ExerciseDifferentiationintermediate

Exercise: comparing derivative formulas

Computing $f'(0.5)$ for $f(x)=x^2e^{-x}$ with the six finite-difference formulas and comparing their errors against the exact value $0.4549$ .

Data and comparison

With $f(x)=x^2e^{-x}$ , step $h=0.25$ and nodes $x=\{0,\,0.25,\,0.5,\,0.75,\,1\}$ , we approximate $f'(0.5)$ . The analytic value is $f'(0.5)=0.4549$ . The table gives each approximation and its absolute error.

Formula	f'(0.5)	Error ε
Progresiva O(h)	0.4563	0.0014
Progresiva 3 puntos O(h²)	0.4801	0.0252
Regresiva O(h)	0.4118	0.0431
Regresiva 3 puntos O(h²)	0.5204	0.0655
Central O(h²)	0.4341	0.0208
Central 5 puntos O(h⁴)	0.4561	0.0012