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 I have included a picture of the question (and the referenced problem 11.5), followed...
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...
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...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
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',...
(2) Follow the steps below to prove Theorem 7.2.8: A bounded function / : [a,b] + R is integrable on (a, b) if and only if Ve > 0, there exists a partition P. of (a,b) such that UC, P.) - LIS, P.) <E. (a) Explain why the existence of the partition P. implies that L(I) = (/), and therefore that is integrable. (b) Conversely, ilis integrable, then there exists partitions P, Q of [a,b] such that UU,P) - L(Q)<E...
state any definitions or theorems used Question 2. In this problem we'll prove that if a<b<c and f is integrable on [a, cl ther it's also integrable on [a,b] and [b, c'. Our approach will be to show that for all ε > 0 there are partitions Q1 and Q2 of [a, b) and [b, c] respectively with Thus, let ε > 0 be given. By our fundamental lemma there exists a partition P of [a, c) such that U...
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...
1. Using the Epsilon Criterion on a function with one discontinuity Consider the function g : [0, 2] + R where g(1) = 5 and g(x) = 1 otherwise. a) Find a partition P of (0, 2) so that U(9, P) - L(9, P) < 1/10. b) Is there a partition of (0, 2) so that U(9,Q) - L(9,Q) < 1/600? If so, find one! c) Suppose e > 0. Construct a partition Pof (0, 2) so that U(g, P.)...
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
2. More Rational Fun (a) Spend two to three minutes in deep meditation on Darboux's Theorem. Pay special attention to the part in bold Darboux's Theorem on Integrability Let A C R" be bounded and let f:A-R be bounded as well. Suppose E is a bounding rectangle of A. Then f is integrable over A and f-1 iff, for every ε > 0, there is a δ > 0, such that for every partition P of E with size Pll...