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Unique identifiers: Demographers study changes in human populations with respect to births, deaths, migration, education level, employment, and income. Climatologists study both the short-term and long-term effects of change in climatic conditions. How are the concepts of average rate of change and instantaneous rate of change used in these two professions to analyse data, solve problems, and make predictions?
This is an expression of rate of change. It is the change in position, in kilometres, with respect to the change in time, in hours. This value can represent an average rate of change or an instantaneous rate of change. However, this expression does not provide any information about your movement at different points during the hour. The rate you are travelling at a particular instant is called instantaneous rate of change.
This is the information that your speedometer provides. In this section, you will explore how the slope of a line can be used to calculate an average rate of change, and how you can use this knowledge to estimate instantaneous rate of change.
You will consider the slope of two types of lines: secants and tangents. The point at which the tangent touches the graph is called the tangent point. The line is said to be tangent to the function at that point. Notice that for more complex functions, a line that is tangent at one x-value may be a secant for an interval on the function.
Imagine that you are shopping for a vehicle. However, like most vehicles, this car loses value, or depreciates, as it ages.
The table below shows the value of the car over a year period. Graph the data in the table as accurately as you can using grid paper. Draw a smooth curve connecting the points.
Describe what the graph tells you about the rate at which the car is depreciating as it ages. Compare the slopes for these intervals and explain what this comparison tells you about the average rate of change in value of the car as it ages. R e f l e c t Determine the first differences for the data in the table. What do you notice about the first differences and average rate of change? Place a ruler along the graph of the function so that it forms a tangent to the point corresponding to year 0.
Move the ruler along the graph keeping it tangent to the curve. What do you think the tangent represents at each of these points? Use the graph to find the slope of the tangent you have drawn.