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Derivatives and integrals of trivial expressions containing polynomials
and trig functions.
Of course, we are aware that a large number of readers may already have
this knowledge. Take a moment to put yourself into one of the following
categories:
1. I know absolutely nothing about derivatives or integrals.
2. I know the basic idea of derivatives and integrals, but probably couldn't
solve any freshman calculus problems with a pencil and paper.
3. I have studied some calculus.
Level 2 knowledge of calculus is su cient for this topic, and our goal is
to move everybody who is currently in category 1 into category 2. If you're
in category 3, our calculus discussions will be a (hopefully entertaining)
review. We have no delusions that we can move anyone who is not already
there into category 3.
11.1.2 Some Helpful Lies about Our Universe
The universe is commonly thought to be discrete in both space and time.
Not only is matter broken up into discrete chunks called atoms, but there
is evidence that the very fabric of space and time is broken up into discrete
pieces also. Now, there is a difference of opinion as to whether it's really
that way or just appears that way because the only way we can interact
with space is to throw particles at it, but it's our opinion that if it looks
like a duck, walks like a duck, quacks like a duck, has webbed feet and a
beak, then it's a good working hypothesis that it tastes good when put into
eggrolls with a little dark sauce.
For a long time, the mere thought that the universe might not be contin-
uous had not even considered the slightest possibility of crossing anybody's
mind, until the ancient Greeks got a harebrained and totally unjustified
idea that things might be made up of atoms. The fact that this later
turned out to be true is regarded by many as being good luck rather then
good judgment. Honestly, who would have thought it? After all, everyday
objects, such as the desk on which one of the authors is currently resting his
wrists as he types this sentence, give every appearance of having smooth,
continuous surfaces. But who cares? Thinking of the desk as having a
smooth, continuous surface is a harmless but useful delusion that lets the
author rest his wrists comfortably without worrying about atomic bond
energy and quantum uncertainty theory at all.
Not only is this trick of thinking of the world as continuous a handy
psychological rationalization, it's also good mathematics. It turns out that
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