Question 3 a) By expanding the integrand give a two term asymptotic expansion of the integral Set e -t5 dt valid for x 0. 6 marks b) By integration by parts give two terms in the expansion of the integral t-52-5t2 e dt in the limit of large x. 6 marks c) Plot a graph of e2xcos(t/5) for t between -51/2 and 51/2. Show on the graph and describe in words how the graph behaves as x . Give a quadratic Taylor expansion of cos(t/5) centred about the maximum in your graph. 8 marks d) Use Laplace's method to give the leading order approximation to 51/2 excos(t/5) dt -57/2 Question 3 a) By expanding the integrand give a two term asymptotic expansion of the integral Set e -t5 dt valid for x 0. 6 marks b) By integration by parts give two terms in the expansion of the integral t-52-5t2 e dt in the limit of large x. 6 marks c) Plot a graph of e2xcos(t/5) for t between -51/2 and 51/2. Show on the graph and describe in words how the graph behaves as x . Give a quadratic Taylor expansion of cos(t/5) centred about the maximum in your graph. 8 marks d) Use Laplace's method to give the leading order approximation to 51/2 excos(t/5) dt -57/2