# Rate of Change, Instantaneous

The instantaneous rate of change is the limit of the function that describes the average rate of change. It has many practical applications, and can be used to describe how an object travels through the air, in space, or across the ground. The changes in the speed of an airplane, a space shuttle, and a car all may be described using the instantaneous rate of change concept. When describing motion, this concept is also referred to as velocity .

There are many other uses of the instantaneous rate of change concept. Chemists use it to examine chemical reactions. Electrical engineers use this concept to describe the changes that occur in the current of an electric circuit. Economists use this concept to describe the profits and losses of a given business.

In order to understand the instantaneous rate of change concept, it is first necessary to understand the meaning of the terms average rate of change and limit.

## Average Rate of Change

The average rate of change describes how one variable changes as the other variable increases by a single unit. For example, Galileo discovered that if the impact of air resistance is ignored, the distance that an object travels

with respect to time when dropped from a given height is described by the following function:

s (t ) = 4.9t 2 where t 0.

In this function, t indicates the amount of time that has passed since the object was dropped. The evaluation of this function, s (t ), describes the total distance that the object has traveled by a given point in time. For example, after 2 seconds, the object will have traveled a total distance of 4.9 (22) or approximately 19.6 meters. This is, of course, assuming that the object has not yet hit the ground.

In order to calculate the average rate of change described by the function s (t ), it is useful to examine a graph of this function. For this discussion, assume that the object is dropped from a height of 240 meters. The graph of the function, s (t ), is shown in (a) of the figure. Notice that the graph begins at 0 seconds and ends at 7 seconds. The curve in this graph suggests that the speed of the object is changing over time. Since this object is dropped from a height of 240 meters (m), after approximately 7 seconds the object will hit the ground (4.9 (72) = 240.1 m).

The average rate of change is the average distance that is traveled in a single second. To determine the average rate of change, a ratio is formed between the difference between the distance traveled and the amount of time that has passed. This can be expressed as m/sec. This same algebraic expression may also used to determine the slope of the line segment between the points (0, 0) and (7, 240). The colored line segment in (a) of the figure has a slope of . The slope of this line represents that average rate of change of the function s (t ). An interpretation of this slope is that on average the object covers a distance of meters every second. If the original function had been a straight line, then the slope of that line would be equal to the average rate of change of that function.

It is important to recognize in the current example that is the average rate of change for a 7-second time interval. The average rate of change can also be calculated for smaller time intervals. For example, the average rate of change may be calculated for each 1-second time interval. These calculations are shown in the first table. A graphical depiction of the average

 AVERAGE RATE OF CHANGE FOR 1-SECOND TIME INTERVALS TimeInterval Calculation Average Rateof Change 0 to 1 4.9 m/sec 1 to 2 14.7 m/sec 2 to 3 24.5 m/sec 3 to 4 34.3 m/sec 4 to 5 44.1 m/sec 5 to 6 53.9 m/sec 6 to 7 63.7 m/sec
 AVERAGE RATE OF CHANGE FOR SMALL TIME INTERVALS For Intervals Ending at 1 Second For Intervals Beginning at 1 Second Time Interval Average Rate of Change Time Interval Average Rate of Change 0.9 to 1 9.31 (m/sec) 1 to 1.1 10.29 (m/sec) 0.99 to 1 9.751 (m/sec) 1 to 1.01 9.849 (m/sec) 0.999 to 1 9.7951 (m/sec) 1 to 1.001 9.8049 (m/sec) 0.9999 to 1 9.79951 (m/sec) 1 to 1.0001 9.80049 (m/sec) 0.99999 to 1 9.799951 (m/sec) 1 to 1.00001 9.800049 (m/sec)

rate of change for each of these 1-second time intervals is shown in (b) of the figure above. The slope of the colored line segments is the average rate of change for each time interval.

Notice that the average of the third column shown in the table is about 34 m/sec or approximately the average rate of change for the 7-second time interval. In other words, the average rate of change across these seven time intervals is equal to the average rate of change across the entire time interval. This will always be true. The average rate of change may be calculated for ½ -second time intervals, ¼ -second time intervals, or even smaller time intervals.

The Limit Concept. Limit refers to a mathematical concept in which numerical values get closer and closer to a given value or approach that value. The limit of the function that describes the average rate of change is referred to as the instantaneous rate of change.

Returning to the previous example, if smaller and smaller time intervals are considered around the same value, the slope of the function s (t ) will approach or get closer and closer to a specific value. For example, smaller and smaller time intervals that all end at 1 second can be identified. As the second table suggests, the average rate of change is getting closer and closer to 9.8 m/sec. Smaller and smaller times intervals that all begin at 1 second can also be identified. As the table suggests, the average rate of change is again getting closer and closer to 9.8 m/sec. The value 9.8 m/sec is therefore the limit of the function that describes the average rate of change.

As the above example illustrates, determining the instantaneous rate of change is cumbersome using the methods of algebra and geometry . In calculus , the derivative concept is examined. Evaluating the derivative of a function at a given point is another way to determine the instantaneous rate of change.

Barbara M. Moskal

## Bibliography

Stewart, James. Calculus: Concepts and Contents. Pacific Grove, CA: Brooks/Cole Publishing Co., 1998.

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