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).
