Newton's method to find the root of the equation:?

Question

Cannot for the life of me remember how to do this...

Use newton's method the find the root of the equation x^4 + x - 1 = 0 in the interval [0,1] correct to six decimal places.

I'm pretty sure this is really easy and i'm just forgetting it. All i need is the set up, the answer is unimportant... Thanks for any help!!

Answer 1

To use Newton's method to find the root of the function f(x) you need to:

(1) differentiate the function and get f'(x)

(2) guess the root - or some approximation of it and use that value r

(3) find f'(r) - this will actually represent the slope of the function at this point (r, f(r)) and can be used as the slope of the tangent

(4) the equation of the tangent can now be found using the slope and the point (r,f(r))

(5) the x-intercept of the tangent is a better guess (r2) so use this value - go back to step 2 and go again

This process converges rapidly to the root of the original function in most cases.

Answer 2

Using a spreadsheet to graph and solve the equation using Newton's method, I found one root:

0.72449195900052