Chapter 2
Models of Growth: Rates of Change
2.2 The Derivative: Instantaneous Rate of Change
2.2.1 Zooming In: Local Linearity
In this section we move from average rates of change to instantaneous rates of change. To begin, we look at the instantaneous rates of change of an object falling near the surface of the earth without significant resistance by the air. This investigation will lead to the first major calculus concept: the derivative.
We resume our study of the falling body problem, for which elementary physics provides a theoretical model (i.e., a formula) for distance fallen, \(s\), as a function of time \(t\):
where \(c\) is a constant that depends on the gravitational force and on the units of measurement. In this investigation, we will measure time in seconds and distance in meters, so \(c\) is approximately \(4.90\) meters per second per second.
Note 1 – Units
The question we address is this: How can we use the formula for distance as a function of time to determine the instantaneous speed of the falling object at any instant of its fall?
The next Activity and several to follow require the use of a graphing tool. You may use your own graphing calculator or click on the Graph button below to use ours. Change the function definition to the one you want to graph. You can "zoom in" on a small portion of the graph by changing the window determined by the ranges for \(t\) and \(y\).
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Use your graphing tool to graph \(s = 4.90 t^2\). What does the graph look like after you zoom in near \(t = 2\)? What value seems reasonable for the instantaneous speed at \(t = 2\)?
Repeat for \(t = 5\).
What you saw in Activity 1 is the emerging straightness of the curves. This straightness you see, as you look at shorter and shorter segments of the curve, is a property of most well-behaved curves.
| Definition The graph of a function \(y = f(t) \) is said to be locally linear at a point \((t_0,y_0) \) on the graph if, in the locality of the point, the curve looks like a straight line. In this case, we also say that the function \(f\) is locally linear at \(t_0\). |
Now there is no problem calculating the instantaneous speed at \(t = 2\). What does the graph look like after you zoom in? What value seems reasonable for the instantaneous speed at \(t = 2\)?
