Sketch the natural domain of f (x, y) = V(2x² +2y² – 8.

Q: Find ∇f(x, y, z) for the function f(x, y, z) = x2 + y2 − 4z and find the direction of maximum…

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Q: Let  f(x, y) = sin(15x2 + 3y2) Calculate the quadratic approximation of f at the point (0,0)

A: Given that,  fx, y=sin15x2+3y2 The quadratic approximation of the function fx, y at the point x0, y0…

Q: Find and sketch the domain of f(x, y) = √√1 − (x + 1)² − y² + 2 ln (x − y + 1).

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Q: Differentiate fx) with respect to g(x) of yx= xsiny

A: To find the derivative

Q: Fundamental Theorem of Calculus.

A: ∫abfx dx

Q: Find fryz from the functions f(x, y, z) = 4x³y² – e²y* ++ 4y – x16 %3D - |

A: To find the third order partial derivative of the function fxyz f(x,y,z)=4x3y2-ezy4+z3x2+4y-x16

Q: Find the number of zeros of f(z) = z' + iz² – z + 2i in the upper half-plane.

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Q: Exercise 4. Show that the function f is not differentiable at (0,0) f(x, y) = x³ + y³

A: We have given that ,  fx , y = x3 + y33 Here , ∂f∂x = 3x23x3 + y323 ∂f∂x = x2x3 + y323 At , x , y =…

Q: 1. Let F(x, y) = 1 0 if x + 2y ≥ 1 if x + 2y < 1 Does F defines the DF in the plane?

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Q: Find the domain of analyticity for g(z) = -y3 – 3x²y + 3y + i(x³ + 3xy² - 3x).

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Q: Let F(x, y) =x2+xyz+z , find at the point (1,1,1)

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Q: Determine the linear approximation of f(x) = x² at x = 1. a. y= 2x +1 O b. y = 2x - 1 O c. y=- 2 x-1…

A: Given: f(x)=x2, x=1 To find: Linear approximation at given point.

Q: x - y -2x Prove by definition that for f(r, y) fy(x, y) = %3D x+ y' (x+ y)2

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Q: xy ,show that: x- y If z = = 0

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Q: Suppose that f(x, y) = y/x³ + 1 at which {(x, y) | 0 < y < x < 1}. %3D D

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Q: Find a point zo where the function f (z) = (z – 12 – 12i) (z – 12 – 12i) is differentiable. What is…

A: Given f(z) = (z-12-12i)(z -12 -12i)¯

Q: Find a point zo where the function f (z) = (z – 8 – 6i) (z – 8 – 6i) is differentiable. What is the…

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Q: Maximize the function f(x, y, z) = x² – y², given that x² + 2y² + 3z² = 1. %3D

A: Here, we can use Lagrange multiplier to maximize f(x,y,z)=x2-y2 with constraint x2+2y2+3z2=1. Let…

Q: Consider the function f(x) = 2x³ + 71° – 8z2 – 8. Enter the antiderivative of f(x) + C

A: To find the antiderivative means integration of the given function

Q: Find ( fx fy, fzf xy f yx fxyz f zzx) for the function: f(x, y, z) = xyz + x²y + z³x + x²y?

A: To find the differentiation of the function

Q: Find a function f (x, y, z) such that ∇f =〈z, 2y,x〉.

A: Given that, ∇f=z,2y,x We have to find a function f(x, y, z) such that ∇f=z,2y,x. Let us consider,…

Q: Show that the function y = x¹/², x > 0, is strictly concave

A: Given that y=x1/2 Differentiate both sides w.r.t x, y'=12x1/2-1y'=x-1/22 Again differentiate w.r.t.…

Q: Consider the function g : Z x Z → Z × Z given by g(x, y) = (x+ 3y, 2x – y). Is g injective? Pr -…

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Q: Integrate f(x, y) = (x + y + 1)- 2 over the triangle with vertices (0, 0), (4 , 0), and (0, 8).

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Find the linear approximation of the function y = x at point xo = 1 .3 О а. у%3D Зх - 2 О Б. у3D 3х…

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Q: Consider the finction f(x,y) = x² + xy – 5x – y+ 1. Then:

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Q: What does it mean for the differentiability of a function if the Cauchy-Reimann equations (Ux = Vy…

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Q: Find fg, fyand f; if f(x,y,z) = x - Vy² + z²

A: Given:     To determine: Partial  derivative  fx, fy, fz

Q: Find whether the function F-(* - y')i + (v² – 3x)j + (=* - xy)k is irrotational.

A: A vector field F is called irrotational if curl (F) = 0.

Q: Find the gradient function for: 1+3x 1) y = %3D x2 +1

A: Given that  y=(1+3x)/(x^2+1) To find the gradient of y

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

A: Solution:-  At what points (x, y, z) in space are the functions in continuous? Let (m,n,p) is the…

Q: Let f (x, y) = 2x³ + 4y³ + 19xy Find D = frx · fyy - fey at the point (2,2)

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Q: Find the lineanzation LlXiy) of f (iy) =x*-5Xy at (2,1). Then use the linearization to approximate f…

A: Here in this question we need to linearize the given function and then approximate the value of…

Q: Find the domain of f(x, y) = Vy+I y(1-x²)

A: Given: f(x,y)=y+1y1−x2 The domain of functions of two variables, f(x,y) are regions from two…

Q: Suppose that f(x, y) = e 42? - ly + 7z + 2y then find the discriminant of f.

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Q: Find the gradient of f(x, y) = 3xty3 + 4x°y® at (1, 2).

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Q: Show that the function d: RxR → R defined by d(x, y) = √x-y

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Q: Find the image of the line z =3 +iy under f(z) = e

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Q: Describe the domain of : Зху 3) f(x, y, z) = 4) f(x, y, z) = 1- x² – y2 – z2 %3D z - Vx

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Q: f(z) = (x²+y)+i(y²-x) is analytic %3D

A: We have, fz=x2+y+iy2-x Compare the equation with fz=u+iv we will get u=x2+y and v=y2-x Now,…

Q: Find the analytic functiopn f (z) = u (x,y) + i ( x, y ) such that u ( x, y)=  4xy3-4x3y+x.

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Q: Found the line of differentiability of f(z)=x2 - 2xy + 3y + i(2xy - y2 +x)

A: Answer

Q: Find ý of the function x2 - e2x+3y - In(xy) = 0. %3D

A: Consider x2-e2x+3y-ln(xy3)=0We differentiate it implicity, we…

Q: 4. Let F : R² → R² defined by F(x,y) = (2x +3y, –x +y). Is F a linear? Why?

A: Given

Q: Integrate the function f (z) = (z - 8i)² from z = 0 to z = 4 + 4i along the line x = y.

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Q: show that the L operator is linear for y" – 4y" + 2y' + y = x2

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Q: Let f(x, y) = (x² +(3))°. Determine the directio %3D 4)

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Q: Describe the domain of : f (x,y) = Ln(2 + x + y)

A: in the question it is asked to write the domain of f(x, y)=ln(2+x+y).

Q: Find f (1, 1) S(x, y) = In( %3D Vx² + y²

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.