f: [o,17 s R. )m =l,2,3, fcx) = 3. otherwise Prove that f t R[O,1]

Q: 1. Use an antiderivative to show that for every contour C extending from a point z₁ to a point 22, 1…

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Q: Prove Theorem (A + B )T = AT + B T

A: Prove Theorem (A + B )T = AT + B T

Q: The displacement function is given by U(A, t) = X(A, t) – A. (6.1) Because X(A,0) = A then it is…

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Q: Let f (x, y) = g(u), where u = x² + y² and g(u) is differentiable. Prove that () = 4u du + ax 2.

A: Given, f(x,y)=g(u)=x2+y2 To prove that, ∂f∂x2+∂f∂y2=4udgdu2

Q: Let T : M2x2(R) → P2(R) be defined by T с d = (a + b) + (2d)x+ bx². Let B = [(6 ") (C ) G ) C ) 0 0…

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Q: Given that F(x, y, z)=sin xi+ cos yj + xzk. Evaluate ſ. F.dr where C is given by r(t)=( r', –f,t) ;…

A: Given that

Q: Let r = xi + yj + zk and let r = ||r||. Prove that

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Give V = C([0, 1]) the inner product defined as follows: (f, g) :| f(t)g(t) dt for all f, g e V.…

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Q: Prove that flx)= (i+x)? + X and diverge Litx)? both

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Q: et T E L(V,W), and U e L(W, X). Prove that UT : V → X is also linear

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Q: Let g(r, y) = ye-, r(t) = In(t² + 1), and y(t) = vt. Find g(r(t), y(t)).

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

Q: KB is ∀xy F(x, y). Prove using resolution-refutation that ∀xy F(y, x).

A: Solution: First, we convert clauses to CNF : (1) ∀xy Fx,y from KB (2) ∀x Fx,Ax (3) Fx,Ax CNF

Q: Find a mat rix P such that p-l 1 0 1 0 %3D 3 2 -1 0 P%3D OA. 3. 13 P = В. 0 1 1 0 P = C. 3 1 1 0 [1…

A: In the question we have to find P matrix.

Q: Determine la derivada direccional de f (x, y) = y In z en el punto P (1, 4) en la dirección de v = i…

A: Topic:- calculus

Q: Let F(x, y,z)=xi + yj+zk, f(x,y,z)=||F(x,y,z)||. Prove that: Vf" = nf"-²F F v(In f) =

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Q: Given that w = f (u, v) And u = xz. , V = dw = X az aw aw Prove that Z - ду

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Q: Given that n ≥ 2 , x ∈ Rn  Prove that there exists a y ∈ Rn in where y ≠ 0, and xy = 0

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Q: Given that vec F =x^ 2 y hat i -(z^ 3 -3x) hat j +4y^ 2 hat k find the k- component of ( nabla* vec…

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Q: Let r(t) = ⟨t, t, t⟩. Compute r'(t) and interpret the result.

A: rt=t, t, t

Q: Compute (S ,g) where f(t)=1– 3r² , g(1)=t-ť'onV = C[0,1]. %3D

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Q: Compute F. Oszs V1-x2 –y? Let E be the hemisphere described by and F=Vz2 + x² +y² i + Vz² + x² + y²…

A: By Gauss divergence theorem:∬∂EF·nds=∭Ediv(F)dvhere givenF=x2+y2+z2( i^+j^ +k^…

Q: Evaluate O F.dT where +(3y %3D y + 1 and C is the triangle with vertices: (0,0), (6,0), (3, 5).

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Q: Let T : V --> W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty. Prove that if s ϵ K,…

A: Given: Let T : V --> W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty The objective is…

Q: Prove that the function f(x1,x2) = In (e*i+= + eV=i+1) is convex over R².

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Q: If f(x, y, z) = In(x7 + y +2)+V + y + z), then find fr, fy and fz at (1, 1, 1). %3D

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Q: if x, y>o such that x Sy then Vã svy Prove by direct method

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Q: Prove that f, uv v • dỉ = – §̟ vũu • dĩ

A: Whenever the force is conservative in nature the work done over the closed loop is always zero. i.e…

Q: a²f If f(x,y) = e" sin (r + y), then prove that дхду дудх %3D

A: Hi! You have posted two non-related subparts. As nothing is specified, we are answering only the…

Q: Let x, y ∈ R. Suppose x2 + y2 = 0. Prove that x = 0 and y = 0.

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Q: Compute the subdifferential of f(x) = max(1-r,0), for r E Rn.

A: As per the question we have to find the subdiffernetial of the function f(x) = max (1-x,0) And we…

Q: Consider f, g E T° (R") and 0 > 0. Show the following results on the subdifferential. (a) a(0f)(x) =…

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Q: (a) Let r = ri+ yj + zk and let r = ||r||. Prove that - (1) a

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Q: For z ∈C, define Tz: C →C by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint,…

A: Firstly, find out Tz*.

Q: Define T:R → R' by T(x, y, z) = (x-5y+4z, y- 6z). a) Is T one-to-one? Justify your answer. b) Is T…

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Q: (a) Let f(r, y) = x - 3ry + 18y, where 3r?y- y- 6x = 0. Prove that f(r, y) has extrema at the points…

A: Calculate partial derivative of f(x,y) with respect to x and y ,

Q: Let z, w e C. Clearly prove that z+w = z + w.

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

Q: Let z, w E C. Show that zw = 0 implies z = 0 or w = 0. %3D

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Q: a. Show that T: 1² → 1² defined by Tx = y = (n), n, = r, is compact where x = (5,) and where0 <r< 1.

A: Show that T: l2→l2 defined by Tx=y=ηj, ηj=rjξj is compact where x=ξj and 0&lt;r&lt;1. Definition:…

Q: Consider ф: C" — C defined by ф(x) %3D х*. a) Kerф %3D (1,-1} b) Kerø = {i,-i} %3D с) Kerф %3D (1,1}…

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Q: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by fX,Y (x, y) = x + y/21…

A: Solution

Q: Let 1 an(r) : u(reie)e ino de %3D Prove that:

A: .

Q: Let r(t) = (3 cost, 5 sin t, 4 cost) . Show that llr(t) II is constant and conclude, using Example…

A: Consider rt=3 cos t, 5 sin t, 4 cos t. rt is always non negative.  Calculate rt rt=3 cos t2+5 sin…

Q: (a) Prove that if f e C (R²) then a (af af %3D

A: This is a problem of the calculus of several variables, higher dimensional calculus.

Q: Prove that (A+B)T=AT+BT

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Q: b) Show that the following wo matricegane orthogonal tker A =-V3 -2VTO 3/T15 15 E/NTS

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Q: let F= Mit Nj + Pk show that Zt)zde +. p人(2人 gYG,Z)ydf t JM(W, Y, 2)dw

A: .

Q: Suppose there are gj: TR continue when j = 1, 2 with 91 92 and T is a cube contained in Rn-1 With…

A: Solution : To prove- ∫BfdV=∫T∫g1(x)g2(x)f(x,y)dy dV(x) B={(x,y) : x∈T,g1(x)≤y≤g2(x)} apply set B…

Q: U R Given: ZR e ZV Prove: A RST AVUT T.

A: The solution is given in the next step.

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.