1.5 Slope of Secants and Average Rate of Change.pdf

Slopes of Secants and Average Rate of Change 1.5 Change occurs in many aspects of everyday life. A person’s height and mass change from birth to adulthood. The distance that a car travels in a period of time changes according to its speed. Temperature fl uctuations occur daily and with each season. The value of a house or antique may increase, or appreciate, over a period of time. The cost of a train ticket varies with the distance travelled. Some things change over a long period of time, while others change in an instant. Being able to understand and predict the rate at which a change occurs can provide useful information. A rate of change is a measure of the change in one quantity (the dependent variable) with respect to a change in another quantity (the independent variable). There are two types of rates of change, average and instantaneous. An average rate of change is a change that takes place over an interval, while an instantaneous rate of change is a change that takes place in an instant. Instantaneous rates of change will be examined more closely in the next section of this chapter. Consider the following scenario. A car leaves Ottawa at 12 p.m. and arrives in Toronto at 4 p.m. after travelling a distance of 400 km. The average rate of change of distance with respect to time—the average velocity—is determined as shown: Average velocity  change in distance ____ change in time  Δd _ Δt  400 _ 4  100 Therefore, the average velocity of the car is 100 km/h. 1.5 Slopes of Secants and Average Rate of Change ? MHR 53 Investigate How can you connect average rate of change and slope? 1. Seismic activity at a certain point on the ocean fl oor creates a wave that spreads in a circular pattern over the calm surface of the ocean. The table shows the radius of the circular pattern during the fi rst 10 s as the wave moves outward. a) Identify the independent variable and the dependent variable. Justify your choice. b) Dete