For an integrable function f(x), there is only one antiderivative F(x). Select one: a. False b....
The integration of an odd integrable function f on a symmetric interval (-a, a) is always zero. Select one: a. False b. True
If F(x)is an antiderivative of f (x), then f (x) = F(x). True False Previous
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except (6)...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except (6)...
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00 Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
exercice 6 6. The goal of this problem is to prove that a function is Riemann integrable if and only if its set of discontinuities has measure 0. So, assume f: a, bR is a bounded function. Define the oscillation of f at , w(f:z) by and for e >0 let Consider the following claims: i- Show that the limit in the definition of the oscillation always exists and that f is continuous at a if and only if w(f;...
Find an antiderivative of the function f(x) = 2x® (3x? +4)? What is a possible antiderivative of the given function? O A. F(x) = 6 (3x® + 4) 3 OB. F(x) = (3x® + 4) 3 OC. F(x) = (3x +4) 3 OD. F(x) = § (3x?+4) 3
Find an antiderivative of the following function. f(x) = ** - Enclose numerators and denominators in parentheses. For example, (a - b)/(1 + n). An antiderivative is Fºx = (1/5)*x^5+2/(15% ab.
Suppose that is integrable on [a,b]. → R is positive and integrable. Show that, f f(x) : [a,
Suppose that g is differentiable, invertible, with g'(x) 0. Suppose that f is any function so that the function h()f(g(a))g'(a) is integrable with anti-derivative H. Prove that f is integrable, and that nog . Is an antiderivative for J.