===== Math 445 HW4 =====

This homework is meant as review for the second-chance exam to be held Nov 26.
It is due Thursday Nov 14th in lecture. 


Review topics

  * **fprintf:** formatted printing function
  * **plotting:** ''plot, semilogy, semilogx, loglog, contour, mesh''
  * **loops:** ''for'' and ''while''
  * **anonymous functions:** defining and using
  * **solving equations:** with ''fzero'' or ''newtonsearch''

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**Problem 1:** Write a Matlab function named ''farenheit2celsius'' that converts
degrees Farenheit to degrees Celsius, according to the formula $C = 5/9 (F-32)$.
Write the function in a file ''farenheit2celsius.m''.

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**Problem 2:** Write a for-loop that prints a Farenheit to Celsius conversion table
for values of //F// between -10 and 120 degrees, in increments of 10 degrees //F//. 
The output should look something like this:

<code>
   0  F == -17.8 C
   10 F == -12.2 C
</code>

etc. Hint: use ''fprintf'' with a formatting codes ''%3d'' and ''%6.1f'' to get the 
numbers to line up nicely. That's three spaces for the integer //F// numbers, and 
six spaces for the floating-point //C// numbers, with 1 digit after the decimal point.

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**Problem 3:** Make a plot of C versus F over the range %%-50 <= F <= 212%%. Put symbols 
on the line where water freezes and boils: a circle for freezing and a square for 
boiling. Label the axes and give the graph an appropriate title. 

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**Problem 4:** Use Matlab's anonymous function facility to define a 
''farenheit2celsius'' function at the Matlab prompt, instead of writing it in a file 
as in problem 1.

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**Problem 5:** Use matlab's ''fsolve'' function or your ''newtonsearch'' function 
and the ''farenheit2celsius'' function from problem 4 to find the temperature 
that has the same numerical value in //F// and //C//. Verify your answer by solving 
the equation with algebra.

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**Problem 6:** Verify your answer from problem 5 by evaluating your ''farenheit2celsius'' 
function at the value found as an answer to problem 5. 

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**Problem 7:** Write a script that produces a multiplication table like this 

<code>
 1  2  3  4
 2  4  6  8
 3  6  9 12
</code>

only for the numbers 1 through 10 in both the rows and columns. Using two nested for-loops and the 
''fprintf'' function with a formatting code like in problem 2 to get uniform spacing.

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**Problem 8:** Find a solution to the system of equations 

<latex>
\begin{align*}
x_1^2 - x_2   &= -2 \\
x_1 \sin(x_2) &=  4
\end{align*}
</latex>

using Matlab's ''fsolve'' function or your ''newtonsearch''.

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**Problem 9:** Deduce the functional relationship ''y(x)'' for the data in this plot.

{{:gibson:teaching:fall-2013:math445:prob9.png?nolink&300|}}

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**Problem 10:** Assume you have a file ''data.asc'' containing the data for the above graph,
with the //x// values in the first column and the //y// values in the second column. Write the
Matlab code that would load the data and produce the above plot.