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8.1 $B:9J,(B

$B87L)$J2r$r5a$a$K$/$$>l9g!"7W;;5!$G?tCM7W;;$r9T$$2r$NMM;R$rD4$Y$k(B $BJ}K!$,$"$k!#(B $BHyJ,J}Dx<0$N?tCM7W;;K!$K$b?'!9$"$k$,!"$3$3$G$O:9J,K!$r>R2p$9$k!#(B

$B:9J,J}$H$O!"4X?t$NHyJ,(B $f'(x)$ $B$r!"(B$h>0$ $B$H$7$F(B

$$\frac{f(x+h)-f(x)}{h}$$ (31)

$B$GCV$-49$($k$3$H$r$$$&!#$3$N:9J,$N(B $h\rightarrow 0$ $B$N$H$-$N6K8B$,(B $BHyJ,$G$"$k$N$G!"(B$h$ $B$,>.$5$1$l$P:9J,$HHyJ,$H$N:9$O>.$5$/!"(B $B$h$C$FHyJ,$O:9J,$G6a;w$G$-$k$3$H$K$J$k!#(B $B<0(B (31) $B$NB>$K$b(B

$$\frac{f(x)-f(x-h)}{h}$$ (32)

$$\frac{f(x+h)-f(x-h)}{2h}$$ (33)

$B$GCV$-49$($k$3$H$b$G$-$k$,!"(B(31) $B$OA0?J:9J,!"(B (32) $B$O8eB`:9J,!"(B (33) $B$OCf?4:9J,$H8F$P$l$k!#(B

$B?^(B 24: $BA0?J!"8eB`!"Cf?4:9J,(B

$BNc$($P!"oHyJ,J}Dx<0$N=i4|CMLdBj(B

$$\left\{\begin{array}{l} \displaystyle \frac{dx(t)}{dt}=x(t) \ \ (t>0), \\ [1zh] x(0)=1 \end{array}\right.$$ (34)

$B$r9M$($F$_$k!#(B(34) $B$NJ}Dx<0$N:8JU$rA0?J:9J,$G(B $BCV$-49$($k$H(B

$$\frac{x(t+h)-x(t)}{h}\approx x(t)$$

$B$H$J$j!"$h$C$F$3$N6a;wJ}Dx<0$O(B

$$x(t+h)\approx (1+h)x(t)$$

$B$H=q$1$k$3$H$K$J$k!#(B$x(0)=1$ $B$HM?$($i$l$F$$$k$N$G!"$3$l$K$h$j(B

$$\begin{eqnarray*} && x(h)\approx (1+h)x(0)=1+h,\\ && x(2h)\approx (1+h)x(h)=(... ... (1+h)x(2h)=(1+h)^3,\\ && \ldots \\ && x(nh) \approx (1+h)^n \end{eqnarray*}$$

$B$H!"(B $t=h,2h,3h,\ldots$ $B$G$N(B $x(t)$ $B$N6a;wCM$,5a$a$i$l$k$3$H$K$J$k!#(B

$B85$NJ}Dx<0(B (34) $B$N??$N2r$O(B $x(t)=e^t$ $B$G$"$k$,!"(B $B>e$N6a;w2r$r(B $x=x_h(t)$ $B$H$9$k$H!"(B$t=mh$ $B$N$H$-$3$NCM$O(B

$$x_h(t)=x_h(mh)=(1+h)^m=(1+h)^{t/h}$$

$B$H$J$k!#:#!"(B$t$ $B$r8GDj$7$F(B $h\rightarrow 0$ $B$H$9$k$H(B

$$(1+h)^{t/h} = \{(1+h)^{1/h}\}^t \rightarrow e^t$$

$B$H$J$k$N$G!"$3$N6a;w2r(B $x_h(t)$ $B$O(B $h$ $B$,>.$5$$CM$N$H$-$O(B $B3N$+$K??$N2r(B $x(t)$ $B$r6a;w$7$F$$$k$3$H$,$o$+$k!#(B

$B:#!">.$5$$@5?t(B $h$ $B$,M?$($i$l$?>l9g!"(B$x_n=x_h(nh)$ $B$r7W;;$9$k<0$O(B

$$\left\{\begin{array}{l} x_{n+1}=(1+h)x_n \ \ (n\geq 0),\\ x_0=x(0)=1\end{array}\right.$$

$B$H$J$j!"$9$J$o$A?tNs$NA22=<0$GM?$($i$l$k$3$H$K$J$k!#(B $B$3$N$h$&$JA22=<0$N7W;;$O!"7W;;5!$G7W;;$9$k>l9g$K$O$`$7$mET9g$,NI$$!#(B $B$3$N$h$&$K!":9J,K!$K$*$$$F$ODL>oA22=<0$K$h$C$F?tCM7W;;$,9T$o$l$k!#(B