Question

Solve sinx=x^2 correct to four decimals using Newton's method with x sub zero=1.

Answer 1

Newton's method was developed to find the roots of a function (i.e. what f(x)=0). As such, when you have something like you have, you need to put it in that form first. So,

sin(x) =x^2 then f(x):=sinx-x2=0.

You can now proceed.
\[x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} \rightarrow x_{n+1}=x_n-\frac{\sin x_n - x_n^2}{\cos x_n -2x_n}\]

Answer 2

So,\[x_1=1-\frac{\sin(1)-(1)^2}{\cos(1)-2(1)} \approx 0.891395995\]and this number becomes your second input, and you keep going (technically, forever).

Answer 3

\[x_2=0.891395995-\frac{\sin (0.891395995)-(0.891395995)^2}{\cos(0.891395995)-2(0.891395995)}\]\[\approx 0.876984844\]

Answer 4

Mathematica spits a terminating output of about 0.8767262153950624.

Answer 5

The other solution is x=0.