Use Newton’s method on to compute when the initial approximation is .

Let be the -th order Taylor polynomial of centered at . What is the smallest value of for which approximates to two decimal places?

The hyperbolic sine is defined by . What is the -th term in the Taylor expansion of around ?

What are the first three non-zero terms in the Taylor series of centred at ?

An algorithm can calculate the universe partition function (a function one can give to a statistical physicist to find out the secrets of our universe) with a precision level in time . The parameter corresponds to a specific choice of hyperparameters in the algorithm. For a given , which does one need to choose to use the fastest version of the algorithm?

Which side lengths of a rectangle with a perimeter of maximize its area?