Rate of Change
The average rate of change of any function is a
that is not new to you. You have studied it in relation to a
That's right! The slope
is the average rate of change of a line. For a line, it was
in the fact that the slope was constant. It didn't change no
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.
Slope of a Secant/Tangent Line (Walter Fendt)
The function is in red. The blue line connects
two points that we want to find the average rate of change (slope of
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
the following formula:
Notice on the graph that the line we are finding the
of crosses the graph twice. Do you remember from geometry what
call a line that crosses a circle twice? You got it!! It's
a secant line! When you
the average rate of change of a function, you are finding the slope of
the secant line between the two points.
1) Find the
rate of change for the function f(x) = 2x2
+ 1. Then find the specific rate of change for x1
= 2 to x2 = 5.
As an example, let's find the average rate of
of the secant line) for any point on a given function.
is finding the general rate of
The general rate of change is good for any two points on the
Find the general rate of change for f(x) = x2
f(x) = x2
f(x + h) = (x + h)2
Therefore, the slope of the
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 x1
= 2 and x2 =
Then in our general answer, we will replace x with x1
and h = x2 - x1.
Replacing these values in the formula yields 2(2) + (4 - 2) = 4 +
2 = 6. Thus, the slope of the secant line connecting the two
of the function is 6. Note that the answer is a positive
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
that you put the values for determining h in the correct order.
already know that slope can be positive, negative or zero.
Now using the same function
find the average rate of change between x1
= -1 and x2 =
The answer is 2(-1) + ( -3 + 1) = -2 + -2 = -4. This means that
secant line is going downhill or decreasing as you look at it from left
2) Find the general
of change for the function f(x) = x3.
Then find the specifice rate of change for x1
= 0 to x2 = 2.
Look for the answers worked
out somewhere below!
Fly into the next
Look down here for the
to the sample problems!!
Specific rate of change = 4(2) +2(5 - 2) = 8
Specific rate of change = 3(0) + 3(0)(2) + (2)2
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