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Formula Matematika

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• 1. MA205 Matematika Teknik I (3 sks) Dosen: Ir. Sihar, MT. Departemen Sistem Komputer – Fak. Teknik Bandung 2003 Referensi: [1]. Ayres, F., Mendelson, E. Calculus - 5th edition. Schaum's Series. McGraw-Hill. 1999. [2]. Edwards, L. Calculus-9th edition. Cengage Learning. 2010. [3]. Simamora, S.N.M.P. “Tuntunan JavaScript”. Dept. T. Informatika, Fak. Teknik. ITHB. Bandung. 2002. [4]. Walker, J. Fundamentals of Physics - 9th edition. John Wiley & Sons. 2003. Formula Matematika Jika diketahui: ax2 + bx + c = 0 maka, x1,2 = ି௕േඥሺ௕ మିସ.௔.௖ሻ ଶ.௔ contoh: ݂ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬ଶ െ 4‫ ݔ‬െ 32 maka: x2 – 4x – 32 = 0 ; (x-8)(x+4)=0 x1 = 8 ; x2 = -4 menggunakan rumus abc: x1 = x2 = ିሺିସሻାඥሺିସሻమ ିሺସሻሺଵሻሺିଷଶሻ ሺଶሻሺଵሻ ିሺିସሻିඥሺିସሻమ ିሺସሻሺଵሻሺିଷଶሻ ሺଶሻሺଵሻ =8 = -4 Kode dalam JavaScript adalah sbb: <script language=JavaScript> aa=window.prompt("Tentukan konstanta a:","0"); a=parseFloat(aa); bb=window.prompt("Tentukan konstanta b:","0"); b=parseFloat(bb); cc=window.prompt("Tentukan konstanta c:","0"); c=parseFloat(cc); b1=Math.pow(b,2); d1=b1-4*a*c; d=Math.pow(d1,0.5); x1=(-1*b+d)/(2*a); x2=(-1*b-d)/(2*a); document.write("f(x) = ",a,"x<sup>2</sup> + ",b,"x + ",c,"<br>"); document.write("x<sub>1</sub> = ",x1,"<br>"); document.write("x<sub>2</sub> = ",x2); </script> 1
• 2. Turunan Trigonometri: ݀ sin ‫ ݔ‬ൌ cos ‫ݔ‬ ݀‫ݔ‬ ݀ cos ‫ ݔ‬ൌ െ sin ‫ݔ‬ ݀‫ݔ‬ Dalam kode JavaScript sbb: <script language=JavaScript> x=window.prompt("Tentukan sin-x atau cos-x:","sin-x"); if(x=="sin-x") { document.write("Turunannya = cos-x"); } else { document.write("Turunannya = -(sin-x)"); } </script> Tabel sudut-sudut Sudut (θo) 0 30 45 60 90 120 180 225 360 istimewa dalam nilai trigonometri sin(θo) cos(θo) 0 1 0.5 0.8660 0.7071 0.7071 0.8660 0.5 1 0 0.8660 -0.5 0 -1 -0.7071 -0.7071 0 1 tan(θo) 0 0.5774 1 1.7321 +∞ -1.7320 -∞ 1 0 Kode dalam JavaScript sbb: Untuk mencari sin(θo) : <script language=JavaScript> derajat=(2*(Math.PI))/360; x=window.prompt("Pilihan sudut?","0"); teta=parseFloat(x); x1=teta*derajat; sinx=Math.sin(x1); document.write(sinx); </script> Untuk mencari cos(θo) : <script language=JavaScript> derajat=(2*(Math.PI))/360; x=window.prompt("Pilihan sudut?","0"); teta=parseFloat(x); x1=teta*derajat; cosx=Math.cos(x1); document.write(cosx); </script> Untuk mencari tan(θo) : <script language=JavaScript> derajat=(2*(Math.PI))/360; x=window.prompt("Pilihan sudut?","0"); teta=parseFloat(x); 2
• 3. x1=teta*derajat; tanx=Math.tan(x1); document.write(tanx); </script> Bentuk eksponen dan logaritma: Bentuk eksponen ab = c 32 = 9 210 = 1024 105 = 100000 Bentuk logaritma log a c = b ⇔ alog c = b 3log 9 = 2 2log 10log 1024 = 10 100000 = 5 Jika e = Bil. Eksponensial, maka: ex = x ௗ ௗ௫ ݁ ௫ = ex dan ‫ ݁ ׬‬௫ ݀‫ ݁ = ݔ‬௫ ܽ ௫ ൌ ݁ ሺ୪୬ ௔ሻ௫ sehingga, untuk a 1 dan u merupakan fungsi diffrensial dari x: ௗ [ܽ ௫ ] = ሺln ܽሻܽ ௫ ௗ௫ ௗ௨ ௗ [ܽ௨ ] = ሺln ܽሻܽ௨ ௗ [log ܽ ௫ ] = ሺ୪୬ ௔ሻ௫ ௗ [log ܽ௨ ] = ሺ୪୬ ௔ሻ௨ ௗ௫ ௗ௫ ௗ௫ ௗ௫ ௗ௫ ଵ ଵ . ௗ௨ Contoh berikut dalam kode JavaScript untuk mendapatkan nilai e (=2.7182818): <script language=JavaScript> document.write("Bilangan eksponensial (e) = ",Math.E); </script> Integral Trigonometri ‫ ׬‬sin ‫- = ݔ݀ ݔ‬cos x ‫ ׬‬cos ‫ = ݔ݀ ݔ‬sin x ሬ Ԧ Untuk θo merupakan sudut di antara dua garis vektor a, dan b (ܽ dan ܾ)maka: Ԧ ሬԦ = ܾ • ܽ = ܽ௫ ܾ௫ + ܽ௬ ܾ௬ + ܽ௭ ܾ௭ = ܽ. ܾ. cos ߠ ሬԦ Ԧ ܽ •ܾ Ԧ หܽ × ܾ ห = ܽ. ܾ. sin ߠ Ԧ ሬԦ 3
• 4. Cramer’s Rule: Jika diketahui: ܽଵ‫ܾ + ݔ‬ଵ ‫ܿ = ݕ‬ଵ dan ܽଶ ‫ܾ + ݔ‬ଶ ‫ܿ = ݕ‬ଶ dimana x dan y masing-masing adalah variabel yang dicari, maka: ‫ݔ‬ ௖ ௕భ ฬ భ ฬ ௖ ௕ ൌ ௔మ ௕మ భ ฬ ฬ భ ௔మ ௕మ dan ‫ ݕ‬ൌ ௔ ቚ௔భ మ ௔ ฬ భ ௔మ ௖భ ௖మ ቚ ௕భ ฬ ௕మ ௖భ ௕మ ି௖మ ௕భ = ௔భ ௕మ ି௔మ ௕భ = ௔భ ௕మ ି௔మ ௕భ ௔భ ௖మ ି௔మ ௖భ 4