to its fulles? How do you know this value?

Next, we would like to find the rate of change of the door's
angle, but we will need another point. To show t2, press the "show
t2" button. Now, drag t1 across the *t*-axis and observe
the behavior of t2. What happens to t2 as you change ?
(Note that the value of is the distance
between t1 and t2). What happens to the values of t2 and d2 as
becomes smaller? What happens when you move
the slider value of t2?

Using set to 0.1, calculate the rate
of change of the door's angle for any time *t*, usign the
values t1, t2, d1, and d2. You can use GSP to perform this calculation
for you. What can you say about the door's motion from the rate
of change? Now, press the "show rate" button. What does
the dotted line represent? What is the dotted line's relationship
to your calculate rate of change? When you move the point t1 along
the *t*-axis what happens to the dotted line? Can you tell
when the door is opening and closeing and the door's rate from
the dotted line?

Now, set and t1=1. What is the rate of change at that time?

Next, select the values t1, d1, t2, d2, , and the rate of change (in that order on the left hand side of the sketch), then choose Tabulate from the Graph menu. Once the table has been constructed, double click the first entry to make it permanent.

Now, change to .01 and notice values change in the table. Double click the table values to make them permanent. What happens to the rate of changa as values of change? Perform the previous action of changing values of and record the subsequent values within the table. What occurs to the rate of change as the value of becomes smaller?

Next, using the button "t->3", which represents the value of t1 at 3 seconds, record the values of as it runs through the values .1 to .000001. What happens?

To proceed further, note that the *rate of change* is
the rate of change between *two* different values of *t*,
while the* instantaneous rate of change* is the *exact*
rate of change at *one* specific time, *t*. Now, since
we need two values to compute the rate of change, we can calculate
the instantaneous rate of change by making our two values 'close'
to each other. To do this, we can take the limit of the rate of
change as the second value, t2, approaches the first, t1. Now,
as you may or may not have guessed, the instantaneous rate of
change is the *derivative* of the function, that is, the
limit of the average rate of change as the interval between t1
and t2 gets closer and closer to zero. Now, what is the derivative
of the door's angle when t1 is 1 second (use the "t->1"
button)?

To conclde the activity, take a few minutes to write down your
own definition of the *derivative* of a function.

This activity was adapted from Scher, D., Steketee, S., Kunkel,
P., & Lyublinskaya, I. (2005). *Exploring precalculus with
The Geometer's Sketchpad*. Emeryville, CA:Key Curriculum Press.

Now that we have an idea of what the derivative of a function is, consider the following mathematical definition:

The *derivative of a function f at a number a*, denoted
by *f'(a)*, is

Or, an equivalent definition, is

However, what benefit is this above definition, if we do not
know what a derivative is in layman's terms. The most basic way
to describe the derivative in one simple term is "slope"
(Adams, Hass, & Thompson, 1998). Consider the following illustration
from *How to Ace Calculus: The Streetwise Guide* (Adams et
al., 1998), to illustrate this concept further:

Let us suppose you have a tranquilized goat, which you are going to carry up a hill. If you begin at the base of the hill, let's say coordinates

x=0andy=0, and climb up the hill, both yourxandycoordinates will increase. If we leth(x)=ybe the function that defines the hill you are walking up, then the derivative ofh(x), is exactly the steepness of the hill at some pointx, denotedh'(x). Now, consider for exampleh'(10)=1/6, that is, after reaching10 feet in thexdirection, you are at a point where the steepness is 1/6. This means that for every foot you travel horizontally, along thexaxis, you must travel 2 inches in the verical direction, or along theyaxis. This is not that steep of an incline. However, ifh'(20)=5, this means that for every 1 foot you travel horizontally, along thexaxis, you must travel 5 feet vertically, along theyaxis. This is quite steep and may require extensive mountainering equipment. Now, ifh'(40)=-3, this would mean that whenx=40, you are heading in a negative direction, that is for every 1 foot in the horizontal direction, you are going downhill 3 feet. At this point it will not be necessary to carry the goat on your back, but rather roll it down the hill.

Now, consider the following two simple facts about derivatives:

- The derivative
*f'(a)*is the instantaneous rate of changeof*y=f(x)*with respect to*x*when*x=a*. Note that the instantaneous rate of change = , where . - Given our first equation, where
*a*is a fixed value. If we want*a*to vary, or change, we will replace*a*by*x*, giving . Using this latter function, we can assign*x*to the number*f'(x)*, creating a new function, called the*derivative of f*.

Adams, C., Hass, J., & Thompson, A. (1998). *How to ace
Calculus: The streetwise guide*. New York: W. H. Freeman &
Co.

Stewart, J. (1998). *Calculus: Concepts and contexts*.
Pacific Grove, CA: Brooks/Cole Publishing Co.

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