P1) In this problem we are going to solve numerically the following differential equation x' (t) = f (x(t)), by an implicit Euler scheme. To do this, remember that if the time step is h, the requested scheme takes the form xn+1=xn+hf (xn+1). we know that for a general f this equation cannot be implemented in a computer. For the that follows the exercise consider f (x) = (1 - x) tanh (x), and that x (0) E (0, 1). (a) To find a solution to the equation g(x)=0, 1. We are told that it is possible through a program as follows: initialize the values I, u and tolerance so that I is less than u while the difference between u and I is greater than the tolerance 2. initialize m as the midpoint of the interval [I, u] if g (m) has the same sign as g (1) then 3. if not 4. assign the value of m to I 5. assign the value of m to u. Implement the algorithm in a programming language of your choice. Justify that the algorithm is valid for solving the problem g (x) = 0 or modify it if you do not agree. (b) Assuming xn known, use the result from the previous part to find an x + solution of x=xn+hf (x), that is, identify who the function g should be. (c) Solve, using the previous parts, the differential equation x(t) (1-x (t)) tanh (x (t)), using the numerical scheme xn + 1 = xn+h (1 - xn + 1) tanh (xn + 1), for 10 different initial conditions x (0) in the interval (0,1), time step h = 0.05, and T=10.