Test: Newton Raphson Method - Civil Engineering (CE) MCQ

The value of derivative of a function f(x) is given as f’(x) = y’/x’. In terms of theta tangent is the ration of opposite side to adjacent side hence y’/x’ = tanθ.

According to the Newton-Rapson method we have;

CALCULATION:
As we know;

Now,

Now putting all the values we have;
x_n+1 = 2 × 0.20 − 0.20² × 4
⇒ x_n+1 = 0.24
Hence option (2) is the correct answer.

Newton Raphson method is also known as Tangent Method. It is carried out by drawing a tangent to the curve at the point of initial guess.

Newton Raphson method:
The iteration formula is 
Calculation:
Given

Iterative formula for Newton Raphson method is given by

Hence x0=6 (initial guess), f(x₀) = 32 and f’(x₀) = 12.
Substituting the values in the equation we get x₁ = 10/3.

Concept: 
Let x0 be an approximation root of f(x) = 0 and x₁ = x₀^th
be the correct root so that f(x₁) = 0 expanding f(x₀^th) by Taylor's series we obtain

Neglecting the second and higher order derivatives we have f(x₀) + hf'(x₀) = 0

A better approximation than x0 is therefore given by x_1, where x₁ = x₀ + h = 
successive approximations are given by x₂, x₃ ....x_n+1 where

For p^th root of a given number N, is not of equation f(x) = x^p - N = 0
Iteration equation:

for cube root, put p = 3

Therefore, option (2) is correct one.

Iterative formula is given by

Hence if f’(x) decreases the value of next iteration approaches the initial one. This decreases the number of iterations in finding out next iterative value.