Question

Q.5. Using Newton Raphson method find the cube root of 41 correct to three decimal place.

Answer 1

Answer:

To find the cube root of 24 using Newton - Raphson method,

we need to solve f(x)=x³−24.

⇒f′(x)=3x²

Notice 3³=27

Therefore the cube root of 24 is slightly less than 3.

We have f(x)=x³−24,f'(x)=3x²

Let us start estimating the root x

Let the first estimation be a=2.9 (slightly less than 3)

Hence the subsequent estimates will be b=a− f′(a)/f(a) ,c=b−f′(b)/f(b).

f(a)=f(2.9)=(2.9)³−24=0.389 and f'(a)=f′(2.9)=3(2.9)²=25.23

Therefore b=2.9− 25.23/0.389 ≈2.88458

Now c=2.88458− f ′(2.88458)/f(2.88458)

=2.88449

Hence the cube root of 24 is 2.884.

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Answer 2

Explanation:

Using Newton-Raphson method compute √41 correct to 4 decimal places.