Example: f (x)=x^{2}-2
Click the "Graph" button (this button also refreshes the graph).
Enter the derivative f '(x) in the text input field marked "f '(x)=".
Example: f '(x)=2x
Select an x_{0} value
Example: x=2
This can be done in either of two ways:
Approximating : f (x)=tan(x/4)-1, f '(x)=sec^{2}(x)/4 (x_{0}=5, n=5)
Which root?: f (x)=sin(x), f '(x)=cos(x)
Horizontal tangent: f (x)=x^{2}-2, f '(x)=2x, x_{0}=0
Diverging: f (x)=4arctan(x), f '(x)=4/(1+x^{2}) (x_{0}=1.5, n=4)
Multiple root: f (x)=x^{3}-3x+2, f '(x)=3x^{2}-3 (double root at x=1)
Under certain rather technical conditions, Newton's Method can be guaranteed to converge quickly to a root x_{r} of f (x), as long as x_{0} is sufficiently close to x_{r}. However, Newton's Method encounters problems for x values near where f '(x)=0 or f ''(x)=0.
One example of a problem is when f '(x_{0})=0 (see the horizontal tangent example above) -- in this case, the tangent line is horizontal, so it has no x-intercept, and there is no x_{1}.
If x_{0} is not close enough to the root x_{r}, then Newton's Method may not converge at all (see the diverging example above). Or Newton's Method may converge, but not to the expected root (see the "which root?" example above).
If the root x_{r} satisfies f '(x_{r})=0, then x_{r} is a multiple root, and even if Newton's Method converges, it will converge more slowly (as in the multiple root example above).