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

4*3^2-7

1.7 - 1.4 - 0.3

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

WCF = 35.74 + 0.6215 T - 35.75 V^{0.16} + 0.4275 \; T \; V^{0.16}

Create variables for temperature T and wind speed V and then using this formula, calculate the WCF for

(a) T = 20 F and V = 0 mph

(b) T = 20 F and V = 10 mph

(c) T = 20 F and V = 20 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 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)