1. Let $f$ be the function such that
$$
f(x)=\left\{\begin{array}{ll}
2 x & \text { if } x \leq \frac{1}{2} \\
2-2 x & \text { if } x>\frac{1}{2}
\end{array}\right.
$$

What is the total length of the graph of $\underbrace{f(f(\ldots f}_{2012 f^{\prime} s}(x) \ldots))$ from $x=0$ to $x=1$ ?
Answer: $\sqrt{\sqrt{4^{2012}+1}}$ When there are $n$ copies of $f$, the graph consists of $2^{n}$ segments, each of which goes $1 / 2^{n}$ units to the right, and alternately 1 unit up or down. So, the length is
$$
2^{n} \sqrt{1+\frac{1}{2^{2 n}}}=\sqrt{4^{n}+1}
$$

Taking $n=2012$, the answer is
$$
\sqrt{4^{2012}+1}
$$