In what concerns the continuous evaluation solving exercises grade during the semester, you should submit until 23:59 of November 2nd
(this exercise will still be available for submission after that deadline, but without counting towards your grade)
[to understand the context of this problem, you should read the class #04 exercise sheet]

In this problem you should submit a function as described. Inside the function do not print anything that was not asked!

[IP032] Come Closer

A numeric method to determine an approximate value for the square root of a number x consists of iteratively computing n estimates according to the following equation: \[ y_n = \frac{1}{2}\Big(y_{n-1} + \frac{x}{y_{n-1}}\Big) \] The initial estimate, \( y_0 \) is a rough value for the square root (e.g.,\( \frac{x}{2} \) ). At each iteration, the error of the estimate is given by the diference between the last two values calculated, that is \( |y_n - y_{n-1}| \). The approximation ends when the allowed error (i.e., a reasonable error threshold) is achieved.

The Problem

Write a function approximate_sqrt(x, max_error) that approximates the square root of a given number x using this iterative method, starting with \( y_0 = \frac{x}{2} \). The approximation should stop when the error is less or equal to max_error. The estimated value should be return by the function.

Constraints

The following limits are guaranteed in all the test cases that will be given to your program:

print( round(approximate_sqrt(2, 0.1),      16) )
print( round(approximate_sqrt(2, 0.05),     16) )
print( round(approximate_sqrt(2, 0.001),    16) )
print( round(approximate_sqrt(2, 0.000001), 16) )

1.4166666666666665
1.4142156862745097
1.4142135623746899
1.414213562373095