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http://www.surin.sru.ac.th/index.php/home/50-mathshowmain/68-calculus
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ҡԡԾ ҹء
Ť ҢѡͧԵʵ觾ѲҨҡժԵ âҤԵ лѭҷҧԡ Ťյ鹡ԴҡͧǤԴѡ ѧ

ǤԴá Ťԧ͹ؾѹ (Differential Calculus) 繷ɮշҴѵҡ¹ŧ ǢͧѺ͹ؾѹͧѧѹҧԵʵ ҧ , ѹͧ ش˹. ɮբͧ͹ؾѹǹçѹ㨨ҡѭҷҧԡ

ǤԴͧ ŤԧԾѹ (Integral Calculus) 繷ɮշçѹ㨨ҡäӹdz鹷ͻҵâͧٻçҧâҤԵҧ . ɮչҿͧѧѹ᷹ٻçҧâҤԵ ɮջԾѹ (Թõ) ѡ㹡äӹdzҾ鹷лҵ

ͧǤԴԴҡѭҷҧѹѺդѹѹ֡ ɮպŰҹͧŤ ԧǷɮշͧº͹繴ҹͧͧ­ѹǡѹ 蹤ǡѹ§ͧҹ (¤ ҡ͹ؾѹлԾѹѧѹѹͧѹСѹ). 㹡͹Ťͤ㨵Ƿɮҧ֡ áǶ֧ͧɮФѹ͹ ֡㹻ѨغѹѡСǶ֧Ťԧ͹ؾѹ͹§ҧ ͧҡҹ¡

͹ ֡Ťҧ´ҵ Դʵ ҧԵʵҡ Ե ɮաѴ

ѵ

鹡ԴͧŤԧԾѹ͹件֧ؤաҳ ѡ繷ѡѹ㹹ͧ鹾 Ըաᨧó 觷öӹdzҾ鹷лҵ Դ ѲԸաù ѲԸաê¤ӹdz 觤¤֧ѺǤԴ㹻Ѩغѹ źԫ ǵѹ ѡѺѺ繼ԴŤʢ ੾Сä鹾ɮպŰҹͧŤ

᫡ ǵѹ
ͷմ źԫ

ա§ѹҹǵѹźԫ 繼鹾ǤԴѡͧŤʡ͹ ԧ 觷˭ش źԫѲѺŤ ͧ¢ͧ ѡѹ 觤Դ֧ѭѡɳ ᷹ǤԴҧԵʵ ҧá §ѹҧźԫ йǵѹ ¡ѡԵʵٴѧ ͡ҡѡԵʵû ҹҹ» 觷餳Եʵѧѧûҹҹ ͧ·ǵѹ ͧǹ¡ҢͧźԫҧѴ ѧѹѧɨ Analytical Society ͧ¢ͧźԫȵɷ 19 ͹ ѹɰҹѹ ǵѹ鹾ǤԴǡѺŤʡ͹ ҧá źԫ繼͹ ءѹ繷͡ѹ 駹ǵѹźԫҧ鹾Ťʴµͧ

繼ѲԪŤʹ͡ҡ õ, Barrow, ҵ, ੾ ҵ 觺ҧѺ¡ͧ ԴŤԧ͹ؾѹ. ѡԵʵǭ ժԵ㹪ǧǡѹѺ źԫ йǵѹ 鹾ѡþ鹰ҹҧҧǡѺ ŤԧԾѹ 繷ѡšѹ㹢й ҡԴ͡ѺѡԪҡêǵѹ

Ťԧ͹ؾѹ

͹ؾѹ (derivative) ͡ҤҤ¹ŧͧ˹ ա˹¹ŧ㹻ҳҡ ҧ͹ؾѹҨ龺áç¹ ٵ ѵ = зҧ/ Ѻѵط͹ѵǤ ѵǢͧس͹ؾѹ͡¹ŧ˹˹ ԪŤʾѲҢ ͨѴáѺѭҷѺ͹繸ҵԡҹ ѵǢͧسҨ¹ŧ

ҡǶ֧´ Ťԧ͹ؾѹ ѵҡ¹ŧ㹢㴢˹ (͹ؾѹ) ҧҢͧѧѹ Ѻâͧѧѹ ԧ ͧ͹ؾѹ Եͧѵǹ㹡¹ŧ (difference quotient). ͹ؾѹ㨢ͧԷʵҾ ͹ͧǵѹ ç = × դŤ ͹ؾѹ˹ ɮ俿ͧ ɮçǧͧ͹䵹 (ѷҾ) Ƕ֧ҢͧŤԧ͹ؾѹ ǡѹѺɮվ鹰ҹͧǧ俿

͹ؾѹͧѧѹ Ƕ֧ҿͧѧѹ㹪ǧ 觷öҨش٧ش Шشش ͧѧѹ ҷشҹ鹡ҿТҹѺ᡹Һ ù Ťѧաûءա ºԸբͧǵѹ (Newton's Method) Ը㹡Ҥҡͧѧѹ ¡ûҳ ѧŤԧ͹ؾѹ ֧ö任ءѺҡ¤Ӷ 觶ͧԹҨԴ ҨŤʨѴ

ŤԧԾѹ

ŤԧԾѹ֡ԸաԾѹ (Թԡ, Integral) ͧѧѹ ҨҡԵͧͧ (¡Եͧѹ) о鹤;鹷׹ᶺҿͧѧѹ ԹõԸշԸ˹㹡Ҿ鹷ҿ ҵͧççк͡

鹰ҹͧŤ

鹰ҹ觤ѴͧŤ հҹҨҡ ǤԴͧѧѹ Ե ѹ෤ԤͧժԵ鹰ҹ ػԧԵʵ ֡Ҿ鹰ҹͧŤ ѡѹ㹪 ԧԧ 觻Сͺ 觤Ѵ к٨ͧɮբͧŤ ɮաѴ ԧѧѹ

ɮպŰҹͧŤ ͧ

ɮպŰҹͧŤ ͹ؾѹСһԾѹԸա÷çѹѹ Ǥ ҧѧѹ繻Ծѹͧѧѹ˹觢 ͹ؾѹͧѧѹҧ ҡѺѧѹ ͡ҡ ѧһԾѹӡѴࢵ¡á˹Ѻҹؾѹ

ɮպŰҹͧŤ¹ٻѭѡɳ줳Եʵѧ: f 繿ѧѹդͧǧ [a, b] F 繻ҹؾѹͧ f ǧ [a, b]

int_{a}^{b} f (x) ,dx = F (b) - F (a)
Ѻء x 㹪ǧ [a, b]
frac{d}{dx}int_a^x f (t) , dt = f (x)

ԧ͹ҡ駹ǵѹ źԫ 繡حᨹ â¼Ѿԧҧҡѧҡҹͧͧ繷ѡ. § ö͹¹ŧ㹿ѧѹ㹪ǧ˹ ҡѵҡ¹ŧ㹢㴢˹ ¡һԾѹͧǹѧ. ɮպŰҹѧԸ㹡äӹdz ԾѹӡѴࢵ ԸշҧժԵ繨ӹǹҡ ͧԸաԵ ¡ҹؾѹ. ɮպѧ͹حҵԧ͹ؾѹ 觤÷Ǣͧѹҧ ѧѹҺ ͹ؾѹͧѹ. ԧ͹ؾѹԷʵ

ûء

þѲСŤ¼᷺ءǹͧԵؤ ѹ繾鹰ҹͧԷʵͺءҢ੾ ԡ þѲͺ ෤Ԥáҧ úԹ ͺ վ鹰ҹҨҡŤ

Ť ԧ͹ؾѹ Ťǡ Ťʢͧ¹ŧ ԧ͹ Ťԧ Ťʡԡѹ ;ԧ͹ؾѹ

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