The Newton-Raphson method can be used to find numerical solutions to equations of the form (f(x) = 0). You need to be able to differentiate (f(x)) to use this method.
The Newton-Raphson formula is:
(xn+1 = x_n - \dfrac{f(x_n)}{f'(x_n)} )
This method uses tangents to find increasingly accurate approximations of a root. The value of (xn+1) is the (x)-intercept of the tangent to the graph at ((x_n, f(x_n))).
mtaimg/images/topics/9/9-101-1.png/mtaimg
[b]uLimitations of this method[/u]/b
If the starting value (x0) is near a turning point, or the derivative at this point (f'(x_0)) is close to zero, then the (x)-intercept of the tangent at (\big(x_0, f(x_0)\big)) will be far away from (x_0).
mtaimg/images/topics/9/9-101-2.png/mtaimg
If the starting value (x0) is at a turning point, then the tangent at (\big(x_0, f(x_0)\big)) will never intersect the (x)-axis, as its gradient is zero.
mtaimg/images/topics/9/9-101-3.png/mtaimg
[b]uThe Newton-Raphson method[/u]/b
(xn+1 = x_n - \dfrac{f(x_n)}{f'(x_n)} )