Game Development Reference
In-Depth Information
the earlier applications of calculus were integrals, even though most calcu-
lus courses cover the “easier” derivative before the “harder” integral.
We first follow in the steps of Newton and start with the physical ex-
ample of velocity, which we feel is the best example for obtaining intuition
about how the derivative works. Afterwards, we consider several other ex-
amples where the derivative can be used, moving from the physical to the
more abstract.
11.4.1
Limit Arguments and the Definition of the Derivative
Back to the question at hand: how do we measure instantaneous velocity?
First, let's observe one particular situation for which it's easy: if an object
moves with constant velocity over an interval, then the velocity is the same
at every instant in the interval. That's the very definition of constant
velocity. In this case, the average velocity over the interval must be the
same as the instantaneous velocity for any point within that interval. In
a graph such as
Figure 11.1,
it's easy to tell when the object is moving
at constant velocity because the graph is a straight line. In fact, almost
all of Figure 11.1 is made up of straight line segments,
8
so determining
instantaneous velocity is as easy as picking any two points on a straight-
line interval (the endpoints of the interval seem like a good choice, but
any two points will do) and determining the average velocity between those
endpoints.
But consider the interval from t
1
to t
2
, during which the hare's over-
confidence causes him to gradually decelerate. On this interval, the graph
of the hare's position is a curve, which means the slope of the line, and
thus the velocity of the hare, is changing continuously. In this situation,
measuring instantaneous velocity requires a bit more finesse.
For concreteness in this example, let's assign some particular numbers.
To keep those numbers round (and also to stick with the racing theme),
please allow the whimsical choice to measure time in minutes and distance
in furlongs.
9
We will assign t
1
= 1 min and t
2
= 3 min, so the total
duration is 2 minutes. Let's say that during this interval, the hare travels
from x(1) = 4 fur to x(3) = 8 fur.
10
For purposes of illustration, we will set
our sights on the answer to the question: what is the hare's instantaneous
velocity at t = 2.5 min? This is all depicted in
Figure 11.4.
8
Mostly because that's the easiest thing for lazy authors to create in Photoshop.
9
The speed chosen for the hare bears some semblance to reality, but for pedagogical
reasons and to make Figure 11.1 fit nicely on a page, the speed of the tortoise is totally
fudged. Sticklers should remind themselves that this is a story with a talking bunny
rabbit and turtle. Oh, and a furlong is 1/8 of a mile.
10
The abbreviation “fur” means “furlongs” and has nothing to do with the fur on the
bunny.
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