# Newtons method

• Jan 13th 2009, 06:52 AM
Emmeyh15@hotmail.com
Newtons method
WORK OUT!

Use Newton's method to solve cos(3x)-2x+1=0. Show the values of each iteration until you have reached the " best" solution your calculator can display
• Jan 13th 2009, 07:30 AM
running-gag
Hi

Let f(x) = cos(3x)-2x+1 with a representative curve Cf

Start from say $\displaystyle x_0 = 0$
Then $\displaystyle x_1$ is the abscissa of the intersection point between the x axis and the tangent to the curve Cf at the point whose abscissa is $\displaystyle x_0$

f'(x) = -3 sin(3x)-2
The equation of the tangent to Cf at the point of abscissa $\displaystyle x_n$ is
$\displaystyle y = (-3 sin(3 x_n) - 2) (x - x_n) + cos(3 x_n) - 2 x_n + 1$

$\displaystyle x_{n+1}$ is given by
$\displaystyle 0 = (-3 sin(3 x_n) - 2) (x_{n+1} - x_n) + cos(3 x_n) - 2 x_n + 1$

$\displaystyle x_{n+1} = x_n + \frac{cos(3 x_n) - 2 x_n + 1}{3 sin(3 x_n) + 2}$

Start for example from $\displaystyle x_0 = 0$ and then use a calculator to iterate and find the best estimation of the solution