PROBLEMS

$3\mathrm{.}13$ The "divide and average" method, an old$-$time metho for approximating the square root of any positive number a can be formulated as

$x=\frac{x+\frac{a}{x}}{2}$

Write a well$-$structured M$-$file function based on the while...break loop structure to implement this algorithm Use proper indentation so that the structure is clear. At each step, estimate the error in your approximation as

$=|\frac{x_{\text{n}\text{e}\text{w}}-x_{\text{o}\text{l}\text{d}}}{x_{\text{n}\text{e}\text{w}}}|$

Repeat the loop until $$ is less than or equal to a specified value. Design your program so that it returns both the result and the error. Make sure that it can evaluate the square root of numbers that are equal to and less than zero. For the latter case, display the result as an imaginary number. For example, the square root of $-4$ would return $2i.$ Test your program by evaluating $a=0,2,10$ and $-4$ for $=1{10}^{-4}.3\mathrm{.}14$ Piecewise functions are sometimes useful when the relationship between a dependent and an independent variable cannot be adequately represented by a single equation. For example, the velocity of a rocket might be described by

$v(t)=\{\begin{array}{cc}10t^2-5t,& 0t8\\ 624-3t,& 8t16\\ 36t+12(t-16{)}^2,& 16t26\\ 2136e^{-0\mathrm{.}1(t-26)},& t>26\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

Develop an M$-$file function to compute $v$ as a function of $t.$ Then, develop a script that uses this function to generate a