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**Problem 1:** Write a function ''x = newtonsearch(f, xguess)'' that finds the solution $x$ of the equation $f(x) == 0$ for an input function ''f'' and an initial guess ''xguess'' using the Newton search algorithm. 

  - Use a ''for'' loop to perform the Newton-search iteration. Take up to ten Newton steps.
  - Use a ''if'' statement inside the ''for'' loop to test if either $|f(x)| < \epsilon$ or $|dx| < \epsilon$. If so, use a ''break'' statement to terminate the iteration and return from the function. For our purposes ''1e-07'' is a decent choice for the value of tolerance $\epsilon$. 
      


**Problem 2:** 

Test your Newton-search algorithm by solving the following problems. Check your answers by 
plugging the answer ''x'' back into ''f'' and verifying that ''f(x)'' is approximately zero.

**(a)** Find an ''x'' for which 

\begin{eqnarray*}
x^3 - 7x - 13 = 0
\end{eqnarray*} 

**(b)** Find the cube root of 54. (Hint: devise an equation whose answer is $x = \sqrt[3]{72}$.)

**%%(c)%%** Find an ''x'' for which $\sqrt{4-x^2} = x \tan x$.

Hint: find good initial guesses for the Newton search by plotting each function and roughly 
estimating an $x$ position at which $f(x)$ is zero. 


**Problem 3:** Use your Newton-search algorithm to solve the following problem:

Utility companies must avoid freezing water mains in cold weather. If we assume uniform soil conditions, 
the temperature $T(x,t)$ at distance $x$ below the surface and time $t$ after the beginning of a cold 
spell is given approximately by

\begin{eqnarray*}
\frac{T(x,t) - T_s}{T_i-T_s} = \erf\left(\frac{x}{\sqrt{2 \alpha t}}
\end{eqnarray*}

where

  * $T_s$ is the constant surface temperature during the cold spell.
  * $T_i$ is the initial soil temperature before the cold spell started. 
  * $\alpha$ is the thermal conductivity of the soil.
  * $\erf$ is the //error function//, which can be computed with the built-in Matlab function ''erf''.

If $x$ is in meters and $t$ is in seconds, the $\alpha = 0.138 \cdot 10^{-6} m^2/s$. Let $T_i=20 C$ and
$T_S = -15 C$ and recall that water freezes at $T = 0 C$. 

Use your Newton-search algorithm to determine how deep a water main must be buried so that it will not 
freeze until at least 60 days' exposure to these conditions.
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