Derivation of sin(x) gives false cos(x)

Hello,

I'm trying to derivate the sin(x) two times to get the cos(x) and the -sin(x). The plot shows a cosine figure, but it doesn't start at 1 and the amplitude is false too.

My code is the following:

clear;


t = 0.05;

x = [0:0.05:2*pi];
y = sin(x);

figure(1)
scatter(x,y,'.')
xlabel('time')
ylabel('distance')

%%

vel = diff(y);

g = 1:length(vel);

for i = 1:length(g)
    
    vel_t(i) = t * g(i);
 
end  

figure(2)
scatter(vel_t,vel,'.')
xlabel('time')
ylabel('velocity')

%%


acc = diff(vel);

h = 1:length(acc);

for i = 1:length(h)
    
    acc_t(i) = t * h(i);
    
end

figure(3)
scatter(acc_t,acc,'.')
xlabel('time')
ylabel('acceleration')

Is it wrong to use x = [0:0.05:2*pi] for defining my period?

Thanks for your help.

[Stephen23]

"Is it wrong to use x = [0:0.05:2*pi] for defining my period?"

No, but those square brackets are superfluous, get rid of them.

"...it doesn't start at 1 and the amplitude is false too."

Hint: take a look at the definition of differentiation. Where in your code do you divide by the change in the independent variable?

Answer 1

[KSSV]

t = 0.05;
x = 0:0.05:2*pi;
y = sin(x);

figure(1)
plot(x,y,'r')
xlabel('time')
ylabel('distance')

%%
vel = gradient(y)./gradient(x);

figure(2)
plot(x,vel,'b')
xlabel('time')
ylabel('velocity')

%%
acc = gradient(vel)./gradient(x);
  
figure(3)
plot(x,acc,'g')
xlabel('time')
ylabel('acceleration')

[Bruce Rogers]

Thank you for your help KSSV, your code gives the right plots. Can you explain why diff() gave me that wrong results?

[Stephen23]

"Can you explain why diff() gave me that wrong results?"

The problem is not DIFF, the problem is that you did not divide by the change in the independent variable (in your code x), which is required by the definition of differentiation. KSSV did that division using ./

Answer 2

[Robert Brown]

Technically, the gradient is extrapolating data!!!

When you have a series of position measurements..... you require 2 position measurements to get to a velocity estimate, and require another position measurement to get to an acceleration measurement.

Assume Time = T1, T2, T3, ...

Position 1 = only know position P1 and T1

Position 2 = know P1 and P2 and T1 and T2

and estimate V1 = (P2 - P1) / (T2-T1) = average velocity getting from P1 to P2,

and corresponding time TV1 = (T2-T1)/2

Position 3 = know P1 and P2 and estimate of V1 (above) and P3 and V2 = (P3 - P2) / (T3 - T2)

and corresponding time TV2 = (T3-T2)/2

and know acceleration 1 A1 = (V2-V1)/(TV2 - TV1) =

((P3 - P2) / (T3 - T2) - (P2 - P1) / (T2-T1)) / ((T3-T2)/2 - (T2-T1)/2)

etc....

[Robert Brown]

Maybe I should have called it V1.5 meaning the average (estimated) velocity between 1 and 2, and T1.5 meaning the time 1/2 of the way between V1 and V2. Then by using V1.5 and V2.5.... you can determine V2, the (estimated) velocity at T2.

If you look back at your code, the diff function caused the velocity measurements to have 1 less data point since you don't know velocity until you know 2 positions, and the diff function caused the acceleration measurements to have 2 less data points, since you don't know acceleration until you know 3 positions and 2 estimated velocities... :-)