81 bukti bukti_limit_ apiq

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Asyiknya pembuktian rumus cepat trigonometri bersama matematika kreatif APIQ.

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81 bukti bukti_limit_ apiq

  1. 1. Inovasi Pembelajaran Matematika Kreatif Pembuktian Limit Terpenting
  2. 2. a b c a < b < c
  3. 3. a b c a < b < c r = 1 r = 1 jika c = x maka a < b < c x cos x < sin x < x cos x < (sin x)/x < 1 1< (sin x)/x < 1 Jadi, (Sin x)/x = 1 Atau, Sin x = x
  4. 4. Keterangan
  5. 5. a b c a < b < c r = 1 r = 1 Jika c = x maka Besar sudut = x/r = x/1 = x Sinx = b /r = b Cosx = d/r = d d = cosx Besar sudut = x = a/cosx
  6. 6. a b c a < b < c r = 1 c = x b = sinx d = cosx d = cosx a = x(cosx)
  7. 7. a b c a < b < c r = 1 x(cosx) < sinx < x d = cosx bagi dengan x, cosx < (sinx)/x < 1 karena x  0 cos(0) < (sinx)/x < 1
  8. 8. a b c r = 1 d = cosx karena x  0 cos(0) < (sinx)/x < 1 1 < (sinx)/x < 1 teori “apit” (sinx)/x = 1 sinx = x
  9. 9. Latihan
  10. 10. Hitunglah Limit x  0 Sin 3x 7x = … = 3x 7x = 3 7 2x Sin 5x = … = 2x 5x = 2 5
  11. 11. Hitunglah Limit x  0 Sin 3x Sin 8x = … = 3x 8x = 3 8 Sin 6x Sin 4x = … = 6x 4x = 3 2
  12. 12. Hitunglah Limit x  0 2x + Sin 3x 3x + Sin 8x = … = 5x 11x = 5 11 3x + Sin 6x 2x + Sin 4x = … = 9x 6x = 3 2
  13. 13. Hitunglah Limit x  0 2x + tan 5x 3x + Sin 8x = … = 7x 11x = 7 11 3x + tan 2x 2x + tan 5x = … = 5x 7x = 5 7
  14. 14. Buktikan Limit x  0 tan x x = 1 tan x = x tan x x = (sin x)/(cosx) x = (sin x)/(1) x = sin x x = 1 tan x x = 1
  15. 15. Inovasi Pembelajaran Matematika Kreatif
  16. 16. Hitunglah Limit x  0 tan 2x + tan 5x Sin 8x - tan 4x = … = 7x 4x = 7 4 Sin 3x - tan 2x Sin 2x + tan 5x = … = 1x 7x = 1 7
  17. 17. Hitunglah Limit x  0 tan 2x . tan 5x Sin 8x . tan 4x = … = 10x 2 32x 2 = 10 32 Sin 3x . tan 2x Sin 2x . tan 5x = … = 6x 2 10x 2 = 6 10
  18. 18. Untuk Limit x  0 Sin x = x = tan x Cos x = Cos 0 = 1
  19. 19. Alternatif Bukti
  20. 20. a b c Luas sektor a < Luas segitiga b < Luas sektor c r = 1 r = 1
  21. 21. a b c Luas a < Luas b < Luas c r = 1 r = 1 sudut = x ½ (cos x) 2 x < ½ cosx . sin x < ½ .1. x (cos x) 2 x < cosx . sin x < x cos x < (sin x)/x < 1/cosx
  22. 22. a b c r = 1 r = 1 sudut = x cos x < (sin x)/x < 1/cosx cos 0 < (sin x)/x < 1/cos 0 1 < (sin x)/x < 1 (sin x)/x = 1 sin x = x
  23. 23. Test Limit
  24. 24. Pengujian limit secara formal memanfaatkan delta epsilon. Tetapi anak-anak lebih suka dan mudah paham menggunakan pengujian dengan kalkulator atau komputer.
  25. 25. Uji dengan kalkulator, Limit x  0 Mis x = 0,001 Sin (0,003) Sin (0,008) = 5,236 13,96 = 0,3750 = 0,3750 Sin 3x Sin 8x = … = 3x 8x = 3 8
  26. 26. Uji dengan kalkulator, Limit x  0 Mis x = 0,001 (rad) 0,002+Sin (0,003) 0,002+Sin (0,008) = 0,00499999 0,00999991 = 0,500 = 0,5 2x + Sin 3x 2x + Sin 8x = … = 5x 10x = 5 10
  27. 27. Uji dengan kalkulator, Limit x  0 Mis x = 0,001 (rad) 0,002+tan (0,004) 0,002+Sin (0,008) = 0,00600000 0,00999991 = 0,600 = 0,6 2x + tan 4x 2x + Sin 8x = … = 6x 10x = 6 10
  28. 28. Inovasi Pembelajaran Matematika Kreatif Terimakasih [email_address] apiqquantum.wordpress.com (022)2008621

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