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====== Math 445 Lab #1 ======

Most of these problems are taken from Attaway chapter 1, both 2nd and 3rd editions. Remember, Matlab's //help// function is your friend.

**Problem 1:** Evaluate these Matlab expressions in your head and write down the answer. 
Then type them into Matlab and see how Matlab evaluates them. If you made a mistake, figure out what it was. 

25/4*4

3+4^2

4\12 + 4

3^2

(5-2)*3


**Problem 2:** Translate these mathematical expressions into Matlab expressions, and then evaluate them. 

$e^{3/4}$

$\sqrt[5]{7}$

$e^{\pi i}$

 cube root of 19

 3 to the 1.2

 tangent of $\pi$


**Problem 3:** Wind chill factor: The WCF supposedly conveys how cold it feels with a given air 
temperature T (degrees Farenheit) and wind speed V (miles per hour). A formula
for WCF is

<latex>
  WCF = 35.7 + 0.6 T - 35.7 V^{0.16} + 0.43 \; T \; V^{0.16}
</latex>

Create variables for temperature T and wind speed V and then using this formula,
calculate the WCF for (a) T = 45 F and V = 10 mph and (b) T = 45 F and V = 0 mph.

**Problem 4:** The geometric mean g of n numbers $x_1, x_2, \ldots, x_n$ is given by

\begin{eqnarray*}
  g = \sqrt[n]{x_1 x_2 \ldots x_n}
\end{eqnarray*}

This is useful, for example, in finding the average rate of return on an investment with varying yearly return. 

**(a)** If an investment returns 15% its first year, 5% its second, and 10% its third, the average rate of return is 
 
\begin{eqnarray*}
  \sqrt[3]{1.15 \cdot 1.05 \cdot 1.10}
\end{eqnarray*}

Compute the average rate of return, expressed as a percent. 

**(b)**Which is better, a steady 5% per year return on investment, or alternating between 0% and 10% year by year?
  
**Problem 5:** The astoundingly brilliant but short-lived mathematician [[http://en.wikipedia.org/wiki/Srinivasa_Ramanujan |  Srinivasa Ramanujan]] devised the following very powerful formula for for $1/\pi$

\begin{eqnarray*}
\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)!\, (1103 + 26390\,k)}{(k!)^4 \, 396^{4k}}
\end{eqnarray*}

You can get an approximation of $\pi$ using only arithmetic operations by evaluating and summing a finite number of terms of this series. What is the numerical approximation of $\pi$ using just the first term ($k=0$)? Using the first and second ($k=0$ and $k=1$)? How many digits of accuracy does each of these approximation have? Be sure to use ''format long''. 

(adapted from a problem in //Introduction to Matlab Programming// by Siauw and Bayen)


**Problem 6:** Translate the following Matlab logical expressions into English.

  x < y
  x < y || y < z
  x <= y && y <= z


**Problem 7:** Translate the following mathematical expressions into Matlab and evaluate for $x=3, y=4, z=5$.

$x < y < z$

$x < y$ and $x < z$ 

$y < z$ or  $x == z$ 


**Problem 8:** Evaluate the following Matlab expressions and explain the results.

  3 == 4
  ~(5 >= 2)
  2 < 3 < 5
  2 > 3 > 5
  2 > 3 < 5
  xor(5 < 6, 8 > 4)
  (3 == 2) + 1
  
**Problem 9:** If P is a logical expression, the law of noncontradiction states that P AND (NOT P) is always false. Use Matlab to verify this for both P false and P true.

**Problem 10:** Let P and Q be logical expressions. De Morgan's rule states that NOT (P OR Q) = (NOT P) AND (NOT Q) and also that NOT (P AND Q) = (NOT P) OR (NOT Q). Demonstrate with Matlab that both these rules hold for all possible combinations of P and Q. 

**Problem 11:** Construct an equivalent logical expression for P OR Q using only AND and NOT. Translate that into Matlab and then test for all possible values of P and Q. Write your demonstration in the following fashion, in order to make it easy to follow

<code matlab>
P=0; Q=0; (my expression) == (P || Q)
P=1; Q=0; (my expression) == (P || Q)
</code>
etc.

**Problem 12:** Construct an equivalent logical expression for P AND Q using only OR and NOT. Translate that into Matlab and then test for all possible values of P and Q. Folow the same kind of pattern as in problem 11.
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