FindtheTaylorpolynomialofdegree2forthefunctionf(x)=3x+ 1at x= 2, together with the associated remainder term. [6 marks] (ii)Usingthesecond-andthird-orderderivativesofthefunctionf(x)youfound inpart(i)above(andifnecessaryalsoitsfourth-orderderivative)establisha formulaforthenth-orderderivativeofthefunctionf(x)whennisaninteger greater than or equal to 2, and hence use this to obtain the Taylor series of the functionf(x) =3x+ 1 atx= 2. [5 marks] NotethatyouarenotrequiredtofindtheintervalofconvergenceoftheTay- lorseriesinpart(ii).Alsonotethatasyoucanonlyfindageneralformula forf(n)(x),n2,itisclearthatyouneedtowritetwotermsoutsidethe summation. (iii)Using the binomial theorem, find the Taylor series forf(x) =3x+ 1 atx= 2 and check your answer against part (ii) above, also establishing the interval of convergence of this Taylor series. [6 marks] Consideraddingandsubtractingthesamenumbertox+ 1.