Prove that d/da E[(X −a)2] = 0 ⇒E(X) = a   by differentiating the integral. Verify using the calculus, that a = E(X) is indeed a minimum.

Q: Evaluate the integral of (3x2 + 9y2) dx dy if the interior limits has an upper limit of y and a…

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Q: Show that [ xdhy – ydy c Vx*+y² is independent of any path of integration which does not pass…

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Q: Calculate the integral of f(æ, y) = 10x over the region D bounded above by y = x(2 – x) and below by…

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

A: Given:

Q: Find the quadratic approximation of f at x = 0 f(x) = sin In (3x + 1) O A. Q(x) = 1 + 3x + 5x 9 B.…

A: The given function is: fx=sin ln3x+1 To find the quadratic approximation of f at x=0

Q: Evaluate the line integral -ydr + rdy along the curve C : y² = 6x from (6, 6) to (0,0). NOTE: Enter…

A: The given line integral is ∫c-ydx+xdy along the curve c: y2=6x from (6,6) to (0,0)

Q: Suppose fhas absolute minimum value m and absolute maximum value M. Between what two values must |…

A: Apply formula for finding correct option

Q: Evaluate the integral. (Use C for the constant of integration.) t dt + 10

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Q: Evaluate the integration of the function. f(x) = 1+ e-x sinan over the fixed interval [a,b] = [0.1]…

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Q: express the length of the curve as an integral but do not evaluate it. r = eθ + 1, 0 ≤ θ ≤ π/2.

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Q: Evaluate the integral. 9. dx J2 (x² - 1 1)3/2 Additional Materials Type here to search

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Q: Show that (2x In yz-5ye") dx-(5e* -x²y)dy +(x²z- +2=) dz, where C runs from (2,1,1) to (3,1,e), is…

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Q: X' 2 -1 Vx * dx X,

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Q: Use Green's theorem to determine the value of the integral where the curve x2 + 4 = 1 traveled…

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Q: Calculate the integral of f(x, y) = 9x over the region D bounded above by y = x(2 – x) and below by…

A: We will evaluate the given integral over the region D.

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

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Q: Let y 5x²8x + 4. If Ax = 0.1 at x = 1, use = differentials to estimate Ay Ay ~

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Q: Evaulate the line integral of c (x^2+y^2) where c is the line segment from (-1,-1) to (2,2)

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Q: Apply Green's Theoren to evaluate the integral. C6x+y) dx + (3xy"+4y) dy ter

A: Given integration is ∮c(6x+y3)dx+(3xy2+4y)dy C : Any simple curve in the plane for which Green's…

Q: Find the local linear approximation of f(x) = In(5 + x) at xo = -4. %3D In(5 + x)

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

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Q: y = sech*x = In(_+ V1- x², sechx = In(- 1+ V1-x² Prove that .... ..

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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 line integral |-ydx + xdy along the curve C : y² = 24x from (24, 24) to (0, 0).

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Q: Compute the line integral of f(x)=2x over the curve y=x² from (0,0) to (1,1). |

A: The line integral of f(x)=2x over the curve y=x2 from (0,0) to (1,1)

Q: Compute the line integral of f(x)=2x over the curve y=x^2 from (0,0) to (1,1)

A: Given query is to find the the line integral for the y = 2x over y = x2.

Q: Evaluate the line integral of the function p(x.y)=y-x,over the straight line Joining the two points…

A: Function p(x, y ) = y - x   points ( 4, 0 ) and ( 0, 4 )  To Evaluate: Integral  of the given…

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

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Q: Find the real number a for which the value of the iterated integral V1- (y- (-2²- ỷ +2y) dx dy a-1 -…

A: Let fx,y=-x2-y2+2y. To determine the maximum value of the function, differentiate f(x,y) partially…

Q: Justify the Fundamental Theorem of Line Integrals for F. dr in the case when F(x, y) = (8x + 8y)i +…

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Q: 1. What should be A for the path independent of (2,4) xy + ) 2Ay – x y? (1,2) Moreover evaluate the…

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Q: Use the local linear approximation of (1 – x)2º at xo O to estimate (0.99)2º. %3D

A: The linear approximation of f(x) at x0 is given by L(x) = f(x0) + f'(x0) (x-x0)

Q: Verify the condition of integrability for the given total differential equatior (3x*y2 – e*z)dx +…

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Q: If ƒ is continuous and f(t) dt = 16, find the integral

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Q: Find the c that satisfies the MVT for Integrals. f(x) = x² [0,4]. over

A: Given: f(x)=x2,  [0,4]

Q: 2.a) Apply the Trapezoid and corrected trapezoid rule, with h = 1, to approximate the integral 1 dx…

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Q: Use Green's Theorem to evaluate the line integral of F (x°, 9x)

A: Given, the line integral of F=x3, 9x around the boundary of the parallelogram in the following…

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

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Q: 2) Find the total differential dz for z = tanhy In(x)

A: Total differential of z is given by dz=(∂z/∂x)dx+(∂z/∂y)dy

Q: Use the property to estimate the best possible bounds of the integral. II 4 sin*(x + y) dA, Tis the…

A: Consider the given integral: ∫∫T4sin4x+ydA T is a triangle enclosed by lines y =0, y = 2x and x =3.…

Q: 5. (a) ( If ryz² + x²yz + ry²z = e*v², compute . (b) Let f(r,y) be some function. Set up the…

A: (a) The equation given in the question is: We need to evaluate  This represents the derivative of…

Q: Show that {zn}nEN is Cauchy, where z, = sint dt. 1+t

A: Sol

Q: 2. Use the Fundamental Theorem of Line Integrals to evaluate fe F.dR, where F(x, y) = (x²y? –…

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

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Q: 2. We have seen that the FTC for line integrals implies fc Vf - dĩ = 0 for any smooth closed path C…

A: FTC for line integrals implies ∫C∇f·dr→=0 for any smooth closed path C and any smooth function f (a)…