Hmm, out Intro to NA professor mentioned that trick with the example of the integral
It's quite brilliant.
Also, yes, if you ran it to a, you would get an approximation of ln(9/8). This is also, I think, the only value of a for which the sequence a[n] converges.
Here I suggest you use your own head.
f[x_] := x^6 + x^5 - 5 x^4 - 4 x^3 + 6 x^2 + 3 x - 1
x = x - f[x]/f'[x]
Well, of course, Newton's method has a number of conditions to be met so that it would converge.
Also, isolating the zeroes might not be a bad idea.
Something like this?
I do not understand. The NewtonRaphson function does not take f as an argument in his code.
It uses f as a global variable, assigned a value outside of the subroutine, so, to call it for a function, you'd have to set f to that function before calling NewtonRaphson. I assume it works without much problem, but that is a silly thing to do...
I try to stay away from argumentation of any sort since I've found they are more often fruitless than not.
To say that an ordered pair (x,y) satisfies the equations given in post #1 means that, when we substitute the values, we get a true equality. If you try to reduce the equations the way you did it, you cannot guarantee that the solutions are the same. It'd be like saying that the equation
Well, the first thing is to notice a pattern and conclude that you get maximum values of g when it satisfies the following:
Since the rational numbers are equivalent calsses so any members of a certain class can represent that class, but numbers of infinite repetition are not rational numbers ,in fact, this is one of the reasons that lead to the extension of the rational numbers into the real numbers.
Wait what? Am I misunderstanding or do you also believe that 1/3≠0.(3)...?
Also, I see no inconsistencies in defining 0.(9) to be equal to 1, and thus see no argument against such a definition... What's more, defining 0.(9) to be anything other than 1 would be inconsistent, cause then the difference 1-0.(9) would be nonnegative, but also less than 1/n, for every natural number n, which would make it have to be 0. So, it's either that, or making it invalid notation.
This question is not impossble!! You only have to be more creative to solve this type of problems. For example, your question did not include any information about the bases of the numbers in the list, so if you choose to take the base to be tridecimal or tetradecimal then 71 will be equivelant to 92 and 99 in the decimal base respectively, and this renders the problem to a trivial one since all the numbers in that list using these bases are more then 100 except "71" also this answer can not be rejected since the base of the numbers in the list is not restricted in the question. Therefore the answer is "71"!!!
What if I take the answer C to be in base 8 and the rest in base 20, though? I think the answer might be C!
This topic has been discussed many times here and you'll get different opinions on it. Since opinions should not affect absolute truth, the solution is to either take 0.(9)=1 by convention, or just define decimal represntation so that 0.(9) is not a possible decimal representation, which is the variant I support.
That is indeed correct, but I feel the approach given on MIF might be a bit more satisfying, as stating that the roots of a quadratic equation are this and that just makes you wonder about how we got that formula, which in turn brings you to a variation of the explanation over there.