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{\large ���wI���K�𓚗�@��T��@2007�N6��5���o�蕪} 

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�S�� ���n����CTA �O�p �~
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\noindent{\bf [1]}\ 
(i) $f(x)= x^x$�Ƃ����ƁA$\log f(x)=x\log x$�ŁA���ӂ�$x$�Ŕ���
���鎖�ɂ��$f'(x)=x^x (1+\log x)$�𓾂�B
���ɁA$g(x)=x^{x^x}$�Ƃ����ƁA$\log g(x)=x^x \log x$�ŁA
���ӂ�$x$�Ŕ������ď�̌��ʂ�p����΁A$g'(x)=
x^{x^x} x^x \{(1+\log x) \log x +\frac{1}{x}
\}$ �𓾂�B
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(ii) �����ł�$a\neq 0$�̏ꍇ�ɂ��Čv�Z����B
$f(x)=\frac{1}{ax^2+bx+c} 
=\frac{1}{a(x-\al)(x-\bt)}$�Ƃ�����B
�A��$\al =\frac{-b+\sqrt{b^2-4ac}}{2a}$,�@ 
$\bt =\frac{-b-\sqrt{b^2-4ac}}{2a}$�B

$b^2-4ac>0$�̂Ƃ��́A$f(x)=\frac{1}{a(\al -\bt)} \left(
\frac{1}{x-\al} -\frac{1}{x-\bt}
\right)$���A
$f^{(n)}(x)=\frac{(-1)^n n!}{a(\al -\bt)}
\left\{\frac{1}{(x-\al)^{n+1}}-\frac{1}{(x-\bt)^{n+1}}
\right\}$�B

$b^2-4ac=0$�̂Ƃ��́A$f(x)=\frac{1}{a (x-\al)^2}$
���A
$f^{(n)}(x)=\frac{(-1)^n (n+1)!}{a (x-\al)^{n+2}}$�B

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\noindent{\bf [2]}\ 
(a) ������A$\displaystyle \lim_{x\to 0}\frac{f(a+x)-f(a)}{x} =f'(a)$
�����݂���B���Ȃ킿�A�C�ӂ�$\e>0$�ɑ΂���$\d>0$�����݂��āA
$0<|x|<\d$�Ȃ��$\displaystyle \left|\frac{f(a+x)-f(a)}{x}-f'(a)
\right|<\e$���݂����B���̂Ƃ��C�ӂ�$h,k\in (0,\d)$�ɑ΂��āA
\begin{eqnarray*}
&&\left|\frac{f(a+h)-f(a-k)}{h+k} -f'(a)\right|\\
&\le&\left|\frac{f(a+h)-f(a)}{h+k}-\frac{h}{h+k} f'(a)\right|
+\left|\frac{f(a)-f(a-k)}{h+k}-\frac{k}{h+k} f'(a)\right|\\
&=&\frac{h}{h+k} \left| \frac{f(a+h)-f(a)}{h} -f'(a) \right|
+\frac{k}{h+k} \left| \frac{f(a)-f(a-k)}{k} -f'(a) \right|\\
&<&\frac{h\e}{h+k} +\frac{k\e}{h+k} =\e
\end{eqnarray*}
�ƂȂ邩���ӂ����藧�B

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(b) �Ⴆ��$f(x)=|x|$�Ƃ���΁A$
\displaystyle \lim_{h\to 0}\frac{f(h)-f(-h)}{2h}=0$
�����A
$f(x)$��$x=0$�Ŕ����\�łȂ��B����͑��ɂ����낢��l������B 

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\noindent{\bf [3]}\ (a) $x\neq 0$�ɂ����ẮA$f(x)$�͔����\�Ȋ֐�($\sin
x, x^{-2}$�Ȃ�)�̘a��ρA�����֐���p���ė^�����Ă��鎖��������\�ŁA����
$f'(x)=2x\sin (x^{-2})-2x^{-1} \cos (x^{-2}) +1$�B

$x=0$�̂Ƃ����l����ƁA
\[
\lim_{h\to 0} \frac{f(h)-f(0)}{h} =\lim_{h\to 0} 
\{h\sin (h^{-2}) +1\}
=1 
\]
��茴�_�ł������\�ŁA����$f'(0)=1$�B

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(b) $f'(0)>0$�͏�Ŏ������B���ܔC�ӂ�$\e>0$�ɑ΂��āA
$n>\frac{1}{2\pi {\e}^2}$���݂����悤�Ȏ��R��$n$��I���
$a=\frac{1}{\sqrt{2n\pi}}$
�Ƃ����B���̂Ƃ�
$a\in (0,\e)$�ł���A�܂�
$f'(a)=1-\sqrt{8n\pi}<0$�ƂȂ邩��A$f(x)$��$(-\e,\e)$��ŒP�������łȂ��B

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\noindent�E[1] (ii)�́A2�����������d���������ǂ����ŏꍇ�������K�v�ł��B

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\noindent�E[3] (a)�ŁA
$f(x)$��$x\neq 0$�Ŕ����\�A����$x\in\R$�ŘA���ł����Ă��A
���_�ł̔����\���Ɋւ�����͕�����܂���B�܂��A��ʂ�
�S�Ă̓_��$f'(x)$��
���݂����Ƃ��āA���ꂪ
$x$�̘A���֐��ɂȂ�Ƃ͌���܂���(�{��͕s�A���ȃP�[�X�ł�)�B 

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\begin{flushright}(�𓚗�쐬�F�O�p)
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