MATH 101/210: Quiz #6March 6, 2014Name:Student Number:Simplify your answers as much as possible. Show your work.This quiz is worth 6 marks.1. [3 marks] Use the Comparision Theorem for Improper Integrals to determine whether the following integral isconvergent or divergent:I=Z∞1x3+ 2x4+x+ 1dx.We find a function to compare the integrand to as follows. For allx≥1,x3+ 2x4+x+ 1≥x3x4+x+ 1≥x3x4+x4+x4=13x≥0.Therefore by the Comparison Theorem, the improper integralIdiverges, since13Z∞1dxxis a divergentp-integral, withp= 1.2. Suppose thatf(0) = 1,f(1) = 3,f(2) = 0,f(3) =-1,f(4) = 0, and-4≤f(4)(x)≤3 for all 0≤x≤4.(a) [2 marks] CalculateS4(i.e., Simpson’s approximation toR40f(x)dxwithn= 4 intervals).