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1.5 Slope of Secants and Average Rate of Change
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
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