A) If S(f.p) is any Riemann sum of f(x):[a,b]-R. Show that there exist a step function o:(a,b]→R. such that fo= S(s.p).

Q: Suppose f is integrable on [-b, b] and f is an odd function-that is, f(-t) = -f(t) for all tE(-b,…

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Q: Verify that fxy = fyx for the following funtion.  f(x,y) = ex + y + 1 fxy = ? fyx = ?

A: Given,            fx,y=ex+y+1

Q: sine : N → [-1, 1] n H sin(n) 14 18 20 22 24 26 28 30

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Q: Suppose that A,B⊂S such that Pr[A|B]=3/7, Pr[A]=3/10 and Pr[B′]=13/20. What is Pr[B|A′]? Write as a…

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Q: For x,y E [0,1], prove that xy(1-x-y)≤1/27

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Q: Let f : [0, 1]→R be defined by f(x) :=x3 and let P :={0, 0.1, 0.4, 1}. Compute L(P, f) and U(P, f).

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Q: Let f:Rx R→ Rx R be given by f(a,b) = (3a + 2b,2a –b) a, b) ERx R. 1) Find f(2,3) and f(– 1,4). 2)…

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Q: B). For what values of r, reR is the function [(x+y+z) ƒ(x,y,z) = { x² + y² +z² 0, continuous on R.…

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Q: Consider the function θ : {0,1}×N→Z defined as θ(a,b) =a−2ab+b. Is θ injective? It is surjective?…

A: According to the given information it is required to check whether the given function is injective,…

Q: Q4. Let A E Rnxm, n ≥ m. a) Use the SVD of A to deduce the SVD of ATA. b) If m = n and A is…

A: Given: The given expression is, A∈ℝn×m , n≥m a) To use the SVD of A to deduce the SVD of ATA b) To…

Q: Explain how the evaluation for each of S. f(x, y) ds, ſ. f(x, y) dx, ſ. f(x,y) dy, and F ( x, y) · d…

A: We have to evaluate each integral. They defines as follows-

Q: Assume that f : R → R is differentiable everywhere and satisfies for all h, x E R: f(x + h) – f(x) =…

A: Let's prove this.

Q: Prove: If f € R(a) on [a, b] and if |f(x)| < M on [a, b] , then | S. f da| < M[a(b) – a(a)].

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Q: EX:- Find the inverse F.T. of F(W) = |n|2+jw/. J Sols du F(W) 2 + jw *j jd FCW) = = dw 2+jw ƒ…

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Q: Prove: If fe R(a) on [a, b] and if a <c < b, then f e R(a) on [a,c] and on [c, b] and S f da = f da…

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Q: Calculate the Riemann sum R(f, P,C) for f(x) = x, P = {1, 1.3, 1.7, 2}, C = {1.2, 1.5, 2}. R(f, P,…

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Q: Let f:R →R ond q: R→ R. Define the pointwise sum f+g as follows! frg= {CX.y): XEIR and y=f(x)+g (xl3…

A: The answer is given as follows

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.