Game Development Reference
In-Depth Information
•
When the position function changes slope at a “corner,” the velocity
graph exhibits a discontinuity. In fact, the derivative at such points
does not exist, and there is no way to define the instantaneous veloc-
ity at those points of discontinuity. Fortunately, such situations are
nonphysical—in the real world, it is impossible for an object to change
its velocity instantaneously. Changes to velocity always occur via an
acceleration over a (potentially brief, but finite) amount of time.
21
Later we show that such rapid accelerations over short durations are
often approximated by using impulses.
•
There are sections on the velocity graph that look identical to each
other even though the corresponding intervals on the position graph
are different from one another. This is because the derivative mea-
sures only the rate of change of a variable. The absolute value of the
function does not matter. If we add a constant to a function, which
produces a vertical shift in the graph of that function, the derivative
will not be affected. We have more to say on this when we talk about
the relationship between the derivative and integral.
At this point, we should acknowledge a few ways in which our explana-
tion of the derivative differs from most calculus textbooks. Our approach
has been to focus on one specific example, that of instantaneous velocity.
This has led to some cosmetic differences, such as notation. But there
were also many finer points that we are glossing over. For example, we
have not bothered defining continuous functions, or given rigorous defini-
tions for when the derivative is defined and when it is not defined. We
have discussed the idea behind what a limit is, but have not provided a
formal definition or considered limits when approached from the left and
right, and the criteria for the existence of a well-defined limit. We feel that
leading off with the best intuitive example is always the optimum way to
teach something, even if it means “lying” to the reader for a short while.
If we were writing a calculus textbook, at this point we would back up and
correct some of our lies, reviewing the finer points and giving more precise
definitions.
However, since this is not a calculus textbook, we will only warn you
that what we said above is the big picture, but isn't su
cient to handle
many edge cases when functions do weird things like go off into infinity or
exhibit “jumps” or “gaps.” Fortunately, such edge cases just don't happen
too often for functions that model physical phenomena, and so these details
won't become an issue for us in the context of physics.
21
For all you math sticklers who object to the vertical lines at the discontinuities where
the derivative is mathematically undefined, the engineer's justification in this and other
similar situations is that the mathematical formula is only a model for what is actually
a physical situation.
Search WWH ::
Custom Search