Given S = 225:
Calculate:√225 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 - 225)/2(1)
x1 = 1 + (1 - 225)/2
x1 =1 + -224/2
x1 = 1 + -112
x1 = 113
Calculate x2
x2 = x1 + (ƒ(x1) - S)/ƒ'(x1)
x2 = 113 + (1132 - 225)/2(113)
x2 = 113 + (12769 - 225)/226
x2 =113 + 12544/226
x2 = 113 + 55.504424778761
x2 = 57.495575221239
Calculate x3
x3 = x2 + (ƒ(x2) - S)/ƒ'(x2)
x3 = 57.495575221239 + (57.4955752212392 - 225)/2(57.495575221239)
x3 = 57.495575221239 + (3305.7411700211 - 225)/114.99115044248
x3 =57.495575221239 + 3080.7411700211/114.99115044248
x3 = 57.495575221239 + 26.791115300322
x3 = 30.704459920917
Calculate x4
x4 = x3 + (ƒ(x3) - S)/ƒ'(x3)
x4 = 30.704459920917 + (30.7044599209172 - 225)/2(30.704459920917)
x4 = 30.704459920917 + (942.76385903517 - 225)/61.408919841833
x4 =30.704459920917 + 717.76385903517/61.408919841833
x4 = 30.704459920917 + 11.688267126077
x4 = 19.01619279484
Calculate x5
x5 = x4 + (ƒ(x4) - S)/ƒ'(x4)
x5 = 19.01619279484 + (19.016192794842 - 225)/2(19.01619279484)
x5 = 19.01619279484 + (361.61558841052 - 225)/38.03238558968
x5 =19.01619279484 + 136.61558841052/38.03238558968
x5 = 19.01619279484 + 3.5920856999197
x5 = 15.42410709492
Calculate x6
x6 = x5 + (ƒ(x5) - S)/ƒ'(x5)
x6 = 15.42410709492 + (15.424107094922 - 225)/2(15.42410709492)
x6 = 15.42410709492 + (237.90307967557 - 225)/30.84821418984
x6 =15.42410709492 + 12.903079675568/30.84821418984
x6 = 15.42410709492 + 0.41827639020405
x6 = 15.005830704716
Calculate x7
x7 = x6 + (ƒ(x6) - S)/ƒ'(x6)
x7 = 15.005830704716 + (15.0058307047162 - 225)/2(15.005830704716)
x7 = 15.005830704716 + (225.1749551386 - 225)/30.011661409432
x7 =15.005830704716 + 0.17495513860214/30.011661409432
x7 = 15.005830704716 + 0.0058295719192393
x7 = 15.000001132797
Final Answer
What is the Answer?
How does the Newton Method Calculator work?
This calculator has 1 input.
What 3 formulas are used for the Newton Method Calculator?
ƒ'(x) = 2x
xn = xn - 1 + (ƒ(xn - 1) - S)/ƒ'(xn - 1)
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
