Hi there, this is quantitative methods questions.

Question 1 (35 marks) Question 3 (10 marks) Evaluate the following: (a) [2x' +3dx [2] Given a series, - dix (b) _ 2x [2] (a) Write down the first four terms of the series. [4] (b) Is the series a geometric series? Why? [2] (c) (4x/ +2x -X x [3] (c) Explain whether the series is convergent. If yes, find its sum to infinity. [4] (d) X VI-4x dx [4] Question 4 (10 marks) (e) [[sec' (4x)] [3- tan (4x) ]'dx [4] Given f (x) = cos x. (1) xedx [5] (a) Find the Taylor polynomial of order 3 generated by f (x) at a = [8] (g) 1+ sin .x -d.x [7] (b) Use your answer in (a) to find an approximate value of cos 30 to 4 decimal places. (h) 3.x3 +1 J2 (x+1)(x-5) dx [8] [2] Question 2 (5 marks) Because of the increasingly important role played by coal as a viable alternative energy source, the production of coal has been growing at the rate of 3.5e"" billion metric tons/year / years from 1970 (which corresponds to { =0). Had it not been for the energy crisis, the rate of production of coal since 1970 might have been only 3.5e" billion metric tons/year / years from 1970. Determine how much additional coal was produced between 1970 and the end of the century as an alternate energy source. Round your answer to four decimal places