A function is measured at two points with some error (for example, the position of an ob...

A function is measured at two points with some error (for example, the position of an object is measured at two times). Let

a. Find E(Z) and Var(Z). What is the effect of choosing a value of h that is very small, as is suggested by the definition of the derivative?

b. Find an approximation to the mean squared error of Z as an estimate of f ^‘(x) using a Taylor series expansion. Can you find the value of h that minimizes the mean squared error?

c. Suppose that f is measured at three points with some error. How could you construct an estimate of the second derivative of f , and what are the mean

and the variance of your estimate?