Solution Notes -- Example Problem Set 3

1.

Prediction is ${\displaystyle f(b)=f(a) + f' (a)(b-a) + \frac{f'' (a)}{2!}(b-a)^{2} + \frac{f''' (a)}{3!}(b-a)^{3}}$(A Taylor series which in effect gives a polynomial expansion of f(t)around t=a, in terms of derivatives of successive powers of t.)

2.

$$\approx \frac{f(t_{i+1})-f(t_{i})}{\Delta t}$$ f'(t)

$$\approx \frac{f(t_{i+2})-2f(t_{i+1})+f(t_{i})}{(\Delta t)^{2}}$$ f''(t)

$$\approx \frac{f(t_{i+3})-3f(t_{i+2})+3f(t_{i+1})-f(t_{i})}{(\Delta t)^{3}}$$ f'''(t)

There are alternative, equally valid answers, which use the symmetric approximation to the derivatives.

3.

If we use $n^{\mbox{th}}$-order Runge-Kutta then the error in a given step will depend upon the step size raised to the $(n+1)^{\mbox{th}}$ power, while the accumulated error will depend upon the step size raised to the $n^{\mbox{th}}$ power

4.

When computing its $n^{\mbox{th}}$-order derivative, the noise in the signal is amplified as the $n^{\mbox{th}}$-power of frequency. Thus for the cases given and for frequency $\omega$, this noise problem will grow as $\omega^{1}, \omega^{2},$ and $\omega^{3}$, respectively.