4. Let fn [0, 1] -> R be a sequence of Riemann integrable functions on pointwise, that is, for every [0,1], [0, 1]. Suppose fn f we have lim fn(x)= f(a) n oo Prove or disprove: f is also Riemann integrable

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A: We have to find (a) using a Riemann sum with right endpoints and n = 8 the value of the R8 and (b)…

Q: 9. Consider the annulus N function on 2 is the real part of a holomorphic function on N. {z € C :1 <…

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Q: 1. Use the definition of the definite integral as the limit of a Riemann sum to evaluate - 4)² dx.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

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A: We will be answering the first question as per the guidelines.

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A: Here we use the formula for definite integral :

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A: Given, ∫017-4xdx Consider n=4 now we find ∆n ∆n=b-an=1-04=0.25 Therefore the intervals are, 0,0.25,…

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Q: 3. Let {gn} be a sequence of integrable functions such that gn → g a.e. with g integrable. Show that…

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Q: 6. Consider the sequence of functions n²a* – sin(nx) fn(x) = x E (0, 00) n2x (i) Show that fn…

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Q: 4. Consider the sequence of functions {fn}=2 where fn: [a, b] R defined for n > 2 by if 0 < x <- fn…

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Q: Consider the sequence of functions n+ cos(x) 3n + sin(x) defined on the R. 1. Find the function f to…

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Q: Compute the definite integral as the limit of Riemann sums. 21) [ x3 dx -3 81 81 C)- 2 A) 27 D) 0 4

A: let I=∫-33x3dx claim- to find the definite integral using limit of riemann sum let fx=x3 where a=-3…

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Q: 3) Write the definite integral corresponding to the Riemann sum lim [4 – 3(x; )² + 6(x;)*] Aæ, [0,…

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Q: 2.1 Show that a sequence of Riemann-Integrable functions fn(r) converges uniformly to a…

A: As per our company guidelines, we are supposed to answer only 1st question at a time. Kindly repost…

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A: Hello. Since your question has multiple parts, we will solve the first part for you. If you want…

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Q: ¹).. Suppose that fn are integrable and ffn du <∞. Show that, if fn 1 f, then f is integrable and f…

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Q: Let the sequence {fn(x)} be defined on [0, 1] by n – n2x; if x E (0, ) fn(x) : otherwise SHOW THAT:…

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Q: 1 and yn+l = V2 + Yn for n > 1. Show (ym) is monotone, and Let yn bounded, and compute its limit.

A: Here, we have a recurrence relation for the sequence. by using that we need to check the sequence is…

Q: 10. Use Riemann sums and a limit to compute the exact area under the curve y = f(x) = 2x² + 1 on the…

A: Given÷

Q: 9. Consider the function f(x, y) = p |xy|. Show that Taylor’s expansion about the point (x, x) is…

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Q: 6. Let g be defined by x² sin ², x * 0, 9(x) = {x² sin ², 0, x = 0. (a) Prove that g is…

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Q: 5. Consider the function if x € Q f(x)= if x # Q. Show that f is not Riemann integrable on [0, 1]. n…

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Q: Exercise 1. Express the limit as a definite integral on the given interval cosxi a) lim E-1 Xị Ax,…

A: We have to find the derivative.

Q: What is the equivalent limit of Riemann Sum expression of the definite integral fo² (4- ²x) dx?…

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Q: Consider the nonsmooth function: 1 f(x) = max x; +;|x||², 1<i<K where x E R", x; is the i-th…

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Q: 14. Let f(x) = 4.x3 + 6x2 + 3. (a) Use a right-hand Riemann sum to approximate the definite integral…

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Q: . Let ƒ : (0, 1) → R be uniformly continuous. Let (Tn) be a sequence in (0, 1). Suppose that lim, s0…

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Q: (B) Let {fn(x)}1 = {r ln x. cos(x – 1)"} be a sequence of meaurable functions defined over [1,2].…

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Q: 17-20 Express the limit as a definite integral on the given interval. x* Ax, [1,3] (x) +4 20. lim E

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Q: 13. Write the definite integral for the Riemann sum: lim 5+ + .2 +. А. (5 + t)*dt В. t dt С. + 5)*dt…

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Q: . Express the following limit of a Riemann sum as a definite integral. 7 lim T(-5+k · n k=1

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Q: 2. Find a function f(x) such that n n 1 n+ i i=1 n n i=1 3. By evaluating an integral, compute the…

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