CHAPTER 4
### Chapter 4 - The Derivative

#### Section 4.3 - Instantaneous Rate of Change

In this section we will take a much
closer look at rate of change and see how we can define the instantaneous rate
of change of a function at any point or value of the independent variable. Let
us begin with a study of the simple
. Functions of the form
, where c is a constant, express direct relationships. This
is because the value of the function, *f*,
is a constant multiple or fraction of the independent variable. Thus f is said
to be **directly proportional** to x. The
graph of the function
looks like

From x = 1 to x = 3,
x = 2, and
f = 9 - 3 = 6. Thus, over this interval of
x = 2,
f equals 6. The *average*
*rate at which f changed* with respect
to x is by definition,
. For each unit change in x, the change in f is 3. This tells us that f is changing three times
faster that x is changing over the
interval from x = 1 to x = 3. We can now look at the interval from x = 2 to
x = 4 where
x equals 2.

Thus

Over this interval, the rate of
change is the same constant, 3. This leads us to conclude that the rate of
change of the function over **any**
interval is a constant, 3. This can be proven by the definition of rate of
change:

The rate of change of the direct
function
is c and is constant
over an interval,
x. The graph of a function
is therefore an
increasing straight line with a constant slope or steepness equal to **c**. For this reason functions of the
form
are also called **linear functions **since their graphs are
straight lines with a** constant** rate
of change.

On the other hand, the graph of the function

is not a straight
line. Unlike a line, the rate of change of

is not the same
constant over any interval
x. To define the rate of change for the function we will have to derive a more precise way of
defining rate of change of function. To do this we will analyze the
function over small intervals of
x

The first point to notice is that
the rate of change of the function varies with x. When x is small, *f* does not change that much as compared
to how much it changes when x is large. We can conclude that the rate of change
of *f* with respect to *x*, is not constant over any interval,
x, but varies with *x*.

To begin our analysis let us divide
up the graph into intervals of
= 2
and study what is happening in each of these intervals separately.

From *x = 0* to *x = 2*,
= 2, the change in f
is f(2) - f(0)= 4.

From* x = 6* to *x = 8*,
= 2, the change in* f * is
f(8) - f(6) = 28.

Clearly, as x
increases the rate of change of the function is increasing and is not a
constant as in our study of the line where the function changed at the same
rate as x increased.

Returning back to the graph of
; we calculated that from*
x = 0 to x = 2* the change in *f* was 4.
We can then conclude that from x = 0 to x = 2 the **average rate of change** of the function over that particular
interval is 2 or:

Remember it is
called average because this rate of change is only valid from x = 0 to x = 2.

Now let us consider the interval
from x = 2 to x = 4 where once again the change in x,
is 2. The change in *f* of the graph is equal to:

Thus the average
rate of change of the function over this interval is equal to

This value is greater than the value
we observed from x = 0 to x = 2. This implies that from x = 2 to x = 4, f(x) is
increasing at a greater rate than from x = 0 to x = 2. This is despite the fact
that in both cases the
= 2. Therefore, the rate of change is not constant over the interval,
but is increasing
with *x*. When we say the rate of
change of *f(x)* from *x = 2* to *x = 4* is 6, it is only the
average value for the given interval since it assumes rate of change is **constant** over that interval.

Let us move on to the next interval
of x = 4 to x = 6. Once again the change in x or
equals 2 but the corresponding change in f(x) is not
the same as before.

For x equal to 4
and = 2, the change in the function over this interval
is

Note that this corresponds to the same value we would get by:

The average rate of change over this interval is
therefore:

This is still larger than the rate of change for the previous interval which was 6.
Remember rate of change , by definition, refers to how much the function
changes with respect to a **change** in
the independent variable. The steepness or slope of the line over the interval
provides a geometric understanding for this concept.

The average rate of changes
calculated over each interval can be used to approximate the graph of *f(x)* from *x = 0 to x = 8*.

This roughly corresponds to the
original graph, but as we can see the rate of change or slope is not constant
through out the interval from *x = 0 to x = 8,*
but increases as x-increases. In order to get more accurate answers we need to
reduce our interval of
=2
to a much smaller one. The idea being we need to analyze our graph over a small
interval, , to see what exactly is going on at each instant
the function is changing.

Here is where we begin our study of
Calculus. We break down and freeze a changing situation into an infinite series of actions and
analyzing what is going on in each individual actions.

**Next section ->**
*
Section 4.4 - The Derivative
*