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: Consider the function f(x, X2, X3, X4) = x+3XX4+2x2 8xxX4 +3x+7xa? O a. Fis positive definite Ob.…

A:

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

A:

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: 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-…

A:

Q: Obtain the Laplace transform of the piecewise continuous function defined by [t, 0st<4 f(t) = 0, t24…

A:

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

A:

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…

A:

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

A:

Q: 1. a) Find the Riemann sum of f (x) = x² – 3x + 2 over [2, 4] for the partition given by 2 = xo < x1…

A:

Q: . Let f : [0, 1] R be defined by |-1, z = ; >50 f(x) = {0 |<a < }; 1, 21 (a) Show that f € R([0,…

A:

Q: Let f(z) be the function defined by Z|2| 2 0 f(z) = if z = 0, if z = 0. 1. Use the e, & definition…

A:

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

A:

Q: ) Let (x, y) €Rx R. Consider the following functions from R2 to R? r(x, y)) = (-y,x) and s(x, y) =…

A: Since you have asked for multiple parts, solving a) and b) , kindly post the remaining as a separate…

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).

A:

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)…

A:

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.…

A:

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)].

A:

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

A:

Q: Let f(x) be any function in C[a, b] with the values given in the following table. Evaluate the…

A: Note - the answer is not in the option.

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…

A:

Q: Sx-2, r<0, x+2, r 0 9. Let f: R R be defined by f(x) = ', then f-(]1,3[) (the inverse image of the…

A:

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…

A:

Q: Find the divergence of the field F. yj - xk F= 1/2 +: 2 OA. 3/2 2 -X O B. 3/2 2 X -y OC. 3/2 O D. 0

A:

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,…

A:

Q: 44. Prove: Let f: X-Y. Then, for any subsets A and B of Y, (a) f-1 [AUB] = f-¹ [A] Uf-¹ [B], (b) A…

A: and are subsets of

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.