Game Development Reference
In-Depth Information
Here are a few examples, using functions we now know how to differen-
tiate:
d
dt sin 3x = 3 cos 3x,
d
dt sin(x 2 ) = 2xcos(x 2 ),
d
dt e cos x+3x = (− sinx + 3)e cos x+3x ,
Examples of the
chain rule
d
dt e sin 3x+sin(x 2 ) = (3 cos 3x + 2xcos(x 2 ))e sin 3x+sin(x 2 ) .
We're going to put calculus from a purely mathematical perspective
on the shelf for a while and return our focus to kinematics. (After all,
our purpose in discussing calculus was, like Ike Newton, to improve our
understanding of mechanics.) However, it won't be long before we will
return to calculus with the discussion of the integral and the fundamental
theorem of calculus.
11.5
Acceleration
We've made quite a fuss about the distinction between instantaneous ve-
locity and average velocity, and this distinction is important (and the fuss
is justified) when the velocity is changing continuously. In such situations,
we might be interested to know the rate at which the velocity is changing.
Luckily we have just learned about the derivative, whose raison d'etre is
to investigate rates of change. When we take the derivative of a velocity
function v(t) we get a new function describing how quickly the velocity is
increasing or decreasing at that instant. This instantaneous rate of change
is an important quantity in physics, and it goes by a familiar name: accel-
eration.
In ordinary conversation, the verb “accelerate” typically means “speed
up.” However, in physics, the word “acceleration” carries a more general
meaning and may refer to any change in velocity, not just an increase in
speed. In fact, a body can undergo an acceleration even when its speed
is constant! How can this be? Velocity is a vector value, which means it
has both magnitude and direction. If the direction of the velocity changes,
but the magnitude (its speed) remains the same, we say that the body is
experiencing an acceleration. Such terminology is not mere nitpicking with
words, the acceleration in this case is a very real sensation that would be
felt by, say, two people riding in the back seat of a swerving car who find
 
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