Professor Dave again, let’s continue with differentiation.
We just introduced the concept of differentiation by looking at tangent lines, so its definition
may be starting to crystallize in your mind, but let’s examine this concept in one more
context just to be thorough.
When we introduced calculus, we talked about how Newton pondered a falling body, perhaps
like his mythical apple.
He knew that if an object was dropped some short distance from rest, meaning with an
initial velocity of zero, that velocity would necessarily increase in the negative direction,
every single instant, until hitting the ground.
He then wondered how one could calculate the velocity of the apple at any particular instant
during the fall.
This is called the instantaneous velocity.
Galileo had already discovered some years prior that the distance travelled by a falling
object, represented by S of T, would be equal to 4.9 t^2, with t in seconds giving a distance
in meters, so Newton could easily use this law to calculate how far something would fall
over a given interval of time.
But this was not the problem at hand.
Newton wanted to know something not about an interval of time, but rather about an instant
in time, like perhaps, the velocity of the apple precisely five seconds after letting it fall.
Well just like we did in the previous tutorial with the secant and tangent lines, let’s
make an approximation.
We can calculate average velocity by taking some distance travelled and dividing by the
time elapsed, as that will be meters travelled over seconds travelling, or meters per second.
And using Galileo’s equation, we can express distance travelled as the difference between
two points in this function.
Say we use the distance travelled after five seconds, or S of 5, and the distance travelled
after six seconds, or S of 6.
If we subtract the former from the latter, we are necessarily describing the distance
travelled during the period from 5 to 6 seconds after being dropped.
That will also mean that the time elapsed is one second, so we just evaluate the function
for these two inputs and subtract, divide by one second, and we get 53.9 meters per second.
So that is the average speed of the apple during that one second interval.
But remember, we don’t want an average speed.
We want the precise speed at a precise instant.
So what do we do?
Well, just like the way we nudged the second point in the line closer to the first until
we got the tangent line for the curve, let’s nudge the end of this time interval closer
and closer to five seconds.
We can use 5 and 5.1 to get the average velocity over this tenth of a second, and we get 49.49
meters per second.
Then we can use 5 and 5.01 to get the average velocity over this hundredth of a second,
and continue to get even narrower from there.
If we were to continue in this manner, we would see the average velocity get closer
and closer to 49, so the instantaneous velocity of the apple precisely five seconds after
being dropped must be 49 meters per second.
This approach feels pretty similar to what we were doing with the tangent line.
In fact, it’s not just similar, it’s exactly the same.
In both cases, we are performing differentiation.
If we draw the graph of 4.9 t^2, all we were doing in shortening that time interval was
nudging the second point in this line closer and closer to the first point sitting at five
seconds, so that we can get that tangent line, and it is the slope of the tangent line that
gave us our instantaneous velocity.
This is equivalent to saying that the rate of change on the distance curve is distance
over time, or velocity.
This fact is no coincidence, and we will come back to this concept later.
For now, let’s continue examining introductory calculus concepts.