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====== Differences ====== This shows you the differences between two versions of the page.

--- gibson:teaching:spring-2015:math445:lab1 [2015/01/19 11:31]
gibson created
+++ gibson:teaching:spring-2015:math445:lab1 [2015/01/21 12:52] (current)
gibson
@@ Line 1: / Line 1: @@
 ====== Math 445 Lab #1 ======
-These problems are taken from Attaway chapter 1, both 2nd and 3rd editions. Remember, Matlab's //help// function is your friend.
+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 expressions in your head and write down the answer.
+**Problem 1:** Evaluate these Matlab expressions in your head and write down the answer.
-Then evaluate them with Matlab. If you made a mistake, figure out what it was.
+Then type them into Matlab and see how Matlab evaluates them. If you made a mistake, figure out what it was.
 /4*4
@@ Line 17: / Line 17: @@
-**Problem 2:** Translate these mthematical expressions into Matlab expressions, and then evaluate them.
+**Problem 2:** Translate these mathematical expressions into Matlab expressions, and then evaluate them.
 $e^{3/4}$
@@ Line 32: / Line 32: @@
-**Problem 3:** Wind chill factor: The WCF conveys how cold it feels with a given air
+**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
@@ Line 43: / Line 43: @@
 calculate the WCF for (a) T = 45 F and V = 10 mph and (b) T = 45 F and V = 0 mph.
-**Problem 4:** Translate the following Matlab logical expressions into English.
+**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
@@ Line 50: / Line 79: @@
-**Problem 5:** Translate the following mathematical expressions into Matlab and evaluate for $x=3, y=4, z=5$.
+**Problem 7:** Translate the following mathematical expressions into Matlab and evaluate for $x=3, y=4, z=5$.
-  $x < y < z$
+$x < y < z$
-  $x < y$ and $x < z$
-  $y < z$ or  $x == z$
+$x < y$ and $x < z$
-**Problem 6:** Evaluate the following Matlab expressions and explain the results.
+$y < z$ or  $x == z$
+**Problem 8:** Evaluate the following Matlab expressions and explain the results.
 == 4
@@ Line 67: / Line 98: @@
   (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 7:** The geometric mean g of n numbers $x_1, x_2, \ldots, x_n$ is given by
+**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.
-\begin{eqnarray*}
+**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
-  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. 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.
-**Problem 8:** 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)!\, (1101 + 26390\,k)}{(k!)^4 \, 296^{4k}}
-\end{eqnarray*}
-You can get an approximation of $\pi$ using only arithmetic operations by evaluating and summing a finite number of terms of
+<code matlab>
-this series. What is the numerical approximation of $\pi$ using just the first term ($k=0$)? Using the first and second
+P=0; Q=0; (my expression) == (P || Q)
-($k=0$ and $k=1$)? How many digits of accuracy does each of these approximation have?
+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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