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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: 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. How could you make this calculation more accurate? Make a conjecture about the slope of the secant between years 1 and 2 in relation to the slope of the tangent. Does your calculation support your conjecture?
R e f l e c t Use the results of this investigation to summarize the relationship between slope, secants, tangents, average rate of change, and instantaneous rate of change. The table shows the volume of helium in the balloon at 3-s intervals for 30 s. What are the dependent and independent variables for this problem?
In what units is the rate of change expressed? What does the slope of the secant represent? Draw a tangent at the point on the graph corresponding to 21 s and calculate the slope of this line. What does this graph illustrate?
What does the slope of the tangent represent? What do you notice? What information would you need to calculate a secant slope that is even closer to the slope of the tangent? In this problem, volume is dependent on time, so V is the dependent variable and t is the independent variable. For the rate of change, V is V expressed with respect to t, or. The slope of the tangent represents the instantaneous rate of change of the volume at the tangent point.
To find the instantaneous rate of change of the volume at 21 s, sketch an approximation of the tangent passing though the point P 21, Choose a second point on the line, Q Notice that as the interval becomes smaller, the slope of the secant gets closer to the slope of the tangent. You could calculate a secant slope that was closer to the slope of the tangent if you had data for smaller intervals.
It corresponds to the slope of the secant connecting the two endpoints of the interval. Instantaneous rate of change refers to the rate of change at a specific point.
It corresponds to the slope of the tangent passing through a single point, or tangent point, on the graph of a function.