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Newtons Method sqrt(50)

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Given S = 50, calculate √50 using the Newtons Method

The square root of a number can be represented by the function ƒ(x) = x2 - S
Taking the Derivative of this, we have ƒ'(x)=2x
Since the square root is always positive, we will start with an x0 value of 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

x6 = 7.0710679840113




What is the Answer?

x6 = 7.0710679840113

How does the Newton Method Calculator work?

Calculates the square root of a positive integer using the Newton Method
This calculator has 1 input.

What 3 formulas are used for the Newton Method Calculator?

  1. ƒ(x) = x2 - S
  2. ƒ'(x) = 2x
  3. xn = xn - 1 + (ƒ(xn - 1) - S)/ƒ'(xn - 1)

For more math formulas, check out our Formula Dossier

What 3 concepts are covered in the Newton Method Calculator?

algorithm
A process to solve a problem in a set amount of time
newtons method
another numerical method for solving an equation f...
square root
a factor of a number that, when multiplied by itself, gives the original number
√x

Example calculations for the Newton Method Calculator

  1. Newton method sqrt(50)

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