Question C 03 (c)
This question is about approximating the value of .
(c) Determine an appropriate function that has a zero at b and apply one iteration of Newton's method (i.e. find x1) to estimate b starting with an initial guess x0 that is the nearest integer to the actual value.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Try to find a nice polynomial function with as a root so that we can apply Newton's method to this function. As a further hint, this about what happens with you square the previous number. Can you find a second degree polynomial with integer coefficients such that is a root?
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
The question is telling us to use Newton's method, which can only be used to estimate roots of functions. Therefore, in order to use Newton's method to estimate b, we must create a function with as a root. Then estimating the root of this function will be the same as estimating the value of b.
One way to construct a function with specific roots is to place the roots into linear factors and then multiply them out to get a polynomial. If we want one of our roots to be , this means we should have at least one linear factor of the form .
Ideally, we want our function to not actually contain the value we're trying to estimate. In order to make that go away, we are going to use the difference of squares, which says:
Putting these two together, we can say:
which has a root at like we wanted.
Now we can finally use Newton's method to iterate and find . The formula for the first iteration of Newton's method is as follows:
We have , , and our starting point should be a value close to b. Based on the previous parts of the question, it makes the most sense to choose .
Putting this all together, we finally get: