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kalkulus

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kalkulus

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  1. KALKULUS I(2 sks) Oleh: MuchammadAbrori, S.Si., M.Kom.

  2. PENDAHULUAN SistemBilangan Real • Himpunanadalahsekumpulanobjek/unsurdengankriteria/syarattertentu. • Himpunansemuabil. asli N = {1,2,3,…} • Himpunansemuabil. bulat Z = {…,-3,-2,-1,0,1,2,3,…} • Bil. rasionaladalahbilygmerupakanhasilbagibilbulatdanbilasli→ Q = {a/b: a Є Z dan b Є N} • Bil. irasional, contoh: , π • Himpbil real R beranggotakanbilirasionaldanrasional, contoh: ½, 1/3, 7/66

  3. Sifat-sifatBilangan Real Untuka, b, c, dЄRberlakusifat: • Komutatif • Asosiatif • Distributif • (i). (ii). (iii). • (i). a.(-b) = (-a).b= -(a.b) (ii). (-a).(-b) = a.b (iii). -a.(-a) = a2 • (i). (ii). (iii). • Hukumkanselasi • Sifatpembaginol

  4. Pertidaksamaan Pertidaksamaan (inequality) adalahpernyataanmatematis yang memuatsatuperubahataulebihdansalahsatutandaketidaksamaan (<, >, ≤, ≥). MenyelesaikanPertidaksamaan Contoh: 1). 2x – 5 < 5x + 7 2). x2 – 5x + 6 > 0 3). x3 – 2x2 – x + 1 ≤ -1

  5. NilaiMutlak (Absolute Value) Nilaimutlaksuatubiladalahpanjang/jarakbiltsbdaribil 0. Didefinisikan: atau Sifat 1: Jikaa ≥ 0, maka |x| = a ↔ x = aataux = -a. Contoh: 1). |x| = 4 berartix = 4 ataux = -4 2). |2x| = 7 ↔ 2x = 7 atau 2x = -7 ↔ x = 7/2 atau x = -7/2 3). |3x – 5| = 6 ↔ 3x – 5 = 6 atau 3x – 5 = -6 ↔ … Sifat 2: Jika a ≥ 0, maka (a). |x| ≤ a ↔ -a ≤ x ≤ a (b). |x| ≥ a ↔ x ≤ -a atau x ≥ a

  6. ContohSoal: 1). Selesaikan |2x – 3| ≥ 7 2). Tentukansemuanilai x sehingga 3). Tentukanpenyelesaianpertidaksamaan

  7. PertidaksamaanBentukAkar • Cara penyelesaiannya: √(ax + b) < c Syarat: ax + b >= 0 Ruaskiridankanandikuadratkan ax + b < c2

  8. Soal Tentukan hp dari: • √(2x -6) < 4 • (x + 2)4(x – 3)5 > 0 • (x – 1)/(x + 2) >= (3 – x)/(x + 2) • |x – 2|2 < 4|x – 2| + 12 • √(x2 – x) < √2

  9. Selang (Interval) Diberikansebarangduabil real a dan b, dengan a < b. Berturut-turutdidefinisikan: • [a,b] = {x| a ≤ x ≤ b} • [a,b) = {x| a ≤ x < b} • [a,∞) = {x| x ≥ a} • (-∞,a] = {x| x ≤ a} • (a,b) = {x| a < x < b} • (a,b] = {x| a < x ≤ b} • (a,∞) = {x| x > a} • (-∞,a) = {x| x < a}

  10. SELESAI

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