1. Let a, b E R with a < b and P { To,Ti,. . . ,Tnf be a partition of the interval a, b] Denote ΔΧ,-2 j-rj-1 for J-1, 2, .. . , n. Consider a function f : [a,Ы-R. What do we need to require from f in order to be able to define the upper and lower Riemann sums of f over P?
5. Let f : [a, b] → R be bounded, a : [a, b] → R monotonically increasing, and P a partition of [a, b]. (a) Define upper and lower Riemann-Stieltjes sums of f with respect to P and a. (b) Let P' be the partition obtained from P by inserting one additional point x' into the subinterval (2k-1, xk] of P. Prove that for the lower and upper Riemann- Stieltjes sums of f we have L(P, f, a) <L(P',...
Let n E Z20. Let a, b є R with a < b. Let y-f(x) be a continuous real- valued function on a, b]. Let Ln and R be the left and right Riemann sums for f over a, b) with n subintervals, respectively. Let Mn denote the Midpoint (Riemann) sum for fover la, b with n subintervals (a) Let P-o be a Riemann partition of a,b. Write down a formula for M. Make sure to clearly define any expressions...
5. Let f : [a, b] → R be bounded, and a : [a, b] → R monotonically increasing, (a) For a partion P of (a, b), define the upper and lower Riemann-Stieltjes sums with respect to a. (b) (i) Define what it means for f to be Riemann-Stieltjes integrable with respect to a. (ii) State Riemann's Integrability Criterion. (C) Suppose f is both bounded and monotonic, and that a is both monotonically increasing and continuous. Prove that then f...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Let f : [a, b] → R and xo e (a,b). Assume that f is continuous on [a,b] \{x0} and lim x approaches too x0 f(x) = L (L is finite) exists. Show that f is Riemann integrable. 1. (20 pts) Let f : [a, b] R and to € (a,b). Assume that f is continuous on [a, b]\{ro} and limz-ro f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into...
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
Problem 10. Let f,g: [a,b] -R be Riemann integrable functions such that f(x) < g(x) for all x E [a,b]. Prove that g(x)
3. Let f: R+R be a function. (a) Assume that f is Riemann integrable on [a, b] by some a < b in R. Does there always exist a differentiable function F:RR such that F' = f? Provide either a counterexample or a proof. (b) Assume that f is differentiable, f'(x) > 1 for every x ER, f(0) = 0. Show that f(x) > x for every x > 0. (c) Assume that f(x) = 2:13 + x. Show that...
Problem 11.11 I have included a picture of the question (and the referenced problem 11.5), followed by definitions and theorems so you're able to use this books particular language. The information I include ranges from basic definitions to the fundamental theorems of calculus. Problem 11.11. Show, if f : [0,1] → R is bounded and the lower integral of f is positive, then there is an open interval on which f > 0. (Compare with problems 11.5 above and Problem...