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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. 

2/3

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:** Are any of your answers for problems 1 and 2 surprising? Which, and why? 

**Problem 4:** 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 = 35.74 + 0.6215 T - 35.75 V^{0.16} + 0.4275 \; 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 

**(b)** T = 45 F and V = 0 mph.

**Problem 5:** 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 for the investor, a steady 5% per year return on investment, or alternating between 0% and 10% year by year?
  
**Problem 6:** 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)
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