Context: For much of this written assignment, we will explore various uses of Taylor polynomials. You may take for granted the following: Given a function f : R - R and its Taylor polynomial Pr expanded around a =0, it is a fact that there is c in between' 0 and I such that f ( @) - Pr(c) = juh(c) (n + 1)! (E) Here f(+1) is the (n + 1)th derivative of f, and recall that PR() = A) (0 ) & 12* = f(0)+ 0)z+ 0,2 + .. . - f()(0) k! 2 n! You may recognize the n = 0 case as the "Mean Value Theorem." Problem 1. (i) Using (E) with n = 1 and f(x) = In(1 + 2), show that In(1 ty) 0. (ii) Using (E) with n = 2 and f(I) = In(1 + 2), show that y - 0.(ini) Use (t) and (it) to deduce that, for every z ( [0, 1], T 2 Hint: lety = c'. Does y satisfy the conditions of (i) and (4)? (iv) Use (ui) to show that 10 In(1 + 2) ) de