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{\large ���wI���K�@��U��@2007�N6��19���z�z} 

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�S�� ���n����CTA �O�p �~
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{\tt http://www.ms.u-tokyo.ac.jp/$\sim$hirachi/courses/sugaku-I-2007/}
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�ȉ��̖����ł���͈͂ŉ���, �U���Q�U���P�R���܂łɃA�h�~�j�X�g���[�V�������̃��|�[�g��o�{�b�N�X�ɒ�o���邱��. 
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{\bf ���} $f(x)$��$(-1,1)$�Œ�`���ꂽ$C^n$�֐��Ƃ���B
$
f(x)=\sum_{j=0}^{n-1} a_jx^j+O(x^n)
$
�̂Ƃ� $F(x)=\int_0^x f(y)dy$��$x=0$�ł�$n+1$���܂ł�Taylor�W�J��
$$F(x)=\sum_{j=0}^{n-1} \frac{1}{j+1}a_jx^{j+1}+O(x^{n+1})
$$
�ł���B
(�� $(\tan^{-1}x)'=(1+x^2)^{-1}$��p����$\tan^{-1}x$��Taylor�W�J�����߂�)

\prob �Q�߉�͂𗘗p���Ĉȉ��̊֐��̌��_�ɂ�����Taylor�W�J�����߂�B

(i)$ \ e^x\sqrt{1-x} $ \qquad ($x^3$�̍��܂Łj

(ii)$ \ \sqrt{\cos x} $ \qquad\ \ \ ($x^4$�̍��܂Łj


\prob  ���̊֐��̌��_�ɂ�����10���̔����W�� $f^{(10)}(0)$ �����߂�.

$$\displaystyle f(x)=\log \frac{1+x^2}{1-x^2} $$



\prob $f$ �� $ (0,\infty)$ �Ŕ����\�Ȋ֐��Ƃ���B���鐳�̎���$c$�����݂��āA�C�ӂ̎��� $x\in (0,\infty) $�ɑ΂��āA$|f'(x)|\ge c$ �𖞂����Ȃ�΁A$\displaystyle\lim_{x \rightarrow \infty} f(x)=\infty$ �ł��邩�܂���
$\displaystyle\lim_{x \rightarrow \infty} f(x)=-\infty$ �ł��邱�Ƃ������B

\prob $f(x)$��$(a,b)$�ŘA������$(a,c)\cup (c,b)$�ɂ����Ĕ����\�Ƃ���B
$\lim_{x\to c}f'(x)$�����݂����$f$��$c$�ɂ����Ă������\�ł��邱�Ƃ������B


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