àà¤Å¤ÙÅÑÊ
á¤Å¤ÙÅÑÊ(Calculus)
1.ÿ X ® a
ÅÔÁÔµ¢Í§¿Ñ§¡ìªÑ¹ÿ à¢Õ¹᷹´éÇÂÿ ÿÿlimÿ ÿf(x)ÿ =ÿ L
ËÁÒ¶֧ÿ x ÁÕ¤èÒà¢éÒã¡Åé a ÿ(x ® a)ÿ áÅéÇÿ f(x) ¨ÐÁÕ¤èÒà¢éÒã¡Åé L
ÿÿÿÿÿ ÇÔ¸ÕËÒÿ ¤èÒÅÔÁÔµ¢Í§¿Ñ§¡ìªÑ¹
(1).ÿ àÍÒ¤èÒ a ä»á·¹ã¹ x ã¹ f(x) ¶éҼŷÕèä´éà»ç¹¨Ó¹Ç¹¨ÃÔ§¤èÒ¹Ñ鹤×Í ¤èÒÅÔÁÔµ
ÿ ÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿÿãËé¾Ô¨ÒóÒÅѡɳТͧ¿Ñ§¡ìªÑ¹ ´Ñ§¹Õé
(2.1)ÿÿ ¶éÒÊÒÁÒöá¡ f(x) ÍÍ¡à»ç¹¼Å¤Ù³¢Í§µÑÇ»ÃСͺä´é ¡çãËéá¡áÅéÇ¢¨Ñ´µÑÇ»ÃСͺÃèÇÁ¢Í§àÈÉáÅÐÊèǹÍÍ¡ ËÅѧ¨Ò¡¹Ñ鹡çàÍÒ¤èÒ a ä»á·¹ x ¶éҼŷÕèä´éà»ç¹¨Ó¹Ç¹¨ÃÔ§ ¤èÒ¹Ñ鹤×ͤèÒÅÔÁÔµ
(2.2)ÿÿ
¡çãËé¹Ó¤Í¹¨Ùࡵ¤Ù³·Ñé§àÈÉáÅÐÊèǹÿ áÅéÇ¢¨Ñ´µÑÇ»ÃСͺ·Õè·ÓãËéÊèǹà»ç¹ÈÙ¹ÂìÍÍ¡ÿ ËÅѧ¨Ò¡¹Ñ鹡çàÍÒ¤èÒ a ä»á·¹ x ¶éҼŷÕèä´éà»ç¹¨Ó¹Ç¹¨ÃÔ§ ¤èÒ¹Ñ鹤×ͤèÒÅÔÁÔµ
2.ÿ ¤ÇÒÁµèÍà¹×èͧ¢Í§¿Ñ§¡ìªÑ¹ÿÿÿ ã¹·Ò§¤³ÔµÈÒʵÃìµÃǨÊͺÇèÒ f ¨ÐµèÍà¹×èͧ·Õè
x = a ËÃ×ÍäÁè¹Ñé¹ÿ µéͧµÃǨÊͺ¨Ò¡¤Ø³ÊÁºÑµÔÿ 3ÿ ¢é͵èÍ仹Õé
1.ÿÿÿ
x® a
ÿÿÿÿÿÿÿ 2.ÿ ÿlim f(x)ÿ ÿËÒ¤èÒä´é
x® a
3. ÿlim f(x)ÿ =ÿ f(a)
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ f(x1-h) - f(x1)
4. ÍѵÃÒ¡ÒÃà»ÅÕè¹á»Å§ ¢Í§ y = f(x) ³ x = x1
h®0
ÿÿÿÿÿÿÿ limÿ f(x+h) - f(x)ÿÿ ¤×Í ÍѵÃÒ¡ÒÃà»ÅÕè¹á»Å§ ¢Í§ y = f(x) ³ x ã´ æ
5.ÿÿÿ ͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹ fÿÿ á·¹´éÇÂÿ f /(x)ÿ ËÃ×Íÿ dy/dx
h®0
ÿ ÿÿÿÿÿÿÿÿÿÿÿh
6.ÿÿÿ ÊÙµÃ㹡ÒÃËÒ͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹
Êٵ÷Õè 2. ¶éÒ y = f(x) = xÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿ dy/dxÿ = f/(x)ÿ = 1
Êٵ÷Õè 3. ¶éÒ y = f(x) = xnÿ àÁ×èÍ n à»ç¹¨Ó¹Ç¹¨ÃÔ§ÿÿÿÿÿ dy/dxÿ = f/(x)ÿ =nxn-1
Êٵ÷Õè 4. ¶éÒ y = f(x) = g(x) + h(x)ÿ ÿÿÿÿÿÿÿ dy/dxÿ = g/ (x) + h/ (x)
Êٵ÷Õè 5. ¶éÒ y = f(x) = g(x) - h(x)ÿÿ ÿÿÿÿÿÿÿ dy/dxÿ = g/ (x) - h/ (x)
Êٵ÷Õè 6. ¶éÒ y = f(x) = cg(x) ÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿ dy/dxÿ = cg/ (x)
Êٵ÷Õè 7. ¶éÒ y = f(x) = g(x) h(x)ÿÿÿÿ ÿÿÿÿÿÿÿ dy/dxÿ = g/(x)h(x)+h/ (x)g(x)
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿh(x)
ÿ
