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Expressions often arise that are too complicated to understand or
analyze directly.  Sometimes these expressions can be replaced by
simpler approximations that are easier to work with.  The wellspring
of many such approximations is Taylor's Theorem:

\begin{theorem*}[Taylor's theorem]
Suppose that $f:\mathbb{R} \to \mathbb{R}$ is $n+1$ times
differentiable on the interval $[0,x]$.  Then
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\[
f(x) = f(0) + f'(0) x + \frac{f''(0) x^2}{2!}
            + \ldots + \frac{f^{(n)}(0) x^n}{n!} + \frac{f^{(n+1)}(z) x^{n+1}}{(n+1)!}
\]
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for some $z \in [0,x]$.
\end{theorem*}

A few terms of Taylor's series often provides an excellent
approximation of the function $f(x)$ for small values of $x$.  Lower
and upper bounds can be obtained by studying the final term.

\subsection*{Example}

Let's find an approximation for $f(x) = \ln(1 + x)$ around $x = 0$.
The derivative is $f'(x) = 1 / (1 + x)$.  Therefore, by Taylor's
theorem:
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\[
\ln(1 + x) = f(0) + x f'(z)
\]
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for some $z \in [0, x]$.  In this range, the maximum value of $f'(z)$
is 1 and the minimum value is $1 / (1 + x)$.  Thus, we have the
bounds:
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\[
\frac{x}{1+x} \leq \ln(1 + x) \leq x
\]
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A plot of all three functions is shown below.
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\includegraphics[height=2.5in]{myplot.eps}

The leading terms of the Taylor series for some important functions
are shown below:

\begin{align*}
e^x & = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \ldots \\
\ln(1 + x) & = x - \frac{x^2}{2} + \frac{x^3}{3}
                 - \frac{x^4}{4} + \frac{x^5}{5} - \ldots \\
\sqrt{1 + x} & = 1 + \frac{x}{2} - \frac{x^2}{8}
                   + \frac{x^3}{16} - \frac{5x^4}{128} + \ldots 
\end{align*}

\subsection*{Rule of 72}

There's an old investing rule that says you can double your money by
investint it for $n$ years at $72 / n$ percent interest.  For example,
you could double your money by investing it for 24 years at $72 / 24 =
3$ percent interest.  This rule isn't exactly right, but it is almost
mysteriously close.  For example, a dollar invested for 24 years at
3\% becomes \$2.03.

In general, a dollar invested for $n$ years at $72 / n$ percent
interest becomes
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\[
m = \left(1 +  \frac{72 / n}{100}\right)^n
\]
%
dollars.  This complicated expression doesn't look much like 2 right
now, but let's try an approximation.  One standard trick when the
variable of interest ($n$ in this case) appears in the exponent is to
analyze the \textit{logarithm} of the expression.
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\begin{align*}
\ln m & = \ln \left(1 +  \frac{72 / n}{100}\right)^n \\
      & = n \ln \left(1 + \frac{72}{100n}\right)
\end{align*}
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Here we're taking a logarithm of 1 plus something small ($72/100n$),
so we can get a reasonable approximation with a few terms of Taylor's
series:
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\begin{align*}
\ln m & \approx n \left(\frac{72}{100n} -
                        \frac{72^2}{2 \cdot 100^2 n^2}\right) \\
      & = \frac{72}{100} - \frac{72^2}{2 \cdot 100^2 n} \\
\end{align*}
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With the approximation done, we can exponentiate again:
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\[
m \approx e^{72/100 - 72^2/ (2 \cdot 100^2 n)}
\]
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This explains the Rule of 72: if you invest a dollar for $n$ years at
$72 / n$ percent interest, then the value of your dollar approaches
$e^{72/100} = 2.05443$ dollars from below as $n$ goes to infinity.  So
the Rule underestimates for small $n$, slightly overestimates for
large $n$, and is fairly accurate for realistic $n$.  A ``Rule for
71'' might actually be more slightly more accurate, but 72 has lots of
convenient divisors, while 71 is a prime.
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