# Use Newton's method to find all roots of the equation correct to eight decimal places.  Use Newton's method to find all roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. (Enter your answers as a comma-separated list.) 5e^(-x^2)sin x = x2 − x + 1 x=__________________?

First sketch the graphs on each side of the equation and approximate the roots of the equation by approximating where the graphs cross

Approximate values for the roots are `x=0.2`  and `1.2`

The formula for Newton's method is

`x_(n+1) = x_n - f(x_n)/(f'(x_n))`

Now, `f(x) = (x^2-x+1) +(-5e^(-x^2)sinx)`

`implies`

`f'(x) = (2x - 1 )+(10xe^(-x^2)sinx -5e^(-x^2)cosx)`

Working on the first root `x approx 0.2` gives the sequence

x0 = 0.2

x1 = 0.17677893

x2 = 0.17721437

x3 = 0.17721451

x4 = 0.17721451 `implies`  root is 0.17721451 to 8dp

Working on the second root `x approx 1.2` gives the sequence

x0 = 1.2

x1 = 1.16247348

x2 = 1.16224076

x3 = 1.16224075

x4 = 1.16224075 `implies`  root is 1.16224075 to 8dp

The roots are 0.17721451, 1.16224075 to 8dp