Newtons Method sqrt(20)

Enter Number

Given S = 20:

Calculate:√20 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 - 20)/2(1)

x1 = 1 + (1 - 20)/2

x1 =1 + -19/2

x1 = 1 + -9.5

x1 = 10.5

Calculate x2

x2 = x1 + (ƒ(x1) - S)/ƒ'(x1)

x2 = 10.5 + (10.52 - 20)/2(10.5)

x2 = 10.5 + (110.25 - 20)/21

x2 =10.5 + 90.25/21

x2 = 10.5 + 4.297619047619

x2 = 6.202380952381

Calculate x3

x3 = x2 + (ƒ(x2) - S)/ƒ'(x2)

x3 = 6.202380952381 + (6.2023809523812 - 20)/2(6.202380952381)

x3 = 6.202380952381 + (38.469529478458 - 20)/12.404761904762

x3 =6.202380952381 + 18.469529478458/12.404761904762

x3 = 6.202380952381 + 1.4889064070926

x3 = 4.7134745452884

Calculate x4

x4 = x3 + (ƒ(x3) - S)/ƒ'(x3)

x4 = 4.7134745452884 + (4.71347454528842 - 20)/2(4.7134745452884)

x4 = 4.7134745452884 + (22.216842289081 - 20)/9.4269490905767

x4 =4.7134745452884 + 2.2168422890814/9.4269490905767

x4 = 4.7134745452884 + 0.23516009981398

x4 = 4.4783144454744

Calculate x5

x5 = x4 + (ƒ(x4) - S)/ƒ'(x4)

x5 = 4.4783144454744 + (4.47831444547442 - 20)/2(4.4783144454744)

x5 = 4.4783144454744 + (20.055300272545 - 20)/8.9566288909488

x5 =4.4783144454744 + 0.055300272544521/8.9566288909488

x5 = 4.4783144454744 + 0.0061742284086824

x5 = 4.4721402170657

Final Answer

x5 = 4.4721402170657

You have 1 free calculations remaining

What is the Answer?

x5 = 4.4721402170657

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Free Newton Method Calculator - Calculates the square root of a positive integer using the Newton Method
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