(a) Write the Maclaurin polynomials po(x), pi(x), p2(x), p3(x), p4(@) for e". Evaluate po(1), pi(1), p2(1), p3(1), p4(1) to estimate e. (b) Let Rn(x) denote the remainder when using p,(r) to estimate er (similar idea of RN we've seen before), so that Rn(x) = e" -pn(x) and Rn(1) = e-Pn(1). Assuming that e = = for integers r and s, evaluate Ro(1), Ri(1), R2(1), R3(1), R4(1). (c) Using the results from part 2, show that for each remainder Ro(1), Ri (1), R2(1), R3(1), R4(1), we can find an integer k such that KRn(1) is an integers for n = 0, 1, 2, 3, 4. (d) Write down the formula for the n Maclaurin polynomial P,(x) for er and the corresponding remainder Rn(x). Show that sn!Rn(1) is an integer. (e) Use Taylor's theorem to write down an explicit formula for Rn(1). Conclude that Rn(1) # 0, and therefore, sn!Rn(1) # 0. (f) Use Taylor's theorem to find an estimate on Rn (1). Use this estimate combined with the result from part 5 to show that (sn!R (1)| 4 se . Conclude that if n is large enough, then (sn!Rn(1)|