# Exercise 2

1. Complete the 8-bit, unsigned addition given below. Show all the steps in both hexadecimal (left) and binary (right).

\begin{aligned} &\textsf{hex} \quad & &\textsf{binary}\\ &\texttt{AF} & &\square\square\square\square\square\square\square\square\\ +&\texttt{72} &+&\square\square\square\square\square\square\square\square\\ \hline &\!\!\square\square &&\square\square\square\square\square\square\square\square \end{aligned}

Is there an overflow?

2. Suppose we want to find an iterative algorithm for raising the number $$\mathsf{a}$$ to a rational power—i.e., for computing $$\mathsf{a^{p/q}}$$ with $$\mathsf{p}$$ and $$\mathsf{q}$$ integer. Convince yourself that the sequence defined by

\begin{aligned} \mathsf{x_0} & := \mathsf{1}\\ \mathsf{x_{n+1}} & := \mathsf{\Bigl(1-\frac{1}{q}\Bigr)x_n} + \mathsf{\frac{a^p}{q}x_n^{1-q}} \end{aligned}

has $$\mathsf{a^{p/q}}$$ as its limit. I suggest you proceed in this way: suppose that $$\mathsf{x_n}$$ is a good guess to the answer; construct a better guess $$\mathsf{x_{n+1}} = \mathsf{x_n} + \mathsf{\delta x} \approx \mathsf{a^{p/q}}$$; solve for $$\mathsf{\delta x}$$ to lowest order in a series expansion.

3. Find the value of $$\mathsf{b}$$ such that the functions

\begin{aligned} \mathsf{t(x)} &= \mathsf{x} - \mathsf{\frac{1}{3}x^3} + \mathsf{\frac{2}{15}x^5}\\ \textsf{and} \ \ \mathsf{r(x)} & = \mathsf{\frac{15x+x^3}{15+b x^2}} \end{aligned}

agree to sixth order. Plot the functions $$\mathsf{t(x)}$$, $$\mathsf{r(x)}$$, and $$\mathsf{tanh}\,\mathsf{x}$$ in the range $$\mathsf{0} \le \mathsf{x} \le \mathsf{2}$$. Also, plot the differences $$\mathsf{t(x)}-\textsf{tanh}\,\mathsf{x}$$ and $$\mathsf{r(x)} - \textsf{tanh}\,\mathsf{x}$$ with the log scale on the y-axis. How do they compare? What do $$\mathsf{t(x)}$$ and $$\mathsf{r(x)}$$ represent?