Army vs Navy: The Long Trip
Secants and Tangents
The following applet demonstrates the relationship between secants and tangents, average rate of change and instantaneous rate of change, and average velocity and instantaneous velocity.
Scenario: Each year hundreds of cadets depart the United States Military Academy (USMA) enroute to the Army vs. Navy game. In the late 90's a group of plebes, freshman, departed USMA in such a rush that they forgot their long overcoats, a part of the next day's required uniform. It was not until arriving at their hotel in Philadelphia did the group realize the mistake. They had to drive back to USMA to get their long overcoats. To pass the time, the group decided to develop a model for the distance away from the hotel as a function of time. The following model was developed:
s(t) = distance from the hotel (miles) after t 30 minute periods
t = time, greater than or equal to zero, (30 minute period)
Using the Applet:
Use the sliders to adjust t1 and t0. As t1 and t0 are varied, a secant between s(t1) and s(t0) and a tangent to s(t) at s(t1) are drawn. The slope of this secant is the average velocity between t1 and t0. The slope of this tangent is the instantaneous velocity at t1 . Adjust t0 such that D t, t1 - t0, goes to zero. As D t approaches 0, the slope of the secant line (the average velocity) approaches the instantaneous velocity or the value of the derivative of the position function.
Points of Interest:
- instantaneous velocity
- average velocity
- the average rate of change is the slope of the secant line
- instantaneous velocity is the limiting value of the average velocities
- instantaneous rate of change is the limiting value of the average rates of change
- the two interpretations of the derivative
Use Applet (author Garrett Heath)