*Introduction:
Average
Rate of Change*
The average rate of change of any function is a
concept
that is not new to you. You have studied it in relation to a
line.
That's right! The *slope*
is the average rate of change of a line. For a line, it was
unique
in the fact that the slope was constant. It didn't change no
matter
what two points you calculated it for on the line. Take a look at
the following graph and we will discuss the slope of a function.
Demo:
Slope of a Secant/Tangent Line (Walter Fendt)
The function is in red. The blue line connects
the
two points that we want to find the average rate of change (slope of
the
blue line). The two points are (x, f(x)) and (x+h, f(x+h)).
To find the slope, the definition is the change in y over the change of
x. Does this sound familiar!! Applying this definition we
get
the following formula:

Notice on the graph that the line we are finding the
slope
of crosses the graph twice. Do you remember from geometry what
you
call a line that crosses a circle twice? You got it!! It's
a *secant line*! When you
calculate
the average rate of change of a function, you are finding the slope of
the secant line between the two points.
As an example, let's find the average rate of
change (slope
of the secant line) for any point on a given function.
This
is finding the general rate of
change.
The general rate of change is good for any two points on the
function.
Find the general rate of change for f(x) = x^{2}

f(x) = x^{2 }
and
f(x + h) = (x + h)^{2}
Therefore, the slope of the
secant
line between any two points on this function is 2x
+ h. To find the specific
rate of change between two given values of
x, is a simple matter of substitution. Let's say we are asked to
find the average rate of change between the points x_{1}
= 2 and x_{2} =
4.
Then in our general answer, we will replace x with x_{1}
and h = x_{2} - x_{1}.
Replacing these values in the formula yields 2(2) + (4 - 2) = 4 +
2 = 6. Thus, the slope of the secant line connecting the two
points
of the function is 6. Note that the answer is a positive
number.
That means what? That's right, you know! The line is going
uphill or increasing as you look at it from left to right. Be
careful
that you put the values for determining h in the correct order.
You
already know that slope can be positive, negative or zero.
Now using the same function
as above,
find the average rate of change between x_{1}
= -1 and x_{2} =
-3.
The answer is 2(-1) + ( -3 + 1) = -2 + -2 = -4. This means that
the
secant line is going downhill or decreasing as you look at it from left
to right.

__Sample problems__

**1) Find the
general
rate of change for the function f(x) = 2x**^{2}
+ 1. Then find the specific rate of change for x_{1}
= 2 to x_{2} = 5.
**2) Find the general
rate
of change for the function f(x) = x**^{3}.
Then find the specifice rate of change for x_{1}
= 0 to x_{2} = 2.

**Look for the answers worked
out somewhere below!**

Fly into the next
section with
Snoopy!

Look down here for the
answers
to the sample problems!!
Specific rate of change = 4(2) +2(5 - 2) = 8
+ 6
= 14
Specific rate of change = 3(0) + 3(0)(2) + (2)^{2}
= 4
Ready for the next
section!
Go back up and hit the yes button!!