f= uv/(u+v)
max error= depends on error in u, and v as compared to the true u,v.
where u and v represent object and image distances respectively,The maximum relative error in f is ?
max error= depends on error in u, and v as compared to the true u,v.
Let's start by differentiating the equation with respect to u:
1/f = 1/u + 1/v
Differentiating both sides with respect to u:
d(1/f)/du = d(1/u)/du + d(1/v)/du
Since the derivative of a constant (1/f) is zero, we can simplify the equation:
0 = -1/u^2 + 0
Now, let's differentiate the equation with respect to v:
1/f = 1/u + 1/v
Differentiating both sides with respect to v:
d(1/f)/dv = d(1/u)/dv + d(1/v)/dv
Again, since the derivative of a constant (1/f) is zero, we can simplify the equation:
0 = 0 + (-1/v^2)
Now, let's solve these equations to find the relative error in f.
Using the given equation 1/f = 1/u + 1/v, we can rearrange it to solve for f:
1/f = (u + v)/(uv)
Now, we can substitute this expression into the equations we derived earlier to calculate the relative error:
Relative error in f with respect to u:
Error_u = (d(1/f)/du)/(1/f) = -1/u^2 / (1/u) = -1/u
Relative error in f with respect to v:
Error_v = (d(1/f)/dv)/(1/f) = (-1/v^2) / (1/u+v/uv) = -1/v
To find the maximum relative error, we need to consider the largest magnitudes of the relative errors. Therefore, the maximum relative error in f is the larger of the two absolute values:
Maximum relative error = max(abs(Error_u), abs(Error_v)) = max(abs(-1/u), abs(-1/v))
So, the maximum relative error in f is the larger of 1/abs(u) and 1/abs(v).