Rates of Change

    This topic explains the difference between the average rate of change and instantaneous rate of change. It also illustrates the importance of the relative rate of change.

    The absolute change is not the same as the average rate of change. Namely, the absolute change is just the differences in the values of at the boundary of the interval namely whereas the average rate of change is the absolute change divided by the size of the interval: The average rate of change is sometimes more useful; for example, suppose you want to know how long it takes to make some money and not just the size of the money made (absolute change). Knowing the rate at which the money is being made, (the average rate of change over a given time interval) is often useful.  

Definition (Average Rate of Change) Suppose is a function of say When a change in the variable is made from to there is a corresponding change to the namely The average rate of change of with respect to is



and is also known as the difference quotient.

Example (Average Rate of Change)  Let Find the average rate of change from to

    Solution. The average rate of change of from to is given by,



which is also the slope of the secant line through and

    In general, suppose an object moves along a straight line according to an equation of motion where is the displacement (directed distance) of the object from the origin at time The function that describes the motion is called the position function of the object. In the time interval from to the change in position is and the average velocity over this time interval is



which is the same as the slope of the secant line through these two points.

Example (Average Velocity)  If a billiard is dropped from a height of 500 feet, its height at time is given by the position function where is measured in feet and is measured in seconds. Find the average velocity over the intervals and

    Solution. For the interval the object falls from a height of feet to a height of



The average velocity is
    


For the interval the object falls from a height of feet to a height of The average velocity is
    


Note that the average velocities are negative indicating that the object is moving downward.

    The difference quotient



is the average rate of change of with respect to over the interval and can be interpreted as the slope of the secant line. Its limit as is the derivative at and is denoted by We interpret the limit of the average rate of change as the interval becomes smaller and smaller to be the instantaneous rate of change. Often, different branches of science have specific interpretations of the derivative.

Definition (Instantaneous Rate of Change) As the average rate of change approaches the instantaneous rate for change; that is,



and is also known as the derivative of at

Example (Instantaneous Rate of Change)  Let Find the instantaneous rate of change at

    Solution. Since the instantaneous rate for change of at is given by,

Example (Estimating the Instantaneous Rate of Change)  Temperature readings (in degrees Celsius) were recorded every hour starting at midnight on a day in April. The time is measured in hours from midnight.






(a) Find the average rates of change of temperatures with respect to time from noon to 3:00 p.m., 2:00 p.m. and 1:00 p.m.

    Solution.  The average rates of change are, respectively,  







(b) Estimate the instantaneous rate of change at noon.

    Solution.  We plot the given data and use them to sketch a smooth curve that approximates the graph of the temperature function. Then we draw that tangent line at the point where and after measuring the sides of the triangle



we estimate that the slope of the tangent line is and so the instantaneous rate of change of temperature with respect to time at noon is about

    Sometimes we are not interested in the instantaneous rate of change and instead we may want a relative rate of change (percentage). For example suppose a student makes a 39 on a test, this would be a very good grade if the score is out of 40 points. However if the score was out of a total of 100 points then the grade is not so good.   

Definition (Relative Rate of Change) Let then the relative rate of change at is the ratio

Example (Relative Rate of Change) Let Find the relative rate of change at and

    Solution. Since The relative rate of change of at is

or

The relative rate of change of at is

or

    Often we are more interested in the relative rate of change of a quantity instead of the instantaneous rate of change. If instance, if you are earning and receive a 5,000 raise, you would probably be very please. However, if you were making you may not be as please since the relative change is not as much. With the pay you only have a



increase whereas with the pay you have a better percentage increase of

  

Example (Comparison Between Different Types of Rates of Change) Let

(a) Find the average rate of change from to

    Solution. The average rate of change of from to is given by,



(b) Find the instantaneous rate of change at

    Solution. Since the instantaneous rate for change of at is given by,

(c) Find the relative rate of change of at

    Solution. The relative rate of change of at is or

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Rates Of Change
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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