f(x + Ax, y) – f (x, y) (а) lim Ar 0 Дх (b) lim Ay 0 S(x, у + Ду) — f(x, y) Ду

Q: (a) Find the solution of the limit using polar coordinate system. x² + y² x2 + y2 lim(x.y)-(0,0)

A: To find the solution of the limit using polar coordinate system lim(x,y)→(0,0)x2+y2x2+y2 We have the…

Q: exist. :) Compute the linearization of f(x, y) at the point (x, y) = (1,1). %3D

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Q: 12. Consider the function (z-y)2 f(x,y) = { if (x, y) = (0,0) if (x, y) = (0,0) a) Does…

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Q: a) Find the directional derivative of a(r, y) = 3-y at (1, 1) in the direction (-3,4). b) Find and…

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Q: Find div F and curl F if F(x,y, z) = x°i – 11j + yzk. - div F= curl F=

A:

Q: (a) Does the limit lim(æ,y)→(1,1) ya² -y exist? x-1 (b) If the gradient of a function f(x, y) at the…

A: we have to find the limit if it exists in part a) lim(x,y)→(1,1)yx2-yx-1 b) we have to find the…

Q: of of Find and for the furction f(xy,z) = x y+ y + xỉ ôx 'ay

A: For the function f( x, y, z ) = x2y + y2 + xz2 . We need to find some partial derivatives.

Q: If f(r, y, z) does lim f(r, y, z) exist? %3D 1² + 2y? + z² ' (z,y,z)¬(0,0,0)

A: To find its limit we have to first convert the function into polar form.

Q: Use the Chain Rule to evaluate the partial derivative ah at the point (q, r) = (1,2), where h(u, v)…

A: ∂h∂q=(∂h∂u×∂u∂q)+(∂h∂v×∂v∂q)∂h∂q=ev×4q3+uev×r2Substitute q=1 , r=2⇒u=q4=(1)4=1and v=qr2=1×(2)2=4

Q: Value of divergence of function (x2y2)î - (z³y 4x) ĵ + 4y² xk at (1, 2, 1) is

A: Given vector is (x2y2)i^ - (z3y-4x)j^ + 4y2xk^

Q: (a) Find the solution of the limit using polar coordinate system. x² + y² x² + y2 lim(x.y)→(0,0)

A: Solution:Given to the find the limitlim(x,y)→(0,0)x2+y2x2+y2In polar co ordinates x=r cos θ   and y=…

Q: Find div F and curl F if F(x, y, z) = 10y³zºi – 8x³z¹ºj – 5xy³k. div F= curl F=

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Q: x'ey Let f(x, y) : Then lim (xy)-(0,0) x2+3y2 f(x,y) (A) exists and equals 0 since f (x,y)…

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Q: Find div F and curl F if F(x, y, z) = x*i – 5j + yzk. - div F= curl F=

A:

Q: Find div F and curl F if F(x, y, z) = x°i – 9j + yzk. - div F= curl F=

A: Divergence of a vector function F(x,y,z) is given by, div(F)=i∂∂x+j∂∂y+k∂∂zF and curl of vector…

Q: (a) Find directional derivative and gradient of f(x, y, z) at indicated point. f(x, y, z)=, z-x u=…

A: To find the gradient of f(x,y,z), use the gradient formula,…

Q: Q5-let F(x,g.2) =x -x 22 @ Fined VE B write the equution of tangent Plane t- the graph of at Point…

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Q: (15) Given two functions u(x + y) and v(x + y), what is Vu × Vv? Don't show it using an example. If…

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Q: Find div F and curl F if F(x, y, z) = xz°i+ 2y°x²j+2z²yk. %3D div F= curl F=

A:

Q: Find div F and curl F if F(x, y, z) = x°i – 6j + yzk. - div F= curl F=

A:

Q: Use the limit definition of partial derivatives to find f,(x, y) and f,(x, y). f(x, y) = x² – 8xy +…

A: We know, fx(x, y) = limh→0f(x+h, y) - f(x, y)h fy(x, y) = limh→0f(x, y+h) - f(x, y)h

Q: Find div F and curl F F(x,y,z)=x^2i-2j+yzk  F(x,y,z)=xz^3i+2y^4x^2j+5z^2yk

A: Given:- i) F(x,y,z)=x2i -2j+yzk   ii )F(x,y,z)=xz3 i+2y4x2j +5z2 yk To find:- div F and curl F of…

Q: (x – 3)² y² [(x – 3)² + y*J° 4. Let f(x,y) for (x, y) = (3,0) and f(3,0) = 0. = %3D lim (.y)(30)…

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Q: Suppose that the Second Derivatives Test is applied to a real-valued function z = f(x, y), whose…

A: Given real value function is z=f(x, y) and second derivatives are continuous on the disk D centered…

Q: Utt Urr 100, x > 0, t > 0. u (x, 0) = u (x, 0) = u (0, t) = 0 lim u (x, t) = 0, x → 00

A: 1. Laplace transform is an integral transform that converts a function of real variable t to a…

Q: Find div F and curl F if F(x,y, z) = x°i – 5j + yzk. - div F= curl F=

A: A detailed solution is given below

Q: se the function to show that fx(0, 0) and fy(0, 0) both exist, but that f is not differentiable at (…

A:

Q: ib Show that the function defined by F(x, y) xy √² + y² O is not differentiable at point (0,0) if…

A: As per the company's guidelines, we are supposed to solve first question for you, kindly repost…

Q: Find the direction of maximum decrease of the function f(x, y, z) =exyz at the point P(1,1,1) and…

A: Given function is

Q: f(x,y) = 25 - x^2 - y^2 at the point P(3, 4). Use the Partial Limit Dervatives

A: Given problem:- f(x,y) = 25 - x2 - y2 at the point P(3, 4). Use the Partial Limit Derivatives

Q: (a) Find the solution of the limit using polar coordinate system. x² + y² x2 + y2 lim (x,y)→(0,0)

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

Q: (2x-y²)² (x,y) + (0,0) Let f(x,y) = x²+y4 2, (x, y) = (0,0) Find the slope of the tangent line to…

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Q: (a) Let f(x, y) be defined on (a, b) × (c,d). (i) Give an example to show that existence of partial…

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Q: Use the function to show that f(0, 0) and f(0, 0) both exist, but that f is not differentiable at…

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Q: ry lim (z.v)¬(0,0) + y² exists.

A: Please refer the attached image for complete solution.

Q: Find div F and curl F if F(x, y, z) = xz°i+ 2y'x²j+ 2z²yk. %3D div F= curl F=

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Q: -{ xy(x²-y²) x²+y² (x, y) ± (0,0) (x, y) = (0,0) f(x, y) = %3D %3D ) Plot f(x, y). Where is f…

A: Given function is: fx,y=xyx2-y2x2+y2,x,y≠0,00                   ,x,y=0,0

Q: Explain why the function is differentiable at the given point. f(x, y) = x³y?, (-2, 2) The partial…

A: L(x,y) = fxx0,y0x-x0+fyx0,y0y-y0+f(x0,y0)

Q: f'(x) = = = = lim h→0 = lim f(x + h) = f(x) h→0 h - h lim h→0 . Ax + Ah - Ax > h lim h→0 h h…

A: We will show more steps to make it easier to understand

Q: Find the linearization L(z, y) of f at the given point. Surface: f(a, y) In(x +9y) Point: (11e5,e)…

A: Given :

Q: Find div F and curl F if F(x, y, z) = xz°i+ 2y'x²j+ 2z²yk. div F= curl F=

A:

Q: (2) This exercise is to convince the student that the extension of the differentiation rules from…

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Q: (a) ze'u lim (x..)(-1,0,4) 6z+ 2y-3z (b) I-4y lim (r,v) (0,0) 7z+ 6y

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Q: 5xy sin ³²x + 3y² 0, f(x, y) (a) Prove that lim T if (x,y) + (0,0) if (x,y) = (0,0) f(x,y) does not…

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Q: Explain why the function is differentiable at the given point. Then find the linearization L(x,y) of…

A: The given function is: f(x,y)=y+sinxyGiven point is:  P=(0, 3)

Q: /alue of divergence of function (x2y2)î - (z³y – 4x) ĵ + 4y2 xk at (1, 2, 1) is

A: Let f = (x2y2)i^ - (z3y-4x)j^ + 4y2xk^

Q: Find div F and curl F if F(x, y, z) = xºi-7j+yzk. div F= curl F=

A: Given that F(x,y,z)=x6i-7j+yzk. The objective is to find divF and curlF.

Q: Use the limit definition of partial derivatives to find f (x, y) and f (x, y). f(x, y) = x2 - 2xy +…

A: Given query is to find the partial derivative.

Q: Find div F and curl F if F(x,y, z) = x°i – 2j + yzk. div F= curl F=

A:

Q: Draw a simple sketch of direction field for y' x2 – y2. Find all equilibrium points and the regions…

A: The sketch of direction field is as follows

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: