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: Let G = D8 = (r, s | r8 = id = s², sr = r²s). : (a) (b) (c) Let N = (4). Show that N ≤G. Find the…

A: Solution:-  (a) To show that N is a normal subgroup of G, we need to verify that for any n in N and…

Q: Letf be a differentiable function over [0, 2]. The values of the constants a and b such that the…

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Q: Let f be a real valued function defined on [a, b] that is differentiable on (a, b). Suppose f′(x) ≤…

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Q: Compute [ f(x,y) dA, where f(x,y) = y²√ and D is the set of (x, y) such that x>0, y> x², and y<10-…

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Q: Let f(z) be an entire function such that |f(z)| < 1 on the unit circle and f(a) = 0 for z - a some a…

A: Here given f(z)≤1 and we have to prove f(z)≤z-α1-α¯z Where z lies on unit circle and f(α)=0 for some…

Q: Let f(x) = Væ. Find a value of c in [6, 14] such that f(c) is equal to the average of f on [6, 14].…

A: Consider the given function.

Q: In a survey, 1204 people were asked which aspect of dining out was most important to them. The…

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Q: 2. Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous

A: Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous.

Q: (c) Let f(z) be an entire function such that f(z) # 0 and f'(z) # 0 for all z e C. Define a function…

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Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

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Q: For r> 0 let A set A. = w:w= exp (¹) where 0 < |z| <r} ; determine the

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Q: Let f(2)= (2+1)|z|². (2.1) Use the Cauchy-Riemann equations to determine where f is differentiable.…

A: The given function fz=z+1z2 We have to find the points for which fz is differentiable and analytic.

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

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Q: Find the value of a,b ∈ R ,so that the following sets of functions are linearly independent for all…

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Q: a) Evaluate the Riemann sum for f(x) = x³ + x² - 6x +2, taking the sample points to be right…

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Q: 3. Let fn(x) = ¹+cos(nx), x € R. Find f(x) so that fn → ƒ pointwise on R, then check whether fn → f…

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Q: Consider the function vec : Cmxn → Cmn such that for any A E Cmxn A*1 vec (A) = = A*2 : A*n Prove…

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Q: Compute the left and right Riemann sums L4 and R4, respectively for f(x) = √4x² on [-2, 2] and…

A: We can find Left and Right Riemann sums .

Q: Let f(x) be a function continuous on [1, 3] and differentiable on (1, 3), such that f(1) = 1, f(3) =…

A: Given that , f(x) be a function continuous on [1, 3] and differentiable on (1, 3), such that f(1) =…

Q: Prove that there exists a unique function f : R → R such that f (0) = 0 and f ′(x) = sin(x)

A: We have to prove that there exists a unique function satisfying given conditions.

Q: Consider F and C below. F(x, y, z) = yzexZi + eײj + xyeײk, C: r(t) = (t² + 2)i + (t² − 1)j + (t² −…

A: In the given question we have to evaluate : a) The function f(x, y, z) .  b) Evaluate the integral.

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: 2. Determine if a function f : [0, 27] → R defined by f(x) = sin 2n is continuous.

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Q: Let f(x) = x2 -2 on the interval [0,2]. Compute the fourth right Reimann sum, call it R4, and sketch…

A: To calculate the right end Riemann sum with 4 equally spaced sub-intervals, and sketch the graph

Q: Let f: RR be a function satisfying f(x + y) = f(x) + f(y) for all x, ye R. If the function f(x) is…

A: Note :   Let r&gt;0 and c∈R ,then vr(c)={x∈R : x-c&lt;r }=(c-r ,c+r) ,which is called r…

Q: c. Let f be differentiable on R. Suppose that f'(x) # 0 for every r. Prove that f has at most one…

A: We can explain this statement by 2 explainations given below

Q: f Let f:(R,1,) - (R,T,), f(x) = x². Find f-(a, b) and then show that is continuous on R.

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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: Consider the function f(x) = x³ + 2x on the interval [2,5]. Given the partition P = {1.8, 2.2, 3.4,…

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Q: Let f be an analytic function on C: |z-1+i=2 with |f] 4 b. [f"(1 – i)| > 2 c. If"(1 – i)| < 2 d.…

A: here option (b) is true.

Q: 8. (p f(A) ≤ f(B), then ACB. Let f: XY. Prove that for all A, B C X, if f is one-to-one and

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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: Define the function S: R3 → R3 according to S(v) = Av,   A= 3 -4 -2 1 2 5 0 11 8 Describe S…

A: According to the question, the function can be written as S: R3 to R3 can be written as S(v) = Av…

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.