Which of the following functions is continuous at (0, 0 )? f( x, y ) = In ( x² + y² ) ху Оb.f (x, у ) 2 x2 + y? ху — х2 х +у +3 Oc f ( x, y ) = In ( 1 Od.f ( x, y ) ху ху — 2у Oe.f ( x, y ) = x2 4x + y2 + 4

Q: (a) Find the points (x ,y, z) in space, where the function f(x, y, z) = Vx² +y -1 Is continuous.

A: It is well known that the equation z=x2+y2 represents a circle with centre at the origin. The…

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Q: Let x and y be differentiable functions of t and let s = 2x2 + y2 be the distance between the points…

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Q: Assume x and y are functions of t. Evaluate dy/dt. 12) x3+y3=9;- dx dt 13) xy + x = 12; dx dt = 4, x…

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Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

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Q: 9. Use the chain rule for multivariable functions to find where f(u, v) = u + v u(x, v, z) = x+y,…

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Q: Let x and y be differentiable functions of t and lets = sqrt(x2 + y2) be the distance between the…

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Q: Define h(0,0) in such a way that h is continuous at origin       h(u,v)=ln(nu2-u2v2+nv2/u2+v2

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Q: Suppose f(x, y) =                             ((xy2) / (x2 + y4)), for (x, y) ≠ (0, 0)…

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Q: What is the linearization L(x, y) of a function f (x, y) at a point (a, b)? How is L(x, y) connected…

A: Linearization L(x,y) of a function f(x,y) is given by the formula

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Q: At what points (x, y, z) in space are the functions continuous?  a. ƒ(x, y, z) = x2 + y2 - 2z2 b.…

A: Continuity: A function f is continuous at the point (a, b, c) if,                            For…

Q: (2) Let f(x, y, z) = xy + yz + x²z². What is the maximum rate of change of f at (1,2, 3)?

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Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y.z) = In (3z° - 4x- 5y- 4) 1…

A: Please rate

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Q: Find fz and fy- f(z, 9) = [" 9eª dt f.(x, y) = f,(x, y) =

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Q: 2. Show that the function g defined by VVT y² + √² 2, is not continuous at (0,0). (z,y) = (0,0)…

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Q: At what points (x,y.z) in space are the functions continuous? a. h(x,y,z) = In (2z - 4x – 5y3 – 1) 1…

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Q: Show that f is continuous at (0, 7) where a cos y + y(e" – 1) if r + 0 f(x, y) = ay if æ = 0.

A: In the given question we have to show that the given function f(x,y) is continuous at the point…

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Q: Find fr and fy. 2e" dt f (x, y) = 2, f.(r, y) = f„,(x, y) =

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Q: (a) Compute the derivative of the function F(x, Y, z) = x² + 2y² – 3z? at the point (1,1, 1) in the…

A: a) Given function F(x, y, z) = x2+2y2-3z2.    Point P(1,1,1).     vector v = i^+j^+k^ .      We have…

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Q: fhow thot te function f depined by f(XrY) = 1 (x,Y) =(1,-リ - 1) %3D x2 +y (x, y) F (I,-1) * t y is…

A: We use the sequential definition of continuity to solve this problem.

Q: prove that xy^2/x^2+y^2 is not differentiable at 0,0 when  xy^2/x^2+y^2 if (x,y) not equal to (0,0)…

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Q: If F(t) = 2xyi + xyz2j + ex+yk, then what is the divergence of F? Select one: A. 2yi + xz2j + ex+yk…

A: The divergence of a vector F=P,Q,R is given by div F=∂P∂x+∂Q∂y+∂R∂z. The given vector is Ft=2xy…

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Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

A: Consider the provided information, fxx0,0=5,fyy0,0=3  and fxy0,0=-4 The Second Derivative Test to…

Q: Suppose that z = f(x, y) is differentiable at the point (27, 81) withf,(27, 81) = 2 and f, (27, 81)…

A: Given that:                            z=f(x, y)fx(27, 81)=2fy(27, 81)=-6x=t3y=t4

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Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

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Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y,z) = In (2z2 – x² – 3y³ – 4)…

A: we want to find the domain of given  function

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Q: At what points (x, y, z) in space are the functions continuous?  a. h(x, y, z) = sqrt(4 - x2 - y2 -…

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Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

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Q: Let flx.y)a gxy? ty? 9x+y (4,y)+(0,0) 1. is f continuOus at (0,01? why?

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: