function lab1soln

%------------------------------------------------------------------------
%%%               ENGINEERING MATHEMATICS IV - MATH 2ZZ3
%%%                             ASSIGNMENT #5:
%%%       TRIPLE INTEGRALS, DIVERGENCE THEOREM AND CHANGE OF VARIABLES
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% Covers:
% - "Numerical Mathematics" by M. Grasselli and D. Pelinovsky,
%    Sections 7.7--7.8.
% "Advanced Engineering Mathematics" by D.G. Zill and W.S. Wright, section 9.15, 9.16, 9.17
%------------------------------------------------------------------------

%------------------------------------------------------------------------
% Instructions:
% - Submit your assignment electronically (via Email) to the
%   address specific to your last name as indicated on the course
%   website; hardcopy submissions will not be accepted.
% - It is obligatory to use the current MATLAB template file available at
%   http://www.math.mcmaster.ca/gabardo/MATH2ZZ3/frames/template.m; 
%   submissions non compliant with this template will not be accepted.
% - Make sure to enter your name and student I.D. number in the
%   appropriate section of the template.
% - Late submissions and submissions which do not comply with
%   these guidelines will not be accepted.
% - All graphs should contain suitable titles and legends.
%------------------------------------------------------------------------
% Written by Vladislav Bukshtynov, 2009


clc; close all; clear all;

% Student information
  
disp('  ------------   Student Information  ------------  ')  

% Please enter your information here
Name = 'Lab 1';
Surname = 'Solutions';
ID = 1234567;

fprintf('  Student:  %s %s (ID: %d) \n', Name, Surname, ID);



disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')


disp('  ----------   Solution to Question #1  ----------  ')  
disp('  ')

% (a)

x = linspace(0,3,10);
y = x;
[X,Y] = meshgrid(x,y);

scrsz = get(0,'ScreenSize');
figure('Name','Direction field and solutions for y'' = x^2/y^2', ...
       'OuterPosition',[1 scrsz(4)/3 2*scrsz(3)/3 2*scrsz(4)/3]);
hold on;
grid on;
title('Direction field and solutions for y'' = x^2/y^2');
quiver(X,Y,Y.^2,X.^2,'Color','black');
xlim([0 3]);
ylim([0 3]);

% (b)

x = linspace(0,3,102);
x1 = linspace(1,3,67); % for the solution through (0,-1)
y1 = (x1-1).^(1/3);
x2 = linspace(2,3,34); % for the solution through (0,-2)

plot(x,(x.^3+8).^(1/3),   ... % (0,2)
	 x,(x.^3+1).^(1/3),   ... % (0,1)
	 x,x,                 ... % (0,0)
	 x1,(x1.^3-1).^(1/3), ... % (0,-1)
	 x2,(x2.^3-8).^(1/3));    % (0,-2)
legend('direction field','soln through (0,2)', 'soln through (0,1)' , ...
       'soln through (0,0)', 'soln through (0,-1)', 'soln through (0,-2)');
legend('Location','EastOutside');

disp('  ')
disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')


disp('  ----------   Solution to Question #2  ----------  ')  
disp('  ')

syms xs;
fs = exp(-xs) - xs;

% because I'm lazy to compute the derivative by hand:
f = matlabFunction(fs);
fp = matlabFunction(diff(fs,xs));

% fake "do ... while" loop
new = 1;
test = true;
while test
    old = new;
    new = old - f(old)/fp(old);
    test = abs(new - old) >= 10^(-5);
end

Answer1 = new;

disp('  ')
disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')


disp('  ----------            Answers         ----------  ')  

disp('  ')
% DO NOT TOUCH THIS PART (RESERVED FOR THE INSTRUCTOR'S USE)

Name 
Surname 
ID



Answer1




% User's Functions Space is below --------------------------