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7
Operators
This chapter is an extended example of an analogy. In the last chapter,
the analogy was often between higher- and lower-dimensional versions of a
problem. Here it is between operators and numbers.
7.1
Derivative operator
Here is a diernt

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6
Analogy
When the going gets tough, the tough lower their standards. It is the creed
of the sloppy, the lazy, and any who want results. This chapter introduces a
technique, reasoning by analogy, that embodies this maxim. It works well
with extreme-

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5
Taking out the
big part
Taking out the big part, the technique of this chapter, is a species of successive
approximation. First do the most important part of the analysis: the big part.
Then estimate changes relative to this big part. This hygieni

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45
4
Picture proofs
Do you ever walk through a proof, understand each step, yet not believe the
theorem, not say Yes, of course its true? The analytic, logical, sequential
approach often does not convince one as well as does a carefully crafted
picture

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3
Discretization
Discretization takes the fundamental idea of calculus
and pushes it to the opposite extreme from what cal
culus uses. Calculus was invented to analyze chang
ing processes such as orbits of planets or, as a one-
dimensional illustrat

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2
Extreme cases
The next item for your toolbox is the method of extreme cases. You can
use it to check results and even to guess them, as the following examples
illustrate.
2.1
Fencepost errors
Fencepost errors are the most common programming mistak

1
Lecture 1: Q
Dimensions
1. Why is an angle dimensionless?
2
2. Why is e3x dx dimensionless? What about et dt?
3. Why is it bad to write h feet?
4. How can we insert dimensions into something dimensionless?
5. What is the signicance of a dimensionless in

3
3
1
Dimensions
Dimensions, often called units, are familiar creatures in physics and engi
neering. They are also helpful in mathematics, as I hope to show you with
examples from dierentiation, integration, and dierential equations.
1.1
Free fall
Dimensi

Solution set 3
Warmups
Warmup problems are quick problems for you to check your understanding; dont turn them in.
1. Draw a picture to show that
(x + y)2 = x2 + 2xy + y2 .
Here is the gure:
xy
y2
x2
xy
x
y
The area of the square is (x + y)2 , and the area

Solution set 2
Warmups
Warmup problems are quick problems for you to check your understanding; dont turn them in.
1. The half-life 1/2 of a radioactive substance is the time until only one-half of the substance
remains. How is 1/2 related to the time unt

Solution set 1
Warmups
Warmup problems are quick problems for you to check your understanding; dont turn them in.
1. Use easy cases to nd (in terms of n) the nal term in the sum
S = 1 + 4 + 9 + .
n terms
The pattern is that each term is a perfect square