Question

In MATLAB write a function that returns the function values and the Jacobian for, x^2 + 9y^2=16 and y - x^2 + 2x = p. Where 0 <= p <= 2.5 The symbolic toolbox can not be used.

Answer 1

clear all;
close all;
x1=-3.9999:0.001:3.9999;

y1= (sqrt(16-x1.^2))/9;

J1 = -x1./(9*sqrt(16-x1.^2)); % Jacobian is the partial derivative of
                              % y = f(x) w.r.t x

figure;plot(x1,y1);
xlabel('x ---->');
ylabel('y ---->');
title('x^2 + 9y^2 =16');

figure;plot(x1,J1);
xlabel('x ---->');
ylabel('J ---->');
title('Jacobian of y = (1/9)*sqrt(16-x^2)');

p=0:0.5:2.5;

x2 = -2:0.001:5;
figure;
for i=1:length(p)
    y2 = p(i) - x2.^2 + 2*x2;
    plot(x2,y2);
    hold on;
end
title('y -x^2 + 2x = p');
xlabel('x ---->');
ylabel('y ---->');

Legend=cell(length(p),1);
for i=1:length(p)
   Legend{i}=strcat('p = ', num2str(p(i)));
end
legend(Legend);

J2 = -2*x2 + 2; %Jacobian independent of p
figure;plot(x2,J2)
xlabel('x ---->');
ylabel('J ---->');
title('Jacobian of y = p - x^2+2x');