Quote of the Day: "All the mathematical sciences are founded on the relations between physical laws and laws of numbers." -- James Clerk Maxwell 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. 1. Collect Homework. 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. Assignment: Read pages 169 - 175 p. 175 (2, 9, 17, 18) |