### Newton's Method

Graphs a function

*f* (

*x*)
and the tangent lines to the graph of

*f* (

*x*) used in Newton's Method to approximate
roots of

*f* (

*x*).

How to use
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Examples
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Other Notes

Try the Derivative Calculator.

**How to Use**

- Enter the function
*f* (*x*) in the text
input field marked "*f* (*x*)="
(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*)=2*x*)
- Select an
*x*_{0} value
(Example:
*x*=2)

This can be done in either of two ways:
- Click and drag the mouse on the graph -- value of
*x*_{0}
corresponds to horizontal mouse position on graph
- Enter the value in the text input field marked
"
*x*_{0}="
and click the
"Graph" button to refresh

- The maximum number of iterations
*n*
can be changed using the "+" and
"-" buttons under
the "*n*=" field

The text input fields marked "*f* (*x*)=" and
"*f* '(*x*)=" can accept
a wide variety of expressions to represent functions, and the
buttons under the graph allow various manipulations of
the graph coordinates.
The text input field marked "*x*_{0}="
can accept any decimal number.
For assistance computing the derivative *f* '(*x*), try
the Derivative Calculator.

**Examples**

**Other Notes**

Newton's Method approximates roots of a function using the iteration
formula
*x*_{n+1}=*x*_{n}-*f* (*x*_{n})/*f* '(*x*_{n}).
So *x*_{n+1} is the *x*-intercept of the tangent
line to the graph of *f* (*x*) at *x*_{n}.
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 happens 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).