## More about Newton's Method

# Newton's method finds an approximate solution to the equation f(x) = 0, using the concept of the *rate of change.*^{ }_{ }

The rate of change r of the values of the function f(x), relative to the variable x, when x changes from x_{0} to x_{1}, is^{ }_{ }

#
This rate of change r can be used to look for a value x_{1} such that f(x_{1}) = 0, by solving for x_{1} the equation_{ }^{ }^{ }_{ }

# We set f(x_{1}) = 0, so^{ }_{ }

# We solve this equation for x_{1}:^{ }_{ }

# So the solution is^{ }_{ }

# We can compute f(x_{0}) for any chosen x_{0}; but how to find r?^{ }_{ }

When the difference x_{1} - x_{0} is rather small (close to 0), then r can be estimated by the value of the derivative of f(x) for x = x_{0}.^{ }_{ }

*r is approximately equal to f'(x*_{0})^{ }_{ }

This gives us a formula, which we can compute, that returns an approximate value of the solution of our equation, f(x) = 0:^{ }_{ }

# How good this approximation is depends on our choice of x_{0}. But the nice thing is that when we get a good approximation once, we can improve it by iterating the whole process,^{ }_{ }

# and so on.^{ }_{ }

Webpage Maintained by Owen Ramsey

Calculus Index