Hey!
I was wondering if I could get some help with solving these multiple choice questions. I haven't see the halving the interval technique and would appreciate a brief explanation (also because it's probably fairly simple)
Also with the domain question, I get it now that I have the answer but I would like a more structured method to solving the inequality. I began by dealing with cos-1(x) and applied that to the whole function but I went wrong somewhere so if anyone could please provide the method that would be amazing.
Thank you
EDIT: I was wondering if someone could provide some more intermediate steps for the following proof, particularly between lines 3 and 4 that would be really helpful. I don't understand what's happened to the denominator . Also with proofs in general, it doesn't matter how we manipulate the function as long as we do the same to the top and bottom - right?
Thanks again
Newton's method works on the basis of the x-intercepts, of the tangent at a given point. That's why in Newton's method we only ever input one point, because we rely on the tangent to do its job for us.
Halving the interval is different. Halving the interval just seeks to make the region that you're interested in (and hence you start with
two points) smaller.
Take f(x)=x^2 - 9. We know that the root is at x=3, but suppose we didn't know that. We could estimate where the root is if we knew that the root was between say, 1.5 and 2.75. Then, we check the average of 1.5 and 2.75, which is 2.125, compute 2.125, and then we see that our interval can be shrunk to (2.125, 2.75).
The convenient thing about halving the interval is that
you can never go outside the region of your initial values. So if your initial interval is (2,4), then you would use halving the interval when you know your root is between 2 and 4.
This is why the answer is x^2 - 5. x^2 - 5 = 0 has a root at sqrt(5), which is 2 point something. Everything else has a root that's outside the interval \( 2< x <4\)