Expert: Ahmed Salami - 8/3/2010

Question
Intersection of curves

a. Does cos 3x ever equal x? Give reasons for your answers.

b. Use Newton's Method to find where.

Answer
Hi Ashley,
a) Yes, cos3x equals x because they both have the same value at certain points. These can be seen as their lines intersect on a graph (in three distinct places).

b) When cos3x = x
cos3x - x = 0
The solution of the equation f(x) = cos3x - x then gives our required points.
If we take an initial guess of (one of the points to be) x = 0.5, Newton's Method states that a better approximation would be;
x = 0.5 - f(0.5)/f'(0.5)
but f'(x) = -3sin3x - 1
so,
x = 0.5 - [cos(1.5) - 0.5]/[-3sin(1.5) - 1]
 = 0.5 - [0.0707 - 0.5]/[-3(0.9975) - 1]
 = 0.5 - [-0.4293]/[-2.9925 - 1]
 = 0.5 - [-0.4293]/[-3.9925]
 = 0.5 - 0.1075
 = 0.3925

You can continue the process until you attain your preferred degree of accuracy or by simply accepting a value when successive approximations dont differ considerably. Note that you require some knowledge of the expressions in order to make good initial guesses for the points.

Regards