2. (Day 1)Taylor Series If we know the value of a function f(x) at a particular point x0, and we also know the derivative ofthat function f'(x) = : (and possibly higher order derivatives, too), we can approximate the function in the vicinity ofx), using a Taylor series (a.k.ar Taylor expansion), for) : M.) + f'(1'u)(1: 37o) + %fn(370)(17 In)? + gimme: m)" + . .. (1) (Sometimes, the Taylor expansion is written as f(x0 + Ax) = f(x0) + f'(xD)Ax + The two formulas are in fact equivalent; to see this, just set x = X0 + Ax in equation (1)) We can write equation (1) in a more concise way, ac no = furl.) + Z $f(")(rvu)(~l7 , mu\") \"=1 - where f(")(x) is the nth derivative of the function f(x) (This formula implies that if you know all derivatives of the function at X0, you can calculate the value of the function x) for all x. Strange, no?) In practice, we are often interested in knowing how a function behaves for values of x close to a given point x0, To get an accurate estimate, we might only need to include a few derivatives, depending on the details of the function, For this problem, we will consider the function _ 1 71+:17' f(-'I') Before you start working on this problem, make a sketch of this function (or plot it with a computer) to familiarize yourself with it. (a) Calculate the first three derivatives off(x) with respect to x. (2 pts) (b) Using your results from part (a) and the formulas given at the beginning of this problem, write a Taylor expansion off(x) about X0 = 0, in powers ofx upto first order, f1(.'17) = (.) -l ()T. (Determine the terms represented by (. r .) with the help of equation (1).) (1 pt) (c) Using your results from part (a), write a Taylor expansion of f(x) about x0 = 0, in powers of x up to second order, f2(;17) = ()+(.t.):17+(.tt)a:2t (1 pt) (d) Using your results from part (a), write a Taylor expansion of f(x) about X0 = 0, in powers of x up to third order, f3(1:)=(rt.)+(t.tzr (lpt) (e) Using a computer, make a plot of f(x) together with the three approximations from part (b)-(d). (I recommend plotting in the range 1