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Suppose the bounded function f on [a, b] is Riemann integrable over a, bj, Show that there is a sequence {A) of partitions of la, b] for which limn→ oo [U(f, Ph)-Lu, Pn] = 0.
κ. If / is continuous ou ¡a,bj xnd F(z) nodt, then F is differnntiable on 1, b, h. If / is integrable un la.บุ, then it has an antiderivative G on ps, bị and J 1(r) dr-G(b) i If i1 is bounded, thea (onlis converget j. If / has infinitely wauy discostinuities on Io., then f s niot integrabie on ja b. G(a)
κ. If / is continuous ou ¡a,bj xnd F(z) nodt, then F is differnntiable on 1, b,...
6. Let f [a, b R be a thrice differentiable function and xo E [a, b]. Show that da
6. Let f [a, b R be a thrice differentiable function and xo E [a, b]. Show that da
4a. (5 pts) Let f, g: [a, b -R be integrable. Show that la, blR, {f (x),g x)) h h (x) max and k[a, bR, k (x) min {f (x),g (x)) integrable. Hint: Observe that, for all a, b e R, max{a, b}= (a+ b+ la - bl) and min{a, b} (a+b-la -bl). are
i? (x, y) dA over the rectangle R - [a, b] x [c, d can Problem 1 Show that the integral be computed in terms of the numbers f(a, c), f(a, d), f (b, c) and f(b, d) 5 marks]
i? (x, y) dA over the rectangle R - [a, b] x [c, d can Problem 1 Show that the integral be computed in terms of the numbers f(a, c), f(a, d), f (b, c) and f(b, d) 5 marks]
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
QUESTION 3 How many unknowns are at the support C (roller)? L F 2 LA 2 B KL- 45°
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
4. Assume f and |fl are both integrable on la, b], a < b, show f(x) dx sf( d. Hint: there are several approaches. One good way is to apply the triangle inequality of series TL T& a where a, E R, ail s on the Riemann sums.
4. Assume f and |fl are both integrable on la, b], a
Can you please also explain how you did it? Thank you.
i? (x, y) dA over the rectangle R - [a, b] x [c, d can Problem 1 Show that the integral be computed in terms of the numbers f(a, c), f(a, d), f (b, c) and f(b, d) 5 marks]
i? (x, y) dA over the rectangle R - [a, b] x [c, d can Problem 1 Show that the integral be computed in terms of the numbers f(a,...