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DIVING TOWER REVIEW
Math Topic:
derivativesKey Terms: secant lines, average velocity, tangent lines, instantaneous velocity
The Math behind the Observations:
I. AVERAGE VELOCITY AND SECANT LINES:
The above diagram shows a static version of the applet. What question is being answered here? If you did the lab exercise you may recognize this picture as the answer to the following question: What is the average velocity in meters per second for the first second of the jump? Answer: -5
How was -5 obtained? This number is the approximate slope of the secant line drawn from t = 0 to t=1. The slope can be estimated by the observing the rise/run from the graph; however, the applet itself indicates the slope as -5.01; "Slope of last line drawn =-5.01". ( Note: the negative sign indicates that the height is decreasing.)
Average velocity = 
= 
= - 5 m/sec
Note: 
= 
= slope of the secant line.
These observations can be generalized as follows:
Average Rate of Change of f (x) over [a,b] = 
= slope of the secant line.
- Instantaneous Velocity and Tangent Lines
This second diagram shows another static view of the applet. What is the corresponding question? In the lab questions you were asked to answer: What is the instantaneous velocity in meters per second at time t = 0.4?
Answer: -4
How was -4 obtained? This number is the approximate slope of the tangent line drawn to the curve when t = 0.4. The slope could be estimated by taking the rise/run of the tangent line with a run of 1.0 seconds (at time 0 the height of the tangent line is about 11 meters and at time one it's about 7 meters, for a "rise" of -4.); once again, the applet itself provides the slope of this tangent line as seen in text right below the graph, " Slope of last line drawn = -4.0".
Instantaneous velocity = slope of tangent line at the point = (approximately) 
= -4.0 meters/sec.
Note: -4.0 is the value of the derivative of the distance function at t = 0.4.
These observations can be generalized as:
Given the graph of f and x = a,
Derivative of f at a = the slope of the tangent line to the graph at a = instantaneous rate of change .
Links to related sites:
The following sites present excellent additional resources- tutorials and Java applets:
USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil