|
|
RADAR
APPLICATIONS Applications
of the Graph of the Derivative Function Real-World Applications: Changes in Volume and Traffic Flow Modeling the Real World:� The concept of derivative is utilized in numerous applications. Real-world problems involving instantaneous rates of change of any kind employ derivatives. However, the derivative can also be applied in a geometric or visual manner. In the following two examples we illustrate the use of the graph of the derivative to real-world situations involving volume and traffic. (These examples are taken from Calculus, Single Variable by Hughes-Hallett, Gleason, et al., Wiley, New York, NY, 2000.) I.
Changes in Volume - Balloon Example: In this example we examine the graph of the derivative to answer related questions about rates of change of volume with respect to a balloon.� We define a function of time, t , to represent the changing volume of air in a balloon. The derivative of this function then represents the rates of change of volume. We then interpret the graph of the derivative function as it relates to observations about the balloon. Suppose a child inflates a
balloon, admires it for a while and then lets the air out at a constant rate.
If� V(t)
gives the volume of the balloon at time t, then
the figure below shows the rate of change of volume, V'(t), as a function of t.
What
observations can we make?
II.
Traffic - Vehicle Motion: Let's consider a vehicle moving along a straight road.
Suppose the vehicle's distance from its starting point at time t is given by f(t). �Then� f'(t) describes the variation in the rate of
change of distance with respect to time. Below are three graphs for f'(t) �( Assume the
scales on the vertical axes are all the same.) : Which of the graphs in the figure below could be f'(t) for the following scenarios ? a) A bus on a popular route, with no traffic b) A car with no traffic and all green lights c) A car in heavy traffic conditions Graphs of the Derivative Functions,
f'(t):
Answers: I. b) �This graph, I, of the derivative illustrates an initial increase in speed that becomes steady and corresponds best to the car with no traffic and all green lights. II. a)� The graph of� f'(t) shown in II, corresponds best to the bus on the popular route. Intervals where f'(t) is zero�� represent the occasions when the bus is stopped picking up passengers. The parabolic sections of same height reflect the fact that the bus has no traffic and is therefore able to reach the same maximum speed at various times throughout the trip. III.
c)� Graph III represents the car in heavy traffic
conditions. Notice that the speed never gets very large. In addition, the
heavy traffic flow is indicated in the numerous changes in speed.�������
|
|
