Numerical Differentiation: The most common numerical methods for approximating the derivatives of a function are based on interpolation. To approximate the klh derivative f(k)(x0) at a point xo, one replaces the function f(x) by an interpolating polynomial pn(x) of degree n ≥ k based on the nearby points x0..........xn (the point x0 is almost always included as one of the interpolation points), leading to the approximation f(k)(x0) ≈ P(k)n(x0).Use this method to construct numerical approximations to
(a) f'(x) using a quadratic interpolating polynomial based on x - h, x, x + h.
(b) f'(x) with the same quadratic polynomial.
(c) f' (x) using a quadratic interpolating polynomial based on x, x + h,x + 2h.
(d) f' (x), f" (x),f" (.x) and f(iv)(x) using a quartic interpolating polynomial based on x - 2h.x - h,x, x + h, x + 2h.
(e) Test your methods by approximating the derivatives of e' and tan x at x = 0 with step sizes h = 1/10.1/100.1/1000.1/10000. Discuss the accuracies you observe. Can the step size be arbi-trarily small?
(f) Why do you need n ≥ k?