Understanding Calculus

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  Table of Contents

  1. Why Study
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  8. Understanding
  9. Derivative
  10. Integration
  11. Understanding
  12. Differentials

  Inverse Functions
  Applications of
  Sine and Cosine
  Sine Function
  Sine Function -
  Differentiation and
  Oscillatory Motion
  Mean Value
  Taylor Series
  More Taylor Series


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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.


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


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