Aproximação Melhor

• \(n=2p+1:\) \(f(t+\varepsilon) -f(t-\varepsilon)\simeq\) \(2\sum_{j=0}^{p} \frac{ \varepsilon^{2j+1} }{(2j+1)!} f^{(2j+1)} (t)=\) \(2 \varepsilon f\prime(t)+ 2 \frac{\varepsilon^3 }{3!}f^{(3)}(t)+...+2\frac{\varepsilon^{2p+1}}{(2p+1)!} f^{(2p+1)}(t)\)
• Isolando \(f\prime(t):\) \(f\prime(t)\) \(\frac{ f(t+\varepsilon) -f(t-\varepsilon) }{ 2 \varepsilon }+\frac{\varepsilon^2}{3!} f^{(3)}(t)+...+\frac{\varepsilon^{2p}}{(2p+1)!} f^{(2p+1)}(t)=\) \(\frac{ f(t+\varepsilon) -f(t-\varepsilon) }{ 2 \varepsilon } +o(\varepsilon^2)\)
  

  Python Listing: ../../../Code/Derivatives.py. def d_f(f,x,eps=1.0E-6): #Better return (f(x+eps)-f(x-eps))/(2.0*eps)

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