Math 137 Week 10 Notes – Dr. Paula Smith
4.8
Newton’s Method
Given a function f(x), the values of x that make f(x) = 0 are called the
roots
of function f.
If f is
a quadratic function, the quadratic formula finds its roots.
There are formulae for finding the
roots of cubic (polynomials of degree 3) and quartic (polynomials of degree 4) functions as well,
but in general finding the roots of f is not simple.
Newton’s (or NewtonRaphson) method for finding the approximate value of a function’s roots
is recursive, that is, we repeat the method over and over until our successive answers agree to as
many decimal places as we like.
It uses an arbitrary value of x to start and the equation of the
tangent line to f at the point x.
Say we choose value x
1
; the equation of the tangent line to f at that point is
y = f(x
1
) + f
′
(x
1
) (x – x
1
).
We let the tangent line stretch till it intersects the xaxis so that y = 0,
and we then solve for x (we are finding the xintercept of the tangent line).
Thus
0 = f(x
1
) + f
′
(x
1
) (x – x
1
)
⇒
x =
x
1
– [f(x
1
) / f
′
(x
1
)].
The new value x is (generally) a better
approximation of a root for f than x
1
was.
But to improve the value still more, we let x
2
= x and
find the tangent line to the function f at the point x
2
, and then solve for the new x.
More mechanically, we use a value to start that close to a suspected root, and the formula
x
n+1
=
x
n
– [f(x
n
) / f
′
(x
n
)] over and over again until successive answers agree.
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 Spring '08
 SPEZIALE
 Math, Quadratic Formula, Derivative

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