Game Development Reference
In-Depth Information
loss” that occurs when we take the derivative. If we know how fast we were
going, we can always figure out how far we traveled. However, we cannot
know where we ended up unless we know where we started. This extra
term x
0
is the “starting point” that the derivative throws out, because any
constant value has a derivative of zero. For this reason, it's not entirely
accurate to refer to “the” antiderivative of v(t), since there is not a unique
function whose derivative is v(t), but infinitely many. All the different
antiderivatives are really just copies of one another, shifted on the graph
vertically according to their particular value of x
0
.
We've stated in a general way that there is some relationship between
the (definite) integral and the antiderivative. So we know that in a certain
sense the operations of integration and differentiation are inverse opera-
tions. The theorem of calculus that summarizes these relationships pre-
cisely goes by an important-sounding name: the fundamental theorem of
calculus. The theorem actually consists of two parts. (Sources don't always
list them in the same order.)
The first part shows how a definite integral may be computed by using
an antiderivative.
Fundamental Theorem of Calculus, Part 1
Let f(t) = F
′
(t). (In other words, F(t) is any antiderivative of f(t).) Then
the definite integral
b
a
f(x) dx can be computed as
b
f(t) dx = F(b) − F(a).
(11.26)
a
Equation (11.26) can seem a bit mystifying in abstract terms, but when
we replace the generic F(t) and f(t) with notation specific to position and
velocity, the first part of the fundamental theorem of calculus seems to state
the obvious:
b
v(t) dt = x(b) − x(a).
a
This says that the cumulative effect of velocity from time a to time b (the
net displacement during that interval), is equal to the difference in the
position at time b and the position at time a.
Notice how any antiderivative will work—it doesn't matter which one.
That's because the constant offset x
0
inside of x(t) cancels itself out when
we do the subtraction x(b)−x(a). To see this, consider the metaphor of the
Search WWH ::
Custom Search