Zeitreihenanalyse - Aufgabe 3

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Show that the simple linear function f(r)=r\, is in the neighborhood of zero approximately equal to the nonlinear function g(r)=\log(1+r) \,

Sketch f(r) and g(r) for -0.1 \le r \le 0.1 \,

> pdf(rpdf)
>
> r=seq(-.1,.1,length.out=500)
>
> plot(r, r, xlab="x", ylab="f(r), g(r)", type="l")
> lines(r, log(1+r), col=2)
>
> legend(-.07,.07, c("f(r) = r", "g(r) = log(1 + r)"), lty=1, col=1:2)

Compare f(0) and f′(0) with g(0) and g′(0).

f(0) = 0; f'(r) = 1; f'(0) = 1 \,

g(0) = 0; g'(r) = \frac{1}{1+r}; g'(0) = 1 \,