�L�n��򥻩w�z�]²�^

�O �O�@�Ӧb [a, b] ��϶��w�q��ȳs��ơA ��i�@�B�A�ڭ̰��]��b (a, b) �}�϶��i�L�C�O �O �b [a, b] ��n��ơC�� x �O (a, b) ��N�@�I�A �ڭ̦Ҽ{ F �b x ��ɼơG

�]��

�N��O��

�b [a, x+h] �϶��n�ȡA��

�N��O��

�b [a, x] �϶��n�ȡA�ҥH�ܩ��a�A

��N��N�O

�b x �M x+h ��n�� (�]�N�O��n)�C �ڭ̦b�o�̤��໡ [x, x+h] �϶��n�ȡA �]�� h �i��O��Ƥ]�i��O�t�ơA�ҥH��@�w x+h �b x ��k��Υ��C

�]�� h �O�n�ͪ�� 0 ��A�ҥH�i�H��]��ܤp�C�S�]�� O�i�L��A�ҥH��b�@�q�ܤp�϶��Ʀ��u�X�G�O�@��u�C �o��u�Y��O��W�A�N�O��C�]��A�Y��O

�N�O

��`��@�ӬO�諸 (�H�W�� s �u�b x �M x+h ��Ҽ{)�C �]��̪��׭z�O��@�˪��A�ҥH�ڭ̰��]�e�̦��ߡC ��A�ڭ̴N��D

�b x �M x+h ��n�A ��P�˰϶��H

��x�έ��n�j�A��P�˰϶��H

��x�έ��n�p�A�]�N�O

�]��

�ڥ��H h ��ܤơA�]�N�O��A��ܶq h �Ө��A

�O�@�ӱ`�ƨ�ơA�ҥH

�t�@�譱�A�]��

�O�s��ơA�ҥH

�{�b�A�i�H�I�s��w�z�F�C�ھڧ��w�z�A�ڭ̱o��

�o�ӵ��G��Ҧ�

��ߡA�ҥH��

�b (a, b) �϶��ɨ�ƴN�O

²�O��

�Ϊ�

�o�N�O�ڭ̪��u²�檩�v�L�n��򥻩w�z (the fundamental theorem of calculus)�C

�e��ҿת��u²�檩�v²��b��̩O�H �]�N�O��A�ҿסu��㪩�v�h�F��H ��A��㪩�N�O��A�� i�L�A�]��O��ȡA�N�i�H�ҩ��W�� (A) ��F�C ��A�n�h��u��ȡv��]�ä��x��A��O�n�h��u�i�L�v��]�N�ݭn�h�@�I��u�ҤF�C �ڭ̧O�B�A�͡C�{�b�A��M�ڭ̥u�ҩ��F²�檩�A�o��]��㪩��ߡG

��] �b [a, b] �϶��O�@�ӳs��ơA�h (A) ��ߡC

�L�n��򥻩w�z��F

�n��ƪ��ɨ�ƬO��

�]�i�H��

�n��P�L��f�B��

�٥i�H��

��u�U��n��ܤƲv�O��ƭȥ��

�o�өw�z��n�P�L��B�A�ٻݭn�ǲߪ̤�[�Ѫ��h�C�C��|�C

��D

�q�g�L�n��򥻩w�z (�ҿת��u��㪩�v)�C
�а� �b [0, 1] ��n��ƬO��H
�а� �b [1, 2] ��n��ƬO��H �p�ߡA�o��D�ئ��A�`�N�n��Ʀb x = 1 �B��O 0�C

�L�n���򥻩w�z�]²�^

���D

�L�n��򥻩w�z�]²�^

��D