calculus

Problem 14. If x0 and x1 are positive numbers and sn = 1 2 (xn + xn??1); prove that the sequence converges. (Hint : use the "Nested closed intervals Theorem") Problem 15. Prove the identity (which appears in the Weierstrass Approximation Theorem): x is a real variable and n is a positive integer, Xn k=0 n k xk(1 ?? x)n??k x ?? k n 2 = x(1 ?? x) n : Problem 16. (TextBook : p.92 No.7) Let P(x) = xn +a1xn??1 + +an where n is an even positive integer, the A's are real , and an < 0. Show that the equation P(x) = 0 has at least two real roots. What more can you say about them? Problem 17. (Textbook, p. 92 No. 9.) Let f(x) = A a2 + x + B b2 + x + C c2 + x ?? 1; where A;B; and C are positive and a > b > c > 0. Discuss the nature of the graph of y = f(x) and explain why the equation f(x) = 0 has exactly three roots x1; x2; x3 , satisfying the inequalities ??a2 < x1 < ??b2 < x2 < ??c2 < x3. 1 2 PROFESSOR CARLOS J MORENO Problem 18. (TextBook p. 104, No 6.) Show that ex = 1 + x + x2 2! + + xn n! + Rn+1 with 0 < Rn+1 < ex xn+1 (n + 1)! ; if 0 < x and jRn+1j < jxjn+1 (n + 1)! ; if x < 0: Problem 19 .Suppose N > 0; x0 > 0, and xn+1 = 1 2 N xn + xn ; n 0: Prove: For n 1, xn > xn+1 > p N and xn ?? p N 1 2n (x0 ?? p N)2 x0 : Note: This is Newton's algorithm to approximate the square root of N starting with the approximation x0 .