Exercises 35 (15) Show that x x0 f(u) du = O x x0 g(u) du if f(x) = O(g(x)) and f and g are integrable on bounded intervals. (16) Prove that if f and g are real-valued continuous functions on a closed and bounded interval [a, b] and g is positive there, then b a f(x)g(x) dx = f(c) b a g(x) dx for some c ∈ (a, b). This is called the first mean value theorem for integrals. (17) a) Prove that if f and g are real-valued continuous functions on a closed and bounded interval [a, b] and f is monotone there, then b a f(x)g(x) dx = f(a) c a g(x) dx + f(b) b c g(x) dx for some c ∈ (a, b). This is called the second mean value theorem for integrals. The second mean value theorem is useful for the estimation of integrals with an oscillatory factor in the integrand. b) Show that 2π 0 f(x) cos(x) dx ≤ |f(0) − f(2π)| if f is a continuous monotone function. (18) Show that lim h→0 b a f(x + h) − f(x) h dx = f(b) − f(a) if f is a continuous function on an open interval containing the closed interval [a, b]. Establish an integration by parts formula for two contin- uous functions, one of which is monotone. (19) Make a guess for the asymptotic behavior of the sum p≤x 1 √ p and use partial summation and estimates of Chebyshev to show that your guess has the right order of growth. (20) Show that pq≤x log(p) log(q) x log(x) where p and q denote primes.

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