Derivatives - Sinosoidal

Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.

< Code 1 >

x = linspace(0,6*pi,80);

y = 1.0 .* sin(0.075 .* x .^ 2);

ylim_y = 4.0 .* [-1.5 1.5];

ylim_yp = 4.0 .* [-1.5 1.5];

ylim_ypp = 4.0 .* [-1.5 1.5];

np = 77;  % the last array index to plot y'(x). Must be between 3 and 77

npp =78; % the last array index to plot y''(x). Must be  between 3 and 77

yp = [];

for i = 2:length(y)-1

  dx = x(i+1) - x(i-1);

  dy = y(i+1) - y(i-1);

  yp = [yp (dy ./ dx)];

end  

ypp = [];

for i = 2:length(yp)-1

  dx = x(i+1) - x(i-1);

  dy = yp(i+1) - yp(i-1);

  ypp = [ypp (dy ./ dx)];

end  

hFig = figure(1,'Position',[300 300 600 500]);

% Calculate the tangent line for y(x)

range = 2.0;

bt = y(np) - yp(np-1) .* x(np);

xt = linspace(x(np)-range,x(np)+range,10);

yt = yp(np-1) .* xt .+ bt;

thickness_f = 1

thickness_t = 2

subplot(3,3,[1 2]);

hold on;

plot(x,y,'k-','LineWidth',thickness_f);

if (npp < 4)

  plot(xt,yt,'b-','LineWidth',thickness_t);

  plot(x(np),y(np),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',8);

end

xlim([x(1) x(end)]);

ylim(ylim_y);

ylabel('\bf \fontsize{20}  f(x)')

grid on;

box on;

hold off;

if (np > 2)

  subplot(3,3,3);

  axis([0 1 0 1]);

  if (npp <= 3)

    tStr = sprintf('Slope at %0.02f',x(np));

    text(0,0.7,tStr,'FontSize',14);

    tStr = sprintf('= %0.02f',yp(np-1));

    text(0.0,0.5,tStr,'FontSize',14);

  end

  set(gca,'Visible','off')

  % Calculate the tangent line for y''(x)

  range = 2.0;

  btt = yp(npp-1) - ypp(npp-2) .* x(npp);

  xtt = linspace(x(npp)-range,x(npp)+range,10);

  ytt = ypp(npp-2) .* xtt .+ btt;

  subplot(3,3,[4 5]);

  hold on;

  plot(x(2:np),yp(1:np-1),'b-','LineWidth',thickness_f);

  if(npp <= 3)

    plot(x(np),yp(np-1),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',8);

  end

  if (npp > 3)

    plot(xtt,ytt,'r-','LineWidth',thickness_t);

    plot(x(npp),yp(npp-1),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',8);

  end

  xlim([x(1) x(end)]);

  ylim(ylim_yp);

  ylabel('\bf \fontsize{20}  f\prime(x)');

  grid on;

  box on;

  hold off;

end

if (npp > 3)

  subplot(3,3,6);

  axis([0 1 0 1]);

  tStr = sprintf('Slope at %0.02f',x(npp));

  text(0,0.7,tStr,'FontSize',14);

  tStr = sprintf('= %0.02f',ypp(npp-2));

  text(0.0,0.5,tStr,'FontSize',14);

  set(gca,'Visible','off')

end  

  subplot(3,3,[7 8]);

  hold on;

if (npp > 3)  

  plot(x(3:npp),ypp(1:npp-2),'r-','LineWidth',thickness_f);

  plot(x(npp),ypp(npp-2),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',8);

end  

  xlim([x(1) x(end)]);

  ylim(ylim_ypp);

  ylabel('\bf \fontsize{20}  f\prime\prime(x)');

  grid on;

  box on;

  hold off;