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(1) Find polynomials g(x), h(x) E Q[x] so that (x4 + 2)g(x) + (x2 – 3x + 3)h(x) = 1 Find all the zeros of x 103 + 3×2 – x + 3 in Q. (2) Show that 220 + (220 – 1)xc6 + (2100 – 1)x2 + 11! is an irreducible polynomial over Q. (3) Exercises 23 page 218: 12, 17, 20, 37 (4) Exercises 26 page 243: 1, 3, 13, 14, 15. Q2 Differentiation of Power Series 10 Points Let f,g:R → R be defined as power series: ~ xan x 2n+1 f(x) = Σ (2n)!’ g(x) = (2n + 1)! n=0 n=0 (a)(4 points) Show that f’ = g and g’ = f on R. (b)(6 points) Show that f? — gʻ = 1. (Hint: take derivative on both sides of the equation.) No files uploaded Q3 Convergence of Taylor Series 12 Points Let f(a) = ln(1 + x) be defined on (-1,00). (a)(5 points) Find the Taylor series of f centered at Xo = 0. (b)(7 points) Using the Lagrange Remainder Theorem to prove the Taylor series of f converges to f when a € (-1/2,0). No files uploaded Q4 Upper and Lower Sum 12 Points Let f : [0, 1] + R be defined as U if x = [0, 1] nQ f(x) = 10 if x € [0, 1] \Q = { (a)(5 points) Show that for any partition P of [0, 1], L(f, P) = 0. (b)(7 points) If P = {0, ì, 1}, find U(f, P). (c)(bonus 4 points) Show that for any partition P of [0, 1], U(f, P) > Ž, hence conclude that f is not integrable on [0, 1]. No files uploaded Q5 Integrability 14 Points Let f : [0,1] → R be defined as 1 if x f(x) = 1 for some n EN otherwise n { 0 Use Theorem 7.2.8(Integrability Criterion) to show that f is integrable on [0, 1]. No files uploaded

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