Limits and DerivativesIntuitive Idea of Derivatives
Let us slow the pebble down and look at the numbers the speed trap was using. Experiments confirm that a body dropped from a height falls metres in the first seconds (as long as air resistance is small). Table 12.1 lists the distance fallen at various times.
Table 12.1
| (s) | (m) |
|---|---|
| 0 | 0 |
| 1 | 4.9 |
| 1.5 | 11.025 |
| 1.8 | 15.876 |
| 1.9 | 17.689 |
| 1.95 | 18.63225 |
| 2 | 19.6 |
| 2.05 | 20.59225 |
| 2.1 | 21.609 |
| 2.2 | 23.716 |
| 2.5 | 30.625 |
| 3 | 44.1 |
| 4 | 78.4 |
We want the velocity at exactly seconds. All that the table can give directly is an
In the first two seconds the body falls from 0 m to 19.6 m, so its average velocity is
Between and it falls from 4.9 m to 19.6 m, so its average velocity is
The second interval is shorter and hugs more closely, so its average is a better stand-in for the velocity at . Shorter still should be better still.
Table 12.2 uses intervals that end at 2 seconds and start at , closer and closer to 2. Table 12.3 uses intervals that start at 2 seconds and end at , closer and closer to 2. Fill in the missing entries from Table 12.1.
Table 12.2 — from to 2 s
| (m/s) | |
|---|---|
| 0 | 9.8 |
| 1 | 14.7 |
| 1.5 | 17.15 |
| 1.8 | 18.62 |
| 1.9 | |
| 1.95 | 19.355 |
| 1.99 | 19.551 |
Table 12.3 — from 2 s to
| (m/s) | |
|---|---|
| 4 | 29.4 |
| 3 | 24.5 |
| 2.5 | 22.05 |
| 2.2 | 20.58 |
| 2.1 | |
| 2.05 | 19.845 |
| 2.01 | 19.649 |
Read both tables from top to bottom. In Table 12.2 the average velocities
So the velocity at must lie between the last entries, 19.551 m/s and 19.649 m/s: the same squeeze you saw in the speed trap. On physical grounds the two sequences must close in on one common value, assuming nothing dramatic happens in the last hundredth of a second. That value is the
Velocity is the
Note
A car's speedometer does exactly this. It cannot wait a minute to see how far the car goes, so it measures the distance covered in a tiny sliver of time and divides. Whenever you read a speedometer, you are reading a derivative.
From Chords to the Tangent
The same squeeze has a picture. Fig. 12.1 plots the distance against the time : the curve , with the point A at .
Pick a later time and its point B on the curve. Go across from A to C, then up from C to B. AC is the time interval , and CB is the distance fallen in that interval, . So the average velocity over the interval, , is exactly the slope of the chord AB.
Bring B closer to A, one step at a time, and watch the slope of the chord. The earlier chords stay behind, faded.
Fig. 12.1 — . The chords AB settle onto the tangent at A as B comes closer.
Label the successive positions , with intervals shrinking to zero. The ratios are the average velocities, so they approach the same limit, while the chords approach the tangent to the curve at A. So the instantaneous velocity at equals the
Two views of one idea: the derivative is a rate of change, and it is also the slope of a tangent. We will come back to both once we know exactly what a limit is.
Speed Round: Average or Instant?
Each body below falls (or rolls) a distance metres in seconds, for some number . Some questions ask for the average speed over a window, and some for the speed at one instant. Look for the pattern in the instant answers: it is the first derivative rule of this chapter in disguise.
Get five right to move on.