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PROBLEM SET 4 (DUE IN LECTURE ON OCT 9 (FRIDAY)) (All Theorem and Exercise numbers are references to the textbook by Apostol; for instance “Exercise 1.15-3” means Exercise 3 in section 1.15.) Problem 1. Suppose a function f : [0, 1] → R is monotonic and satisfies the equations 1 for 0 ≤ x ≤ 31 2 f (3x) f (x) = 12 for 13 ≤ x ≤ 23 1 + 1 f (3x − 2) for 2 ≤ x ≤ 1. 2 2 3 R1 R 1/3 R 2/3 R 1 R1 Compute 0 f (x)dx. (Hint: compute 0 f, 1/3 f, 2/3 f in terms of 0 f .) Problem 2. Compute the integral Z 9q √ 1 + xdx. 0 √ (Hint: Use the same technique used to integrate x in lecture (Example 5 in Section 2.3).) Problem 3. Let p be a polynomial. Define functions q and r on the reals by the indefinite integrals Z x q(x) = p(t)dt 0 and Z r(x) = x q(t)dt. 0 Prove that Z x (x − t)p(t)dt r(x) = 0 for any real x. (Hint: check this first for p(x) = xn , then prove it for arbitrary polynomials.) Problem 4. (a) Let x1 < x2 < · · · be an infinite increasing sequence of real numbers. Let X = {xn | n ∈ N}. Prove that if sup X exists, then sup X is a limit point of X. (b) Give an example of a set of reals S ⊆ R such that sup S exists but is not a limit point of S. Problem 5. Let m and n be positive integers. Compute the limit 1 − xm . lim x→1 1 − xn Problem 6. Consider the function f : R → R defined by ( x for x ∈ Q f (x) = 0 for x ∈ / Q. At which real numbers is f continuous? 1