Math 1910, Prelim 3November 27, 2012, 7:30 PM to 9:00 PMYou are NOT allowed calculators, the text or any other book or notes. SHOW ALL WORK!Writing clearly and legibly will improve your chances of receiving the maximum credit thatyour solution deserves. Please label the questions in your answer booklet clearly. There are5 questions.Write your name and section numberon each booklet you use. You may leave when you havefinished, but if you have not handed in your exam booklet and left the room by 8:45pm, pleaseremain in your seat so as not to disturb others who are still working.1. (20 points) Supposef(x) = 6x2+ 2x+ 1.(a) CalculateR10f(x)dxexactly.SolutionZ10f(x)dx=Z10(6x2+ 2x+ 1)dx= 2x3+x2+x10= 2 + 1 + 1 =4.(b) CalculateT1the estimate forR10f(x)dxfor one interval for the Trapezoidal Rule.SolutionT1=12(1-0)[f(0) +f(1)] =12[1 + 9] =5.(c) Calculate the error estimate forT1, which isK2(b-a)3/(12N2), whereK2is themaximum of the absolute value of second derivative forf(x) on the interval [a, b],andNis the number of subintervals.Solutionf0(x) = 12x+ 2,f00(x) = 12.Thef00(x) is constant, hence its maximum value is same as that,K2= 12. Withb= 1,a= 0 andN= 1, the error estimate is:E= 12(1-0)3/(12) = 1(d) Calculate the actual error|R10f(x)dx-T1|. How does this compare to the errorestimate in Part (c)?