Assume f(x) = ) anx" converges on (-R, R). Show that F(r) = o n+1 is defined m%3D0 n+1 on (–R, R) and satisfies F' = f on (-R, R).

Q: SUppose F(x) = dt. Find the formda 13 F (x ³) expressed using an integral. ? F (x* )= ]

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Q: Let xn -> 0 and (yn) be bounded. Prove xnyn -> 0.

A: A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k,…

Q: Provide a function f, continuous on (0, 1), such that for all r where f'(x) exists, f'(x) > 0 but f…

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Q: Suppose f(x) is continuous on [3, 8] and -5 < f'(x) < 3 for all x in (3, 8). Use the Mean Value…

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Q: Draw a graph for following f' > 0 on (-∞,1) and f"l > 0 on (-∞, 1) U (1, ∞).

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Q: What would be an example of a function that is not integrable on [2, 10] and why is it not…

A: Consider a function: y=1x-2……1

Q: Suponga que f(r) = -2 y que esta función se define en intervalo [-0.5,1]. El mínimo absoluto de…

A: Answer to the question is as follows:

Q: Determine the average rate of change of f(x)=2x+1 on the interval 4<x<5. a) f Ax b) Af Ax = 2 O d) 4…

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Q: Let f(t) 5t4 - 2t + 6et. Then f'(t) = =

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Q: oncave up on (-16/1. creasing on (4/11, 4) a local max at x = 4. an inflection point at x

A: Function: f'(x)=-22x+8(x-4)3f''(x)=44x+64(x-4)4f(x) is continuous in its domain (-∞,4)∪(4,∞)

Q: Is the following function analytic in |z|2 : (z^2-1)/(z-(sqrt(z^2+4))

A: Given:We have to find the following function is analytic or not. =Ƶ2-1Ƶ-Ƶ2+4

Q: Use the given graph of f(x) to find the intervals on which f'(x) > 0. 20+ 15 10 5+ -7 6 -5 3 2 -1…

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Q: Given the graph of ƒ' (x) below, where the Dom f (x) = (-∞, ∞) . Find ALL interval(s) for which f""…

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Q: Show that if F: AB and G, H F"" (GUH) = F"" (G) U f" (H) and F" (Gn H) = F²¹ (6) ^ F-1 (H) are…

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.