`4(e^(-x^2))sin(x) = x^2 - x + 1` 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.
To solve equation using Newton's method apply the formula,
Plug in f(x) and f'(x) in the formula,
See the attached graph for getting the initial values of x . The curve intersects the x axis at `~~` 0.20 and 1.1.
Let's solve for the first zero x_1=0.2, carry out iteration until we have same approximations at decimal places.
Now we have two approximations that have same decimal places,
Now let's solve for second zero x_1=1.1,
So the roots of the equation to the eight decimal places are 0.21916367 , 1.08422461