In this lab you will
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.
Your Newton search algorithm should
1. Use a
for loop to perform the Newton-search iteration . Take up to ten Newton-search iterations.
2. Use an
if statement inside the
for loop to test if either or for some specified tolerance .
3. If either of those conditions is true, use a
break statement to terminate the iteration and return from the function. For our purposes is a decent choice.
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
(a) Find an for which .
(b) Find the cube root of 72 by devising and solving an equation of the form whose solution is . Is there a simpler way to calculate in Matlab? Do that, and compare your answers.
(c) Find an for which .
Hint: find good initial guesses for the Newton search by plotting each function and roughly estimating an position at which 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 at distance below the surface and time after the beginning of a cold spell is given approximately by
If is in meters and is in seconds, the thermal conductivity of soil is . Let and , and recall that water freezes at . 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.