CS100M Spring 2006
Project 2
Due Thursday 2/15 at 6pm

Submit your files on-line in CMS before the project deadline. See the CMS link on the course webpage for instructions on using CMS. Both correctness and good programming style contribute to your project score.

You must work either on your own or with one partner. You may discuss background issues and general solution strategies with others, but the project you submit must be the work of just you (and your partner). If you work with a partner, you and your partner must register as a group in CMS and submit your work as a group.

Objectives

Completing this project will help you learn about developing algorithms and about loops and selection. Compared to Project 1, this project is more open-ended. Any assumptions you make should be stated using comments.

Do not use arrays (vectors) for this project.

1. Rolling Dice

For this problem, you need to write two programs (Matlab scripts): oneTurn.m and averageRolls.m.  

Craps is a popular casino game based on rolling a pair of dice.  Here, we're not interested in the gambling aspects of the game; instead, we want to estimate the expected number of rolls for a single turn.  Here are the rules for how many rolls take place.

• If the first roll is 2, 3, 7, 11, or 12 then no more rolls take place.
• If the first roll is anything else then this value is the point. Rolls continue until the point is rolled again or until a 7 is rolled.

Write a program (a Matlab script) called oneTurn.m that simulates a single turn.  As it runs, it should print the value of each roll. After all the rolls are done it should print the number of rolls that occurred.  Here's an example of one way to arrange the program's output.

>> oneTurn
8 4 4 5 7 -> 5 rolls

Write a separate program called averageRolls.m that uses the first program over several turns to determine the average number of rolls per turn.  Your program should ask the user for the number of turns, run oneTurn.m the specified number of turns, and then report the average number of rolls per turn.

• You'll want to use the built-in function rand to simulate the dice rolls.  Recall that rand(1) creates a single random number in the range 0 to 1. Note that the 1 in the parentheses indicates that one number is to be generated --- it has nothing to do with the range.  To get the value for a single die, use ceil(6*rand(1)); this makes each of the values 1 through 6 equally likely.
   
• Submit two files to CMS: oneTurn.m and averageRolls.m.

2. Approximation via Taylor Series

The Taylor series for e^x is given by the sum 1 + x/1 + x²/2 + x³/6 + ... + x^k/k! + ... where k! is 1*2*...*k.  We can use this series to approximate e^p for various values of p.

For this problem, write a program (a Matlab script) called approxE.m to see how many terms are necessary to get a good approximation.  Your program should request two inputs from the user: p (the exponent) and epsilon (the desired accuracy).  Your program should then determine and report the number of terms of the Taylor series that are necessary to get within epsilon of the correct value for e^p.  We'll use Matlab's built-in function exp as our definition of accurate; in other words, we assume that exp(p) is the correct value for e^p.  Note that two values a and b are within epsilon of each other if |a-b| < epsilon.

• To receive full credit your solution should involve a single loop and it must be a while-loop.
   
• Avoid the use of extremely small values for epsilon.  A floating point number, as used within a computer, is an approximation to a real number.  As you add smaller and smaller numbers to a sum, it's possible for the sum to stop changing due to number-representation errors within the computer.  Epsilon values near 10^-5 should work fine for exponents between, say, 0 and 10.
   
• Submit one file to CMS: approxE.m.

3. Twinkle, Twinkle Little Star

Create a program (a Matlab function) called star.m that draws a star. Your function should have two arguments: points (the number of points) and inc (the increment). For instance, for points=5 and inc=2, five points are equally spaced around a circle and then each point is connected to the point two forward from it; this creates a traditional five-pointed star. For points=5 and inc=1, your program should draw a pentagon. Example output for star(7, 2) appears below.

• We have given you a partially completed version of star.m.  We have provided the setup code for the graphics window.  You need to add code for drawing the appropriate points and lines. 
   
• Be sure to draw both the points and the connecting lines. 
   
• Recall that plot(x, y, 'ro') draws the point (x, y) as a red circle and that plot([x1, x2], [y1, y2], 'b-') draws a blue line from (x1, y1) to (x2, y2).  Feel free to modify the way lines and points are drawn, but be sure that the resulting picture is clear.
   
• Design your code to use a helper function: angle2xy.m.  Here is its function header.

  function [x, y] = angle2xy (angle)
  % Return (x,y) coordinates given angle (in degrees) on a unit circle

• Submit two files to CMS: star.m and angle2xy.m.