Use Newton's Method to solve # 2sinx = 2 - x #?
1 Answer
# x = 0.704577 #
Explanation:
We want to solve:
# 2sinx = 2 - x #
Let:
# f(x) = 2sinx + x -2 #
It is important to understand that we cannot use Newton's Method to determine the existence of a root, or establish the number of roots of
graph{2sinx + x -2 [-10, 10, -10, 10]}
From which we can be confident that there is a single root,
# { (x_1,=x_0,), ( x_(n+1), = x_n - f(x_n)/(f'(x_n)),n gt 1 ) :} #
Therefore we need the derivative:
# \ \ \ \ \ \ f(x) = 2sinx+x-2 #
# :. f'(x) = 2cosx+1 #
So our iterative formula is:
# { (x_1,=x_0,), ( x_(n+1), = x_n - (2sinx+x-2)/(2cosx+1), x gt 1 ) :} #
Then using excel working to 8dp with
We get convergence to
We could equally use a modern scientific graphing calculator as most new calculators have an "Ans" button that allows the last calculated result to be used as the input of an iterated expression.
And we conclude that the solution is (to 6dp):
# x = 0.704577 #