##### How to Use
Enter the function f (x) in the text input field marked "f(x)=".

Example: f (x)=x2-2

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 x0 value

Example: x=2

This can be done in either of two ways:

• Click and drag the mouse on the graph -- value of x0 corresponds to horizontal mouse position on graph.
• Enter the value in the text input field marked "x0=" and click the "Graph" button to refresh.

##### Examples
Square roots of 2: f (x)=x2-2, f '(x)=2x

Approximating : f (x)=tan(x/4)-1, f '(x)=sec2(x)/4 (x0=5, n=5)

Which root?: f (x)=sin(x), f '(x)=cos(x)

Horizontal tangent: f (x)=x2-2, f '(x)=2x, x0=0

Diverging: f (x)=4arctan(x), f '(x)=4/(1+x2) (x0=1.5, n=4)

Multiple root: f (x)=x3-3x+2, f '(x)=3x2-3 (double root at x=1)

##### Other Notes
Newton's Method approximates roots of a function using the iteration formula xn+1=xn-f (xn)/f '(xn). So xn+1 is the x-intercept of the tangent line to the graph of f (x) at xn.

Under certain rather technical conditions, Newton's Method can be guaranteed to converge quickly to a root xr of f (x), as long as x0 is sufficiently close to xr. 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 '(x0)=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 x1.

If x0 is not close enough to the root xr, 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 xr satisfies f '(xr)=0, then xr is a multiple root, and even if Newton's Method converges, it will converge more slowly (as in the multiple root example above).