Use newton method to approxumate the solution cos x= x^2-4 to six decimal places

Question

How I should find solve this problem?

Also, I want to know what the graph looks like.

Answer 1

If you plot this with something like Desmos, you will get an inverted parabola with the vertex at (0,5). You will also notice that the curve crosses the x-axis near -2 and 2. So the approximation is near -2 or 2.

x_n+1=x_n-f(x)/f '(x0

f(x) = cos(x)-x²+4

f '(x) = -sin(x)-2x

If you let x₁=2, x will settle around 1.9140206 in 3 iterations.

x₂=x₁-f(x₁)/f'(x₁)

x₂=2-f(2)/f'(2)

x₃=....

x4=....

If you let x₁=-2, x will settle around -19.140206

Answer 2

desmos.com/calculator/ywlpuyuvfx

Newton's Method:

desmos.com/calculator/swpcl6zq7k