http://www.ck12.org Chapter 2. Derivatives
- Guess the first approximation to a solution of the equationf(x) =0. A graph would be very helpful in finding
the first approximation (see figure below). - Use the first approximation to find the second, the second to find the third and so on by using the recursion
relation
xn+ 1 =xn−ff′((xxnn)).Example 3:
Use Newton’s method to find the roots of the polynomialf(x) =x^3 +x− 1.
Solution:
f(x) =x^3 +x− 1
f′(x) = 3 x^2 + 1.Using the recursion relation,
xn+ 1 =xn−ff′((xxnn))=xn−x(^3) n+xn− 1
3 x^2 n+ 1.
To help us find the first approximation, we make a graph off(x). As Figure 11 suggests, setx 1 = 0 .6. Then using
the recursion relation, we can generatex 2 ,x 3 ,....