Given S = 50:
Calculate:√50 using the Newtons Method
Build Newtons Method
The square root of a number can be represented
ƒ(x) = x2 - S
Take the Derivative of this
ƒ'(x) = 2x
Since the square root > 0, start with x0 = 1
Calculate x1
x1 = x0 + (ƒ(x0) - S)/ƒ'(x0)
x1 = 1 + (12 - 50)/2(1)
x1 = 1 + (1 - 50)/2
x1 =1 + -49/2
x1 = 1 + -24.5
x1 = 25.5
Calculate x2
x2 = x1 + (ƒ(x1) - S)/ƒ'(x1)
x2 = 25.5 + (25.52 - 50)/2(25.5)
x2 = 25.5 + (650.25 - 50)/51
x2 =25.5 + 600.25/51
x2 = 25.5 + 11.769607843137
x2 = 13.730392156863
Calculate x3
x3 = x2 + (ƒ(x2) - S)/ƒ'(x2)
x3 = 13.730392156863 + (13.7303921568632 - 50)/2(13.730392156863)
x3 = 13.730392156863 + (188.52366878124 - 50)/27.460784313725
x3 =13.730392156863 + 138.52366878124/27.460784313725
x3 = 13.730392156863 + 5.0444177849648
x3 = 8.685974371898
Calculate x4
x4 = x3 + (ƒ(x3) - S)/ƒ'(x3)
x4 = 8.685974371898 + (8.6859743718982 - 50)/2(8.685974371898)
x4 = 8.685974371898 + (75.446150789269 - 50)/17.371948743796
x4 =8.685974371898 + 25.446150789269/17.371948743796
x4 = 8.685974371898 + 1.4647838975668
x4 = 7.2211904743312
Calculate x5
x5 = x4 + (ƒ(x4) - S)/ƒ'(x4)
x5 = 7.2211904743312 + (7.22119047433122 - 50)/2(7.2211904743312)
x5 = 7.2211904743312 + (52.145591866571 - 50)/14.442380948662
x5 =7.2211904743312 + 2.1455918665711/14.442380948662
x5 = 7.2211904743312 + 0.14856219858747
x5 = 7.0726282757437
Calculate x6
x6 = x5 + (ƒ(x5) - S)/ƒ'(x5)
x6 = 7.0726282757437 + (7.07262827574372 - 50)/2(7.0726282757437)
x6 = 7.0726282757437 + (50.022070726849 - 50)/14.145256551487
x6 =7.0726282757437 + 0.022070726849137/14.145256551487
x6 = 7.0726282757437 + 0.0015602917323416
x6 = 7.0710679840113
