have no idea how to do this
Use the Newton-Raphson method once to find an improved solution to $$f(x) = x \times \cos (x) - \frac{ 1 }{ x }$$ . Using an Initial value of $$x _{0} = 4.005^{c} (radians)$$

Question

have no idea how to do this
Use the Newton-Raphson method once to find an improved solution to $$f(x) =  x \times \cos (x) - \frac{ 1 }{ x }$$ . Using an Initial value of $$x _{0} = 4.005^{c} (radians)$$

anonymous · Answer

https://en.wikipedia.org/wiki/Newton's_method

ray10 · Answer

I believe it is similar to the Approximation technique but I don't know the method on going about it

anonymous · Answer

Note $f'(x)=\cos x-x\sin x+\dfrac1{x^2}$$$x_0=4.005\x_1=x_0-\frac{f(x_0)}{f'(x_0)}=4.005-\frac{4.005\cos(4.005)-\frac1{4.005}}{\cos(4.005)+4.005\sin(4.005)-\frac1{4.005^2}}\approx3.246$$

anonymous · Answer

Oops I messed up the signs.
$$x_1=4.005-\frac{4.005\cos4.005-\frac1{4.005}}{\cos4.005-4.005\sin4.005+\frac1{4.005^2}}\approx5.166$$

ray10 · Answer

Yes that is what I came to but I could not acquire the approximation.
I placed those values into my calculator and it wouldn't come out as a decimal value, instead it came out as I typed it :S

ray10 · Answer

does it lie in the fact that the value is in radians? do I need to change my calculator mode?

anonymous · Answer

http://www.wolframalpha.com/input/?i=solve+x+cos+x+-+1%2Fx+%3D+0+using+newton%27s+method+starting+at+x_0+%3D+4.005 Click 'Show Details' in the 'Steps' section and you'll see it agrees with me

anonymous · Answer

Ray10 maybe your calculator is in some 'exact' mode

ray10 · Answer

ah I see, so I had a look and now I tried it again, it came out as what you got too! :)
thank you for that! You helped me get my head around that, I was going all over the place with my working out before. One last question; Can this "Newton Method' be seen as the Approximation formula?

anonymous · Answer

Newton's method is most definitely an iterative approximation algorithm; if you look at the Wolfram Alpha output, you'll see it even shows an 'iteration diagram'.

anonymous · Answer

It belongs to the larger study known as numerical analysis. https://en.wikipedia.org/wiki/Numerical_analysis

jhannybean · Answer

Doyou know the calculator method of solving Newton's Method?

jack1 · Answer

@oldrin.bataku mate what did newton use it for...?

ray10 · Answer

I saw that just then @oldrin.bataku  , it makes a lot more sense
@Jhannybean do you know the way to solve it through calculator?

anonymous · Answer

Newton used a special case of it to determine the roots of higher-degree polynomials. Here's a good look at how it works: https://upload.wikimedia.org/wikipedia/commons/e/e0/NewtonIteration_Ani.gif

anonymous · Answer

It uses local information at a point to try and find x-intercepts, so it's known as locally convergent. http://en.wikipedia.org/wiki/Local_convergence

jack1 · Answer

cheers @oldrin.bataku 
slaters @Jhannybean

ray10 · Answer

@oldrin.bataku  thank you a lot for your help!! :)
I have to go now, been on a way too long study session!
You really helped me out!