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`:

${\displaystyle s=c{t}^2}$

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?

Activity 1   This activity requires that you use a graphing tool. A graphing calculator will work fine, or select a computer tool with one of the buttons at the right.

1. Use your graphing tool to graph `s = 4.90 t^2`. What does the graph look like after you zoom in? What value seems reasonable for the instantaneous speed at `t = 2`?

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 = ftext[(]t text[)]` is said to be locally linear at a point `text[(]t_0,y_0 text[)]` 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`?

Checkpoint 1