2. Taylor Series (25 pts). As a physicist, you may often want to make a polynomial approximation of a function as it may be easier to deal with than the real thing. One thing that should be in your tool-belt is the Taylor Expansion. We will use it often in this class. Recall that the formula for the Taylor Series Expansion is given by: f (2) = f(a) + f' (@) 1 ! ( 2 - a ) + - f (a) 2 ! (x - a ) + 1 f" (a) 3 ! (xx - at... (a) Calculate the first four terms of the Taylor Series for: f (x) = et about a = 0. (b) The Taylor series is most accurate close to its expansion point, a. Plot the function f (x) = et and your Taylor series expansion for the first 2 terms, and the first 4 terms all on the same plot. Limit the x-range of your plot from 0 to 2. What happens as you add more terms? (c) We often truncate our Taylor expansions to the first two terms. In this example, if we only kept the first two terms, how close to the expansion point would we have to be to maintain 95% accuracy