## 9.3 The Newton-Raphson method

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:

$$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

This method uses tangents to find increasingly accurate approximations of a root. The value of $$x_{n+1}$$ is the $$x$$-intercept of the tangent to the graph at $$(x_n, f(x_n))$$.

Limitations of this method

If the starting value $$x_0$$ 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 $$x_0$$.

If the starting value $$x_0$$ 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.

Important
The Newton-Raphson method

$$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$
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