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3 2001 $BG/4pAC?tM}(B I

$BA0@a$NJ}K!$O87L)@-$O$"$k$,!">ZL@$OI,$:$7$b3X@8$K$H$C$F4JC1$G$O$J$/!"(B $B$=$3$G@bL@(B ($BM}2r(B) $B$NN.$l$,@Z$l$F$7$^$$!"I,$:$7$b6a;w$K$h$k@bL@$,(B $B$&$^$/$$$C$?$H$O;W$($J$$!#(B

$B$h$C$F:#G/$O$=$N$h$&$J>ZL@$O $y=f(x)$ $B$N(B $x=a$ $B$N6a$/$G$N(B $y$ $B$NCM$O!"(B$x=a$ $B$G$N@\@~(B

$$y-f(a)=f'(a)(x-a)$$

$B$G6a;w$G$-$k!#$9$J$o$A!"(B

$$f(x) \approx f(a)+f'(a)(x-a) \mbox{\ \ ($x\approx a$\ $B$N$H$-(B)}$$ (8)

$B$G$"$k$3$H$K$J$k!#1&JU$O(B $x$ $B$N(B 1 $B $B$N(B 1 $B $B$H8F$P$l$k!#(B

$$f(x) \approx f(a) \mbox{\ \ ($x\approx a$\ $B$N$H$-(B)}$$ (9)

$B$,(B 0 $B $B$b$A$m$s!"(B0 $B/$J$/6a;w$,NI$$!#(B 2 $B/$J$/$J$k!#$=$N(B 2 $B $B0J2<4JC1$N$?$a(B $a=0$ $B$H$9$k!#(B

1. $B$^$:$O!"(B0 $B $a=0$ $B$N$H$-!"(B(9) $B$N8m:9$O(B $f(x)-f(0)$ $B$G$"$j!"$3$l$O$b$A$m$s(B $x\rightarrow 0$ $B$N$H$-$K(B 0 $B$K<}B+$9$k$,!"(B $B$3$l$H(B $x$ $B<+?H$H$rHf3S$7$F$_$k$H(B
   
   $$\frac{f(x)-f(0)}{x}$$
   
   
   $B$OJ,;R!"J,Jl$H$b$K(B 0 $B$K<}B+$9$k$N$G!"%m%T%?%k$NDjM}$K$h$j(B
   
   $$\lim_{x\rightarrow 0}\frac{f(x)-f(0)}{x} =\lim_{x\rightarrow 0}\frac{f'(x)}{1}=f'(0)$$
   
   
   $B$H$J$k!#$D$^$j!"$3$l$O(B
   
   $$f(x)-f(0) \approx f'(0)x \mbox{\ \ ($x\approx 0$\ $B$N$H$-(B)}$$
   
   
   $B$G$"$k$3$H$r0UL#$9$k!#$3$N(B $f(0)$ $B$r1&JU$K0\9`$7$?$b$N$,(B (8) $B$G$"$k!#(B

2. $B $f(x)-f(0)-f'(0)x$ $B$rD4$Y$F$_$k!#(B$x$ $B$HHf3S$9$k$H(B $B%m%T%?%k$NDjM}$K$h$j(B
   
   $$\lim_{x\rightarrow 0}\frac{f(x)-f(0)-f'(0)x}{x} =\lim_{x\rightarrow 0}\frac{f'(x)-f'(0)}{1}=0$$
   
   
   $B$H$J$k$N$G!"(B$x$ $B$h$j$:$C$H>.$5$$(B ($BEvA0$G$b$"$k(B)$B!#(B $x^2$ $B$HHf3S$7$F$_$k$H(B
   
   $$\lim_{x\rightarrow 0}\frac{f(x)-f(0)-f'(0)x}{x^2} =\lim_{x\r... ...}{2x} =\lim_{x\rightarrow 0}\frac{f''(x)}{2} =\frac{f''(0)}{2}$$
   
   
   $B$H$J$k!#$3$l$O(B
   
   $$f(x)-f(0)-f'(0)x \approx \frac{f''(0)}{2}x^2 \mbox{\ \ ($x\approx 0$\ $B$N$H$-(B)}$$
   
   
   $B$r0UL#$9$k!#$3$l$K$h$j(B 2 $B
   
   $$f(x)\approx f(0)+f'(0)x+\frac{f''(0)}{2}x^2 \mbox{\ \ ($x\approx 0$\ $B$N$H$-(B)}$$
   
   
   $B$,F@$i$l$k!#(B

3. $BF1MM$K!"$3$N<0$N8m:9(B
   
   $$f(x)-f(0)-f'(0)x-\frac{f''(0)}{2}x^2$$
   
   
   $B$r(B $x^3$ $B$HHf3S$9$k$3$H$G(B
   $$\begin{eqnarray*}&& \lim_{x\rightarrow 0}\frac{\displaystyle f(x)-f(0)-f'(0)x-\f... ...$B0$rF@$k6K8B$N7W;;$(B$f$\ $B$r(B $f'$\ $B$H$7$?$b$N$r(B 3 $B$G3d$C$?$b$N(B)}\end{eqnarray*}$$
   
   
   
   
   
   $B$,F@$i$l$k!#$3$l$K$h$j(B 3 $B
   $$f(x)\approx f(0)+f'(0)x+\frac{f''(0)}{2}x^2+\frac{f'''(0)}{6}x^3 \mbox{\ \ ($x\approx 0$\ $B$N$H$-(B)}$$
   
   
   $B$,F@$i$l$k$,!"$3$l$r7+$jJV$9$3$H$G%F%$%i!
   $$f(x)\approx f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3 + \cdots \mbox{\ \ ($x\approx 0$\ $B$N$H$-(B)}$$
   
   
   $B$,F@$i$l$k!#(B