Let's play a game. I am thinking of two differentiable functions cat(x) and dog(x). Both functions can be written as power series cat(x) = > (nx", dog(x) = > dux". n=0 n=0 Both power series converge for all r E R which means that both functions are defined everywhere. I am also telling you that cat(0) = 0 and dog(0) = 1 and that dog' (x) = cat(x) and cat'(x) = dog(x). (a) Find cat(r) and dog(x) by calculating the general terms on and dn of their power series. Hint: To figure out what the coefficients of the two power series are use the fact that two power series are equal if and only if corresponding coefficients are equal and note that cat"(x) = cat(x). [10 marks] (b) Write the functions et and e- in terms of of cat (x) and dog(x). [5 marks] (c) Conversely, express cat(x) and dog(r) as a combination of er and e-I. [5 marks] (d) Using Mathematica or another piece of software plot cat(x) and the first four different partial sums of its power series in the interval [-2, 2].1