Ť

Ť Ң˹觢ͧԵʵ觾ѲҨҡժԵâҤԵ ԴҡѹͧͧǤԴѡ ǤԴá Ťԧ͹ؾѹ (Differential Calculus) ͷɮշҴѵҡ¹ŧ ǢͧѺ͹ؾѹ觾ٴ֧ѧѹҧԵʵ, , Фѹͧ ش˹ ǤԴͧ ŤԧԾѹ (Integral Calculus) ǢͧѺǤԴ㹡Թõ ǤԴ鹰ҹǡѺ鹷¡ҿͧѧѹ ǤԴǡѺҵ

ҡͧǤԴԴ ѧѹԹ ͸Ժ´ ɮպŰҹͧŤ Щй 㹡͹ŤʤáǶ֧ͧҧ͹ ֡㹻ѨغѹѡСǶ֧Ťԧ͹ؾѹ͹

ѵ (History)
鹡ԴͧŤԧԾѹ͹件֧ؤաҳ ⴫ ѡ繷ѡѹ㹹ͧ鹾 Method ofExhaustion 觷öӹdzҾ鹷лҵ Դ ѲԸաù ѲԸաê¤ӹdz觤¤֧ѺǤԴ㹻Ѩغѹ źԫ ǵѹ ѡѺѺ繼ԴŤʢ ੾Сä鹾ɮպŰҹͧŤ ա§ѹҹǵѹźԫ繼鹾ǤԴѡͧŤʡ͹ ԧ 觷˭شźԫѲѺŤʤͧ¢ͧ ѡѹ觤Դ֧ѭѡɳ᷹ǤԴҧԵʵ ҧá §ѹҧźԫ ǵѹ ¡ѡԵʵٴѧ͡ҡѡԵʵûҹҹ» 觷餳Եʵѧѧûҹҹ ͧ·ǵѹ鹤ͧǹ¡Ңͧ źԫҧѴ ѧѹѧɨ Analytical Society ͧ¢ͧźԫȵɷ 19 ͹ ѹɰҹѹҹǵѹ鹾ǤԴǡѺŤʡ͹ ҧá źԫ繼͹ءѹ繷͡ѹҷ駹ǵѹźԫҧ鹾Ťʴµͧ繼ѲԪŤʹ͡ҡ Barrow, Descartes, de Fermat, Huygens Wallis੾ de Fermat 觺ҧѺ¡ͧ ԴŤԧ͹ؾѹ ѡԵʵǭ Kowa SekiժԵ㹪ǧǡѹѺ źԫ йǵѹ ѧ鹾ѡþ鹰ҹҧҧǡѺ ŤԧԾѹ 繷ѡšѹ㹢й ҡԴ͡ѺѡԪҡêǵѹ

Ťԧ͹ؾѹ (Differential Calculus)͹ؾѹ (derivative) ͡ҤҤ¹ŧͧ˹ ա˹¹ŧ㹻ҳҡ ҧ͹ؾѹ龺áç¹ٵ ѵ=зҧ/Ѻѵط͹ѵǤ ѵǢͧس͹ؾѹ͡¹ŧ˹˹ ԪŤʾѲҢͨѴáѺѭҷѺ͹繸ҵԡҹ ѵǢͧسҨ¹ŧ

ҡǶ֧´ Ťԧ͹ؾѹ ѵҡ¹ŧ㹢㴢˹ (͹ؾѹ) ҧҢͧѧѹ Ѻâͧѧѹ ԧͧ͹ؾѹԵͧѵǹ㹡û¹ŧ ͹ؾѹ㨢ͧԷʵҾ ͹ͧǵѹ ç= x դŤ ͹ؾѹ˹ ɮ俿Ңͧ зɮçǧͧ͹䵹 (ѷҾ) Ƕ֧ҢͧŤԧ͹ؾѹ ǡѹѺɮվ鹰ҹͧǧ俿

͹ؾѹͧѧѹǶ֧ҿͧѧѹ㹪ǧ 觷öҨش٧شШششͧѧѹ ҷشҹ鹡ҿТҹѺ᡹Һ ù Ťѧաûءա ºԸբͧǵѹ Ը㹡Ҥҡͧѧѹ¡ûҳ ѧŤԧ͹ؾѹ֧ö任ءѺҡ¤Ӷ觶ͧԹҨԴҨŤʨѴ

ŤԧԾѹ (Integral Calculus)
ŤԧԾѹ֡ԸաһԾѹ ( Թԡ,Integral ) ͧѧѹ ҨҡԵͧͧ(¡Եͧѹ) о鹤;鹷׹ᶺҿͧѧѹ ԹõԸշԸ˹㹡Ҿ鹷ҿ о鹷 лҵâͧ蹷çзçк͡

鹰ҹͧŤ (Foundation)
鹰ҹ觤ѴͧŤհҹҨҡǤԴͧѧѹԵ ѹ෤ԤͧժԵ鹰ҹСػԧԵʵ ֡Ҿ鹰ҹͧŤѡѹ㹪͡ԧԧ 觻Сͺ¹觤Ѵк٨ͧɮբͧŤ 蹷ɮաѴ Сԧѧѹ

ɮպŰҹͧŤ (Fundamental theorem of calculus)
ɮպŰҹͧŤʡҡ͹ؾѹ СһԾѹ ԸաáѺѹ ҡ´ ҹؾѹöӹdz¡һԾѹӡѴࢵԧ͹ҡ駹ǵѹźԫ 繡حᨹâ¼Ѿԧҧҡѧҡҹͧͧ繷ѡ §ö͹¹ŧ㹿ѧѹ㹪ǧ˹觨ҡѵҡ¹ŧ㹢㴢˹ ¡һԾѹͧǹѧ ɮպŰҹѧԸ㹡äӹdzһԾѹࢵԸշҧժԵ繨ӹǹҡ ͧԸաԵ ¡һҹؾѹɮպѧ͹حҵԧ͹ؾѹ 觤÷Ǣͧѹҧ ѧѹҺ ͹ؾѹͧѹ
ԧ͹ؾѹԷʵ
ɮպŰҹͧŤ 1: ͹ؾѹ(Ծѹ(f(x))) ͧ x x
ɮպŰҹͧŤ 2: Ծѹ(f(x)) b 繢ͺࢵ a 繢ͺࢵҧ ҡѺ F(b)-F(a) F(x) 繻ҹؾѹͧ f(x)

ûء (Application)
þѲСŤ¼᷺ءǹͧԵؤ ѹ繾鹰ҹͧԷʵͺءҢ੾ ԡ þѲͺ ෤Ԥáҧ úԹ ෤ͺ վ鹰ҹҨҡŤ

Ť ԧ͹ؾѹ Ťǡ Ťʢͧ¹ŧ ԧ͹ Ťԧ Ťʡԡѹ ;ԧ͹ؾѹ


Koalarӵ鹩ѺҨҡwww.learner.org::

VictorJ.Katz繹ѡԵʵѺԭͧMAA(MathematicalAssociationofAmerica:)ҧա֡25372538仺·ԷDistrictofColumbiaҢͧҫժѵԢͧԪҤԵʵ:ͧѺõվӹѡHarperCollins..1993

Ӷҷ͹ؾѹͧѧѹ䫹֧դҡѺѧѹ䫹?ͤӶ˵͹ؾѹͧѧѹ᷹ਹ֧դҡѺѧѹ૤᤹¡ѧͧ?ӵͺ˹觫觤س¹ԪŤʡ͡ҧҹö٨㹤繨ԧǤ٢ͧسҨо٨Ӷáҡͧ͹ؾѹй蹡س =1

ö٨Ӷͧ¡顮͹ؾѹͧѧҡҨйѧѹ⡳ԵԧâҤԵѹ觨зسͧǢͧ͹ؾѹԧâҤԵ觢ж͹Ѻ仾ԨóһѵԢͧѧѹ⡳ԵҹѺѹͧѵԪŤʡзسǡѹ

ءѹѡԨóҿѧѹ䫹пѧѹ⡳ԵԵ繿ѧѹԧŢ(numericalfunctions)ͧӹǹԧ觤ҨԨóͧѴ訹з觶֧ؤͧ㹪ǧҧȵɷ18ҿѧѹ䫹١Ԩó繡ͧ鹵çǧյ˹鹵çǢͧѺǢͧѺ줹ͧ(ͧ䫹֧դҡѺ줢ͧ䫹˹Ѻѧѹѹͧѧѹ䫹)ѧҡٻҹسy=sintt繡Ѵͤдǡ㹡þԨóբͧǧҡѺ1˹㹷ӹͧǡѹx=cost

ҵͧ͹ؾѹͧѧѹ䫹ºѺ㹪ǧȵɷ18÷Ẻ¤Ҩѵǹͧ¹ŧդҹҡͧy=sintºѺt觹ҡ蹡ѹѧ¹ŧҡͧtö᷹¡Ҵٻǧդҹҡѧǧش줢ͧtҾԨó᷹ਹѧ繴ҹçҡͧٻҡҧѧ鹴觹᷹dy᷹¹ŧҡͧ䫹ͧҡٻ¡Ѻٻ˭¡ͧöʴ = ¹աٻ˹ = ͧ

ԧâҤԵ͹ؾѹͧѧѹ䫹᷹ਹѧǹҡ繤á㹡õվԴѧҡö֧ͧRogerCotes(16821716)繺óҸԡ㹡þ駷2ͧNewtonsPrincipiaѺûҡǢͧѧ㹵ԪŤ繤á鹤˹ѧͷANewTreatiseofFluxions(1737)ThomasSimpson(17101761)ժ§ҨءѹҡͧSimpsonѺҤԹԡԧŢ¡ûҳẺSimpson˹㹡ͧ͡㹻ѧɫ龺Ѻͧ÷ٹ鹢ͧ¹дѺҧԪҤԵʵ͡õҴѧǷSimpsonҪԡҤԵʵSpitalfields觡ͧSimpsonѺþԨóҧҧҧҡҪԡءҨж١ӶԧԵʵԧѪҪԡҹͷЪѡɳзºдǡشҷз

ҨNewtonLeibnizԨó͹ؾѹͧѧѹ⡳Եǡҡǹ㹡äԴ͹ءѧԧ͹ؾѹҧLeibnizٻҾǡѹѺٻԧ͹ؾѹػ = ػͧҡx= dx= =

dtҧѧ

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