Assigned: Tuesday, September 6
Due: Tuesday, September 13

Relevant reading: Lecture Notes 1. From Hunt, Chapter 2 (don't bother for now with the sections about symbolic computation and matrices) and the first six pages of Chapter 5. From Moler, Sections 1.1 and 1.7.  From Driscoll: Chapter 1, Sections 2.1-2.4, and Section 5.1.

You will need to turn in (among other things) a  file contaning the MATLAB commands that you type in order to solve the problems below, so once you complete a problem, you should select, copy and save a transcript of your session, or, better yet, create M-files.  (See "What to Hand In" below for more detail.)

1. Harmonic series and summation order

The harmonic series


is divergent---the sum does not approach a finite limit..

(a) Evaluate the sum of the first one million terms of this series.  To do this, you will need to use the various techniques available in MATLAB for manipulating vectors:

• First create the vector (1,2,....,10000000) and assign it to a variable.
• Then create the vector of reciprocals (1,1/2,1/3,...,1/1000000) by using componentwise division. 
  
• If v is a vector, then sum(v) is the sum of all the components of v. (This is not true if v is a matrix with more than one row and more than one column.)
  

(b) Now evaluate the sum again, but this time evaluate it as


(c)The answers are close, but not exactly equal.  Again, you will need to use format long to see this. Which one do you think is more accurate?  We will return to this question next week, when we discuss the floating-point representation of numbers in the computer.


2. Plot the lemniscate of Gerono, pictured above.  Parametric equations for the curve are given by


If you try to plot this in the most straightforward way, you are likely to encounter a difficulty connected with the limits on the parameter t, and not be able to produce a smooth plot like the one above.  How can you fix this?

3. Plot a "conical helix"

 (I don't know if that's the right term, but it should be clear from the picture what I mean!) The idea is similar to what we did in the Lecture Notes to produce the cylindrical helix.  You need the following fact about so-called cylindrical coordinates: A point at height z, a distance r from the z-axis, and whose projection onto the (x,y)-plane makes an angle of u with the positive x-axis, has ordinary (Cartesian) coordinates (r cos(u), r sin(u), z).  In the helix, we had r=1, but here r varies linearly from the apex of the cone to the base.


4. Plot a conical surface.




 You need a parametrization in terms of two parameters, which you should choose as the distance r from the z-axis and the polar angle u, as above.