Taka (matematik)

Dalam matematik, taka iya nya nilai ti disemak fungsyen (tauka turu) lebuh argumen (tauka indeks) semak enggau sekeda nilai.^[1] Taka fungsyen beguna amat ba kalkulus enggau analisis matematik, lalu dikena meri definisyen ngagai kontinuiti, derivatif, enggau integral. Konsep taka turu digeneralisasyenka agi ngagai konsep taka jaring topologi, lalu bekaul rat enggau taka enggau taka terus dalam teori kategori. Taka inferior enggau taka superior meri generalisasyen konsep taka ti kelebih agi relevan lebuh taka ba sesebengkah titik ti engka nadai.

Notasyen

Dalam formula, taka fungsi selalu iya ditulis baka tu:

\lim _{x\to c}f(x)=L,

lalu dibacha "taka f ari x lebuh x semak enggau c sama enggau L". Reti nya nilai fungsyen f ulih digaga sepeneka ati semak enggau L, enggau chara milih x chukup semak enggau c. Tauka, fakta ti madahka fungsyen f semak enggau taka L lebuh x semak enggau c kekadang ditanda ngena anak panah kanan (→ tauka $\rightarrow$ ), baka:

f(x)\to L{\text{ as }}x\to c,

ti betulis " $f$ ke $x$ nuju ngagai $L$ lebuh $x$ nuju ngagai $c$ ".

Sejarah

Nitihka Hankel (1871), konsep moden pasal taka bepun ari Proposisyen X.1 ari Elemen Euclid, ti nyadika pelasar Chara penanggul ti ditemu dalam Euclid enggau Archimedes: "Dua iti magnitud ti enda sebaka benung ditetapka, enti ari ti besai agi bisi dikurangka magnitud ti besai agi ari setengah iya, lalu ari ti tinggal magnitud ti besai agi ari setengah iya, lalu enti proses tu diulang belama, dia deka bisi tinggal sekeda magnitud ti kurang ari magnitud ti mit agi ti ditetapka."^[2]^[3]

Grégoire de Saint-Vincent meri definisyen keterubah taka (ujung) siri geometri dalam karya iya Opus Geometricum (1647): "Ujung pengerembai nya pengujung siri, ti nadai pengerembai ulih dijapai, taja pan ukai enti iya diteruska dalam infiniti, tang ti ulih disemak iya semak agi ari segmen ti diberi."^[4]

Dalam Scholium ngagai Principia dalam taun 1687, Isaac Newton bisi definisyen ti terang pasal taka, madahka "Nisbah ti pemadu tinggi nya... ukai nisbah kuantiti ti pemadu tinggi, tang taka... ti ulih disemak sida enggau naka pemayuh nyentukka bida sida kurang ari sebarang kuantiti ti diberi". ^[5]

Definisyen moden pasal taka nya datai ari Bernard Bolzano ti, dalam taun 1817, ngaga pelasar teknik epsilon-delta dikena meri definisyen fungsyen ti terus. Taja pia, pengawa iya mengkang enda ditemu bala pakar matematik bukai nyentukka tiga puluh taun udah iya mati.^[6]

Augustin-Louis Cauchy dalam taun 1821,^[7] lalu ditangkanka Karl Weierstrass, ngeformalka definisyen taka fungsyen ti dikelala enggau nama definisyen (ε, δ) taka.

Notasyen moden ti ngengkahka anak panah ba baruh lambang taka nya ketegal G. H. Hardy, ti udah mantaika iya dalam bup iya A Course of Pure Mathematics dalam taun 1908.^[8]

Kereban sanding

↑ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8.
↑ Schubring, Gert (2005). Conflicts between generalization, rigor, and intuition: number concepts underlying the development of analysis in 17th-19th century France and Germany. New York: Springer. pp. 22–23. ISBN 0387228365.
↑ "Euclid's Elements, Book X, Proposition 1". aleph0.clarku.edu.
↑ Van Looy, Herman (1984). "A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sancto Vincentio (1584–1667)". Historia Mathematica (in Inggeris). 11 (1): 57–75. doi:10.1016/0315-0860(84)90005-3.
↑ Rowlands, Peter (2017). Newton and the Great World System (in Inggeris). World Scientific Publishing. p. 28. doi:10.1142/q0108. ISBN 978-1-78634-372-7.
↑ Felscher, Walter (2000), "Bolzano, Cauchy, Epsilon, Delta", American Mathematical Monthly, 107 (9): 844–862, doi:10.2307/2695743, JSTOR 2695743
↑ Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (Ninth ed.). Brooks/Cole, Cengage Learning. ISBN 978-0-547-20998-2.
↑ Miller, Jeff (1 December 2004), Earliest Uses of Symbols of Calculus, archived from the original on 2015-05-01, retrieved 2008-12-18

[1] Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8.

[2] Schubring, Gert (2005). Conflicts between generalization, rigor, and intuition: number concepts underlying the development of analysis in 17th-19th century France and Germany. New York: Springer. pp. 22–23. ISBN 0387228365.

[3] "Euclid's Elements, Book X, Proposition 1". aleph0.clarku.edu.

[4] Van Looy, Herman (1984). "A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sancto Vincentio (1584–1667)". Historia Mathematica (in Inggeris). 11 (1): 57–75. doi:10.1016/0315-0860(84)90005-3.

[:17-5] Rowlands, Peter (2017). Newton and the Great World System (in Inggeris). World Scientific Publishing. p. 28. doi:10.1142/q0108. ISBN 978-1-78634-372-7.

[6] Felscher, Walter (2000), "Bolzano, Cauchy, Epsilon, Delta", American Mathematical Monthly, 107 (9): 844–862, doi:10.2307/2695743, JSTOR 2695743

[Larson-7] Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (Ninth ed.). Brooks/Cole, Cengage Learning. ISBN 978-0-547-20998-2.

[8] Miller, Jeff (1 December 2004), Earliest Uses of Symbols of Calculus, archived from the original on 2015-05-01, retrieved 2008-12-18

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