Quote of the Day: 
"The shortest distance between two points is under construction."
           -- Bill Sanderson

Objectives:
    The student will determine the Average Rate of Change 
    	of a function over a given interval.
    The student will determine the Instantaneous Rate of 
	Change of a function at a given point.
    The student will determine the average velocity of a given function
        over a given time interval.
    The student will determine the instantaneous velocity of a given function
        at a particular time.

1. Bellringer:

  A. Problem:
	You make a roundtrip from Winchester to Harrisonburg,
	a distance of 60 miles one way (actually, any distance 
        will work in this problem).  If you travel down at 
        30 m.p.h., and return at 60 m.p.h., what was your 
        average m.p.h. over the whole trip?

	Use the following table to help answer the problem:

2. Average Rate of Change vs. Instantaneous Rate of Change             
   Slope of a secant line vs. Slope of a tangent line
   Average velocity vs. Instantaneous velocity

     Slope is a rate of change of y with respect to x.

     Average rate of change takes place over an interval 
	while instantaneous rate of change takes place at an 
	instant.

     Average velocity occurs over a specific time interval 
	while instantaneous velocity is what is shown on your 
	speedometer.
 

If a secant line is drawn between two points, P and Q, 
on a curve y = f(x), and if Q is allowed to move along 
the curve toward P, then the secant line will rotate 
toward a limiting position (which is the tangent line 
at P).

3. Examples:

   In Mount Rainier National Park, here are some of the 
   hiking trails in the Longmire area:

   (1)	Trail of the Shadows: 
        .7 miles round-trip; 
        average hiking time is 30 minutes.

   (2)  Carter Falls/Madcap Falls:
        2.2 miles round-trip; 500' elevation gain; 
        average hiking time: 2 hours.
 
   (3)  Rampart Ridge:
        4.6 mile loop; 1,339' elevation gain; 
        average hiking time: 2.5 hours.

   (4)  Comet Falls and Van Trump Park:
        5.6 miles round-trip; 2,200' elevation gain; 
        average hiking time: 4 hours.

   (5)  Wonderland Trail to Indian Henry's:
        6.7 miles one way ; 2,400' elevation gain; 
        average hiking time: 4 hours.             

   (6)  Wonderland Trail to Paradise:
        6 miles one way; 2,700' elevation gain; 
        average hiking time: 3 hours.

   (7)  Eagle Peak Saddle:
        7.2 miles round trip; 2,955' elevation gain; 
        average hiking time: 5 hours.        

   Determine the AVERAGE RATE OF CHANGE OF THE DISTANCE WITH RESPECT
   TO TIME of some of the trails.

   Determine the AVERAGE RATE OF CHANGE OF THE ELEVATION WITH RESPECT  
   TO THE HORIZONTAL DISTANCE (you must solve for this 
   horizontal distance since the information given is the 
   elevation and the distance traveled).

4. Displacement, Average Velocity, and Instantaneous Velocity

    One of the most important concepts in calculus is the study of motion.
    Speed is how fast an object is going. 
    The speed and the direction make up the velocity (it is a vector quantity).
    Displacement is the change in position (the difference between the initial and final position).

5. Assignment: 
      Read pages 165 - 175 
      p. 176 (1, 9,  19, 20)

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