# Alexander's question via email about Newton's Method

#### Prove It

##### Well-known member
MHB Math Helper
Apply three iterations of Newton's Method to find an approximate solution of the equation

$\displaystyle \mathrm{e}^{1.2\,x} = 1.5 + 2.5\cos^2{\left( x \right) }$

if your initial estimate is $\displaystyle x_0 = 1$.
Newton's Method solves an equation of the form $\displaystyle f\left( x \right) = 0$, so we need to rewrite the equation as

$\displaystyle \mathrm{e}^{1.2\,x} - 1.5 - 2.5\cos^2{\left( x \right) } = 0$

Thus $\displaystyle f\left( x \right) = \mathrm{e}^{1.2\,x} - 1.5 - 2.5\cos^2{\left( x \right) }$.

Newton's Method is: $\displaystyle x_{n+1} = x_n - \frac{f\left( x_n \right) }{f'\left( x_n \right) }$

We will need the derivative, $\displaystyle f'\left( x \right) = 1.2\,\mathrm{e}^{1.2\,x} + 5\sin{\left( x \right) }\cos{\left( x \right) }$.

I have used my CAS to do this problem:

So after three iterations the root is approximately $\displaystyle x_3 = 0.81797$.