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

$WCF = 35.7 + 0.6 T - 35.7 V^{0.16} + 0.43 \; T \; V^{0.16}$

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 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

P=0; Q=0; (my expression) == (P || Q)
P=1; Q=0; (my expression) == (P || Q)

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.