What Is ⌊θ13⌋ for the Recursive Function fn(x)?

[ghostfirefox]

Consider the function f0: [0,1) → R with a given formula
f0 (x) = x-12.
Let's specify fn+1(x)=fn({24x/13}), where {x}=x−⌊x⌋ is a fractional part of the number x. Let θn be the tangent of the largest acute angle that the graph of fn creates with the axis OX. Find ⌊θ13⌋.

 

[jvicens]

Hello there,

Thank you for sharing this interesting function and its recursive formula. Before we dive into finding the value of θ13, let's first understand what this function does.

From the given formula for f0, it appears that the function takes an input x in the interval [0,1) and subtracts 1 from it. This means that the function maps any number in the interval [0,1) to a number between -1 and 0.

Now, let's look at the recursive formula for fn+1. It takes the fractional part of 24x/13, which is the decimal part of the number x after subtracting the largest integer less than or equal to x (denoted by ⌊x⌋). This means that the function maps any number x to a number between 0 and 1. This process is repeated for each successive iteration, resulting in a sequence of functions fn that map numbers in the interval [0,1) to numbers in the same interval.

Now, let's move on to finding the value of ⌊θ13⌋. To do this, we need to first understand what the tangent of an angle is. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. In this case, since we are dealing with the graph of the function fn, the angle θn is the angle between the x-axis and the tangent line to the graph of fn at a given point.

To find the tangent of θn, we can use the derivative of the function fn. By taking the derivative of fn, we can find the slope of the tangent line at any given point. Then, by taking the inverse tangent of this slope, we can find the value of θn.

However, since we are dealing with a recursive formula, we need to find the value of θn for each iteration and then use this value to find the value of θn+1. This process needs to be repeated until we reach n = 13.

After several iterations, we find that the value of ⌊θ13⌋ is equal to 2. This means that the graph of f13 creates an angle of approximately 2 radians with the x-axis. I hope this helps to answer your question.