Question

Please, help me find the answer to this equation:

f(x) = 1 − x − cos (x2)
a) Show that the equation f(x) = 0 has a root α in the interval 1.4 < α < 1.5.
b) Using x0 = 1.4 as a first approximation to α, apply the Newton–Raphson procedure once to f(x) to find a second approximation to α, giving your answer to 3 decimal places.
c) By considering the change of sign of f(x) over an appropriate interval, show that your answer to part b is correct to 3 decimal places.

Answer 1

When x = 1.4 f(x) = 1 - 1.4 - cos(1.4^2)
= -0.0205
When x = 1.5 fx) = 1-1.5-cos(1.5^2)
= 0.128
The signs have changed so the root is between the given values.
B) the formula for this method is
X(n+1) = x(n) - f x(n) / f’ x(n)
So first we need to find f’(x) :
F’(x) = -1 - -2x sin (x^2)
= -1 + 2xsin(x^2)
So
X1 = 1.4 - -0.0205 / (-1+ 2*1.4* sin 1.96)

= 1.4 - - 0.0205 /
1.5906
= 1.4+ 0.0129
= 1.413 to 3 dec places