The technique of approximating the real zeros of a function is called Newtons method.

Newtons method, also called as Newton-Raphson method, is a root finding algorithm  that uses the first few terms of

the taylor series of a function .

If  is a function continous on and diffrentiable on where and .

Newtons method uses tangent lines to approximate such that .

(1). Newtons approximation method formula : .

(2). Make an initial estimate that is close to .

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From the initial a new estimate can be obtained.

Newton Formula : .

.

This is second estimate.

Similarly perform newton approximation for the obtained value .

Newton Formula : .

.

This is third estimate.

Repeated application of this process is called Newtons method.

(3).If is within the desired accuracy, let serves as a final approximation, otherwise perform Step 2 again and calculate a new approximation.

Each sucessive application of this procedure is called Iteration.

Hence Newtons method of approximation is described.