1. Taylor series We have seen in the lectures how the y = 1/V1 v factor (where vo is in units of c) plays a significant role in special relativity, appearing in the Lorentz transformations and elsewhere. (a) (5 points) Calculate the Taylor series expansion of the expression for 7 to ap- proximate to 4th order (i.e. up to v.). Recall that the expression for the Taylor series for f(x) is f"(xo) f(x) = (x xo)" (1) n! n=0 where the expansion is about some conveniently-chosen value xo, and f"(x) in- dicates the nth derivative of function f(x). Show your working. (b) (3 points) Demonstrate how adding terms successively improves the estimate for y, by plotting (with your favourite plotting package) the approximate values against the exact expression, over a range of vo. We use and recommend the open-source Python programming language http://python.org. (c) (2 points) Hence determine (approximately) at what values of vo the 2nd (4th) order approximations are correct to 1%. 1. Taylor series We have seen in the lectures how the y = 1/V1 v factor (where vo is in units of c) plays a significant role in special relativity, appearing in the Lorentz transformations and elsewhere. (a) (5 points) Calculate the Taylor series expansion of the expression for 7 to ap- proximate to 4th order (i.e. up to v.). Recall that the expression for the Taylor series for f(x) is f"(xo) f(x) = (x xo)" (1) n! n=0 where the expansion is about some conveniently-chosen value xo, and f"(x) in- dicates the nth derivative of function f(x). Show your working. (b) (3 points) Demonstrate how adding terms successively improves the estimate for y, by plotting (with your favourite plotting package) the approximate values against the exact expression, over a range of vo. We use and recommend the open-source Python programming language http://python.org. (c) (2 points) Hence determine (approximately) at what values of vo the 2nd (4th) order approximations are correct to 1%