Example Problem Set 3

1.

Suppose we wish to compute a predicted value for some continuous, differentiable function f(t) at future time t=b, based only upon knowledge of its value now at time t=a, which is f(a), and its first three derivatives now, namely f'(a), f''(a), and f'''(a). How would you compute such a prediction for the future value f(b)? Give a formula or explain the sorts of terms that it would include and why they appear.

2.

In practical computing problems with numerical data that is discretely sampled at periodic points in time, derivatives defined in terms of continuous limits must be replaced by discrete estimates. Provide finite-difference expressions for each of the three derivatives f'(t), f''(t), and f'''(t) on the discrete set of samples

assuming that this sequence of samples are separated by a constant interval $\Delta t = t_{i+1}-t_{i}$.

3.

If you were trying to compute numerically the solution to a continuous-time first-order differential equation using the Runge-Kutta method of integration, how would you expect the error of your solution in the worst case to depend upon the stepsize $\Delta t$?

4.

Real-world signals are generally accompanied by noise. Using the Differentiation Theorem of the Fourier Transform, comment upon how noise is amplified as a function of its spectral composition when computing the first-order, second-order, and third-order derivatives of the signal.