Show that the functions x and x2 are orthogonal in P5 with inner product defined by (5), where xi = (i − 3)/2 for i = 1, . . . , 5.

Q: Find the integration norm of the given polynomial on C(0, 1]. f(x) = -6x – 4 || f|| = Ex: 1.23

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Q: Compute the double integral of (x – 2y) Væ + 2y o i over the quadrilateral with vertices (0, 0), (4,…

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Q: Evaluate the integral. 2 dt 1- t2 V

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Q: (2) Let F(x, y) = (cos(y)+2x, -x sin y+1). Show that F is path independent by finding %3D a…

A: For the given vector function to be independent, it must be derived from a potential scalar function…

Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

A: Given:

Q: 3. Show that the line integral 4. x + y)dx + dy C is independent of path and evaluate this integral…

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Q: 2) Use Green's Theorem to compute f F df where F= (3x +4y)i + 2xyj and C is the triangular path from…

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Q: dy is If y = In then cosh z

A: Given,          y=lnxcoshx

Q: Evaluate the line integral §. xy²dx + 2x²y dy over triangular path with vertices (0,0). (2,2) and…

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Q: Given that f xdx = 1/2 and f²dx = compute the integral [*(1-x)²dx.

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Q: -JIL e √x² + y² dv, where E is the region that lies inside the cylinder x² + y² = 4 and between the…

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Q: *) Use the Wronskian to determine whether the functions y₁ = sin(3x) and y₂ = cos(4x) cos(4x) are…

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Q: 2. Consider the space L ([1,2)) of complex square integrable functions on the interval [1, 2).…

A: Consider the  equation

Q: Let y1 = 9cos(2x) and y2 = 2cos2 (x) – 2sin? (x). Calculate the Wronskian, W(y1, Y2). Give your…

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Q: dx 2 -o x+4x+5

A: The general form of a complex number is given by , where a, b are two real numbers and . Any complex…

Q: Show that the integral S. (3x²y + y)dx + (x3 + x)dy is independent of path, if it is, then evaluate…

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Q: Consider the following integral 3dx V(16 + 4.x2 )5

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Q: Evaluate the given line integral. I (y, –x) · dR, where C consists of 4y = x² from (-2, 1) to %3D C…

A: Line integration of the function

Q: Show that y^3dx+3xy^2dy=0 is separable and exact

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Q: Consider the definite integral. (8x- – 1) e*x²-x dx Let u = 4x2 - x. Use the substitution method to…

A: Answer and explanation is given below

Q: Let z(a, y) = 7x – 6y where x = - 7s – 4t & y = – 2s + 8t. %3D %3D az Calculate as az by first…

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Q: 2) Use Green's Theorem to compute f F. df where F=(3x +4y) i + 2xyj and C is the triangular path…

A: Use green's theorem

Q: 2) Use Green's Theorem to compute f F df where F=(3x + 4y)i + 2xyj and C is the triangular path from…

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Q: Find the product of the integral 1- S x²cos(xy)dxdy

A: We have to change order of integration first.

Q: Derive the Simpson rule with f(x) = ax^2 + bx + c going through points xi−1, xi, and xi+1. Determine…

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Q: Evaluate the given integral where C is the directed line segment that joins the point 1 to the point…

A: The given problem is to evaluate the given path integral over given path of line segment joining…

Q: Find the line integral of F =9xy*ï +(x² + 3),yj over the path L.

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Q: Let xbe inner Product and ||x1|=|ly|| = 1 x+y Lix.y ? why?

A: Given that X is an real inner product space. x ,y be two unit vector from X then need to show that…

Q: Find the solution in the form of Fourier integrals: U₁-3 = 0, uz (0, t) = 0, |u(x, t) bounded as…

A: We have the following partial differential equation : ut - 3uxx = 0  , x,t > 0 With boundary…

Q: (a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by…

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Q: 7. Use the Fundamental Theorem of Line Integrals to evaluate S(6xy, 3x² + 6y²) · dr where C is any…

A: Line integral theorem

Q: Evaluate the improper integral t* e-2t 3+e-4t at

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Q: Find the Selution of (1-t +3t)4 (t) di - , (2) In which IL) denoter Bessel function

A: Leibniz integral rule : ddx∫bxaxFtdt=∫bxax∂∂xFtdt+Faxddxax-Fbxddxbx   Linear Volterra Equation of…

Q: Compute the path integral of F = ⟨ y , x ⟩ along the line segment starting at    ( 1 , 0 ) and…

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Q: Let {yk } be the DFT of {x; }, where x = [0, –1, 0, 1] y3 = O i -1 -i 1

A: Here, x0=0, x1=−1, x2=0 and x3=1 We have to find y3 Now, y3 is given by…

Q: Show that the following line integral is path independent an (3, 2) S cosh(+ X (0,0) (8xdx+6ydy)

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Q: is Theorem 1.6. The line integrals of f in D are independent of path if and only if f has an…

A: Given: The line integrals of f in D are independent of the path if and only if f has an…

Q: 2. Let z = sin(xy) + x siny,x=u² + w² and y = uw. 2a. Compute Vz= when u = 0 and w = 1.

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Q: Verify Green's Theorem by evaluating both integrals v² dx + x² dy dA ax ду for the given path. C:…

A: This question is related to vector calculus solved in step 2.

Q: #1. Evaluate (2x – 5y) dA, R where R is the parallelogram with vertices (0,0), (2,3), (–1,4) and…

A: The given region R is a parallelogram with vertices as shown in below graph. We have to use change…

Q: (1) Show that F(x, y) : (2x + y, x - y) is path independent and use the fundamental = theorem of…

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Q: 3. Show that the line integral | (r + y)dx + xdy is independent of path and evaluate this integral…

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Q: Evaluate the path integral between the points A = (1,3) and B = (2,5) of: B *(x²y + 2y)dy, with y =…

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Q: O Evaluate the path integral between the points A = (1,3) and B = (2,5) of: (x²y + 2y)dy, with y =…

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Q: Use the fundamental theorem of calculus for path integrals to evaluate f.(yz2, xz2, 2.xyz). dr where…

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Q: Use the Integral Test to determinOwheher the scrie sure to show that the hypothescs of the Intogr al…

A: Given

Q: Suppose w = In(æ² + y² + z²) , Compute dw/du and dw/dv in two different ways: ue3" sin(9v) , y :…

A: GIvenw=ln(x2+y2+z2) where x=ue3vsin(9v) , y=ue3vcos9v and z=ue3vTo find∂w∂u and ∂w∂v

Q: Compute the following Cauchy principal value integral: dx р.v. x4 – 1 п/2 -T/2 Зп /2 -27

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Q: Evaluate the integral dz dy dr. -2 Jo 5-722-7y2 32

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Q: (B) Compute div F and curl F for F= x²yi- (z3-3x)j + 4y2k %3D

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