How do I find and display specific y-values on a spline plot?

The task was to take data points, compute a spline of the points, then calculate the first and second derivate of the spline curve. I have found and plotted the curves correctly, but am having trouble with displaying specific values on the derivate curves. How can I input specific x-values in to the first and second derivate curves to give me y-values, then display them?

code:

close all x = linspace(-3, 3, 9); y = [2, 3, 4.1, 4.5, 2, 2, 3, 3, 3];

%x = linspace(0, 4, 6); %y = [2, 1, .4, 0.08, 0, .2];

xx = linspace(-3,4); pp = spline(x,y); yy = ppval(pp, xx); plot(x, y, '.', 'markersize', 10); hold on; plot(xx, yy, 'b');

dpp = fnder(pp); plot(xx,fnval(dpp,xx),'k'); hold on; ddpp=fnder(dpp); plot(xx,fnval(ddpp,xx),'r'); hold on; y_val=fnval(dpp,xx); y2_val=fnval(ddpp,xx); fprintf('At x = %d, f(x) = %.4f, f''(x) = %.4f\n\n',xx,y_val,y2_val)

Answer 1

[Prateekshya]

Hello Dylan,

To evaluate and display specific y-values for the first and second derivatives of a spline at given x-values, you can use the fnval function to compute these values. Then, you can display the results using fprintf or annotate them directly on your plot. Here is a step-by-step guide on how to modify your code to achieve this:

• Define Specific x-values: Decide which specific x-values you want to evaluate.
• Evaluate the Derivatives: Use fnval to compute the values of the first and second derivatives at these specific x-values.
• Display the Results: Use fprintf to print the results or annotate them on the plot.

Here is how you can modify your code:

close all;

% Define data points
x = linspace(-3, 3, 9);
y = [2, 3, 4.1, 4.5, 2, 2, 3, 3, 3];

% Define a finer grid for plotting the spline
xx = linspace(-3, 4, 100); % Use more points for smoother plots

% Compute the spline
pp = spline(x, y);

% Evaluate the spline and plot it
yy = ppval(pp, xx);
plot(x, y, '.', 'markersize', 10);
hold on;
plot(xx, yy, 'b');

% Compute and plot the first derivative
dpp = fnder(pp);
plot(xx, fnval(dpp, xx), 'k');

% Compute and plot the second derivative
ddpp = fnder(dpp);
plot(xx, fnval(ddpp, xx), 'r');

% Define specific x-values for evaluation
specific_x_values = [-2, 0, 2]; % Example x-values

% Evaluate and display the derivatives at specific x-values
for i = 1:length(specific_x_values)
    x_val = specific_x_values(i);
    y_val = fnval(pp, x_val);
    dy_val = fnval(dpp, x_val);
    ddy_val = fnval(ddpp, x_val);
    fprintf('At x = %.2f, f(x) = %.4f, f''(x) = %.4f, f''''(x) = %.4f\n', x_val, y_val, ...
        dy_val, ddy_val);
    
    % Annotate the plot
    plot(x_val, dy_val, 'ko', 'MarkerFaceColor', 'k'); % First derivative
    plot(x_val, ddy_val, 'ro', 'MarkerFaceColor', 'r'); % Second derivative
    text(x_val, dy_val, sprintf('%.2f', dy_val), 'VerticalAlignment', 'bottom', ...
        'HorizontalAlignment', 'right');
    text(x_val, ddy_val, sprintf('%.2f', ddy_val), 'VerticalAlignment', 'bottom', ...
        'HorizontalAlignment', 'right');
end

hold off;

The output you get looks like:

I hope this helps!