(Solved): 2. (a) (i) Use the linear approximation formula Ay f'(x) Ax or f(x+x) f(x) + f'(x) Ax with a ...

2. (a) (i) Use the linear approximation formula Ay? f'(x) Ax or f(x+x)? f(x) + f'(x) Ax with a suitable choice of f(x) to show that e²1+0² for small values of 0. -1/4 (ii) Use the result obtained in part (a) above to approximate [¹/4 do. -1/4 (iii) Approximate [¹/4 e de using Simpson's rule with n = 8 strips. How does the approximate answer in (iii) compare with the approximate answer in (ii)? (b) If A dollars are initially invested in a bank account which pays yearly interest at the rate of x%, then after n years the account will contain An= Ao(1+x/100)" dollars. The amount of money in the account will double (i.e. An = 2 Ao) when n = log 2 log(1+x/100) (i) Use the linear approximation formula given above (in part (a)(i)) with a suitable choice of f(x) to show that x log(1+x/100) ? 100 (ii) Hence, show that the number of years n for the sum of money to double is given approximately by 100 log2 70 n? I (This is known as the "Rule of 70".)