Introductory Programming
Fall 2006

Today:

1) follow the instructions in notes09.txt to find the roots
   of the 6th Chebyshev polynomial

2) work on the duck problem


Eval 04 Solution
----------------

Many of you did a good job on the first part and had trouble
with the second part.

function y = cheby6(x)
    y = 32 * x^6 - 48 * x^4 + 18 * x^2 - 1;
end


function plot_cheby6(low, high)
    X = linspace(low, high, 100);
    for i=1:length(X)
        Y(i) = cheby6(X(i));
    end
    plot(X, Y)
end


Notice that part two is an example of

BEFORE:  create the vector of x values

DURING:  for each x value, use cheby6 to find the corresponding y value

AFTER:   plot Y versus X


A common source of confusion was how to iterate through the
elements of a vector:

    for i=1:length(X)
        Y(i) = cheby6(X(i));
    end

In this loop, the values of i are the indices, from 1 to length(X)

Y(i) and X(i) are the elements of the vectors.




The Duck Problem
----------------

The density of a duck, rho, is 0.3 g / cm^3 (0.3 times the density of water).

The volume of a sphere with radius r is 4/3 pi r^3.

If a sphere with radius r is submerged in water to a depth d, the
volume of the sphere below the water line is 

       volume = 1/3 pi (3r d^2 - d^3)           as long as d < r

An object floats at the level where the weight of the displaced
water equals the total weight of the object.

Assuming that a duck is a sphere with radius 10 cm, at what depth does
a duck float?


Suggestions:


1) Write an equation relating rho, d and r.

2) Rearrange the equation so the right-hand side is zero.

   Our goal is to find values of d that are roots of this equation.

3) Write a MATLAB function that evaluates this function.  Test it,
   then make it a quiet function.

4) Make a guess about the value of d0 to use as a starting place.

5) Use fzero to find a root near d0.

6) Check to make sure the result makes sense.  In particular,
   check that d < r, because otherwise the equation we used
   doesn't work!

7) Try different values of rho and r and see if you get the
   effect you expect.

   What happens as rho increases?  Goes to infinity?  Goes to zero?

   What happens as r increases?  Goes to infinity?  Goes to zero?