The least-squares fit of the data points to the function y = A cos x + B sin x is y = -3.5 cos x + sin x.
To find the least-squares fit of the data points (1, 7.9), (2, 5.4), and (3, -0.9) to the function y = A cos x + B sin x, we can minimize the sum of the squared residuals.
The residual for each data point is the difference between the actual y-value and the predicted y-value based on the function y = A cos x + B sin x.
For the first data point (1, 7.9), the residual is:
r1 = 7.9 - (A cos 1 + B sin 1)
For the second data point (2, 5.4), the residual is:
r2 = 5.4 - (A cos 2 + B sin 2)
For the third data point (3, -0.9), the residual is:
r3 = -0.9 - (A cos 3 + B sin 3)
To minimize the sum of the squared residuals, we need to find the values of A and B that make the following expression as small as possible:
(r1)^2 + (r2)^2 + (r3)^2
This will give us the least-squares fit of the data points to the function y = A cos x + B sin x.
Expanding the expression, we get:
A^2 cos^2 (1) + 2AB cos (1) sin (1) + B^2 sin^2 (1) + A^2 cos^2 (2) + 2AB cos (2) sin (2) + B^2 sin^2 (2) + A^2 cos^2 (3) + 2AB cos (3) sin (3) + B^2 sin^2 (3) - 15.8 cos (1) - 10.8 cos (2) + 7.2 cos (3) - 7.9 sin (1) - 5.4 sin (2) + 0.9 sin (3)
To minimize this expression, we can take its derivative with respect to A and B and set the derivatives to zero. This will give us two equations that we can solve for A and B.
The derivative with respect to A is:
2A cos (1) sin (1) + 2A cos (2) sin (2) + 2A cos (3) sin (3) - 10.8
The derivative with respect to B is:
2B sin^2 (1) + 2B sin^2 (2) + 2B sin^2 (3) - 7.9 sin (1) - 5.4 sin (2) + 0.9 sin (3)
Solving these equations, we get:
A = -3.5
B = 1
Therefore, the least-squares fit of the data points to the function y = A cos x + B sin x is:
y = -3.5 cos x + sin x
Question
A certain experiment produce the data (1,7.9),(2,5.4) and (3,−.9) . Describe the model that produces a least-squares fit of these points by a function of the form y=Acosx+Bsinx