Let g be the function: g(x] = exp(x2 + 2x]:/1+ 2x. (a) Jorts is trying to find the Taylor expansion for 9. He takes 5 minutes to work out its first derivative. 15 minutes to work out its second derivative. 45 minutes to work out its third derivative. tripling in time for each subsequent derivative. How long does Jorts take to work out the first ten derivatives? Give your answer in days. to 4 decimal points. [6 marks] Jean suggests a more subtle approach. (b) Expand 1.141 + 2x up to and including the term in x3 using the binomial series. [5 marks] (c) Expand exp(x2 + 2x] up to and including the term in x3 using the Taylor series about x = 0. [6 marks] (d) Using (b) and (It). show that the Taylor expansion for g is: _ 5 292 2173 g(x)1+2x+ 8x + 48x +... [4 marks] (e) Using (d). give an estimate for In\" 90:] six. Explain briefly why this estimate is probably correct to 4 decimal places. [4 marks]