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(Calculus)

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1.ÿ

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ÅÔÁÔµ¢Í§¿Ñ§¡ìªÑ¹ÿ à¢Õ¹᷹´éÇÂÿ ÿÿlimÿ ÿf(x)ÿ =ÿ L

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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

(2).ÿ àÍÒ¤èÒ a ä»á·¹ã¹ x ã¹ f(x)áÅéÇ»ÃÒ¡¯¼ÅÍÍ¡ÁÒã¹ÃÙ»

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 ¶éÒᡵÑÇ»ÃСͺäÁèä´é à¹×èͧ¨Ò¡ f(x) ÁÑ¡ÍÂÙèã¹ÃÙ»

¡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 ËÒ f(a)ÿ ä´é

x® a

ÿÿÿÿÿÿÿ 2.ÿ ÿlim f(x)ÿ ÿËÒ¤èÒä´é

ÿ

x® a

3. ÿlim f(x)ÿ =ÿ f(a)

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ÿ

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3.ÿ ÍѵÃÒ¡ÒÃà»ÅÕè¹á»Å§â´Âà©ÅÕ蠢ͧ yÿ ËÃ×Í f(x)ÿ 㹪èǧ x1ÿ ¶Ö§ x1+hÿÿ ¤×Í

ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ f(x1-h) - f(x1)

ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿ h

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4. ÍѵÃÒ¡ÒÃà»ÅÕè¹á»Å§ ¢Í§ y = f(x) ³ x = x1

ÿÿÿÿÿÿÿ

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ÿÿÿÿÿÿÿ limÿ f(x+h) - f(x)ÿÿ ¤×Í ÍѵÃÒ¡ÒÃà»ÅÕè¹á»Å§ ¢Í§ y = f(x) ³ x ã´ æ

ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿÿ h

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5.ÿÿÿ ͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹ fÿÿ á·¹´éÇÂÿ f /(x)ÿ ËÃ×Íÿ dy/dx

h®0

¶éÒ ÿy = f(x)ÿÿ à»ç¹¿Ñ§¡ìªÑ¹·ÕèÁÕâ´àÁ¹áÅÐàù¨ìà»ç¹ÊѺ૵¢Í§à«µ¨Ó¹Ç¹¨ÃÔ§àÃÒàÃÕ¡ limÿÿÿÿÿÿÿÿ f(x+h) - f(x)ÿÿ ·ÕèËÒä´éÇèÒ Í¹Ø¾Ñ¹¸ì¢Í§¿Ñ§¡ìªÑ¹ f ·Õè x

ÿ ÿÿÿÿÿÿÿÿÿÿÿh

6.ÿÿÿ ÊÙµÃ㹡ÒÃËÒ͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹

Êٵ÷Õè 1. ¶éÒ y = f(x) = cÿ à»ç¹¤èÒ¤§·Õèÿÿÿ dy/dxÿ = f/(x)ÿ = 0

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)

Êٵ÷Õè 8. ¶éÒ y = f(x) = ÿÿÿÿg(x) ÿÿÿÿÿÿÿ ÿÿàÁ×èÍ h(x) ¹ 0

ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿh(x)

ÿÿÿÿÿÿÿ dy/dxÿ = g/(x)h(x) - h/(x)g(x)

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