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

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…

Q: Let f(z) = u(x, y)+iv(x,y) be entire such that u(x, y) 4v(x, y). Show that f is constant. %3D

A: Here we use CRE equation to prove this

Q: Let F(x, y, z) = 6x?i – cos(xy)(i + 5). Calulate the divergence: div F(x, y, z) =

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Q: Let f(x, y) = - 4xy² – 2x² + 8y². Find and classify the critical points of f.

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Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

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Q: Solve for g'(x)?

A: Given:

Q: Determine the value of m such that the function *- y - 2 E + 2 vỹ VI – Vy ; z +y f(x, y) = is…

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Q: KB is ∀xy F(x, y). Prove using resolution-refutation that ∀xy F(y, x).

A: Solution: First, we convert clauses to CNF : (1) ∀xy Fx,y from KB (2) ∀x Fx,Ax (3) Fx,Ax CNF

Q: Find the linearization of f(x, y) = V1+x - y? at (4, 1), and then use it o approximate f(3.96,…

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Q: Find the linearization L(X,Y) of the function  f(x,y)=y-1/x+1 at(0,0).

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Q: For the function f(x, y) show that fx(0, 0) and fy(0, 0) both exist but that f is not differentiable…

A: We have given;

Q: Evaluate f fs 2² dS, where S is the portion of x2 + y² + z² = 2 in octant 1.

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Q: For the function f(x,y,z) = In (5x + 7y + 1Oz), find f,, f, and f,.

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Q: Find a function v(x, y) such that, given u = 2x(1– y), f(z) = u + jv is analytic in z.

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Q: Let f : R² → R with f(x,y) = x²+ y. Prove that f is differentiable at the point (0, 0)

A: Given function is f(x,y)=x2+y Since we know that, differentiability of function of two variable at…

Q: find the divergence of F. F(x, y, z) = x²i + y²j+z²k div F = 2(x + y+ z) div F = 3yz? + 2y sin z +…

A: Topic = Derivative

Q: Find the local linearization of f(z) =z² at 9. lo(z) = %3D

A: Given : f(x) = x2   x0 =  9

Q: Consider the function h : Z × Z → Z ×Z defined by h(x, y) = (2xy, 7x – 3y). %3D (a) Is h injective?…

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Q: Let H(x) integral ,for -2 ≤ x ≤2.Use geometry to evaluate H(2).

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

Q: Find the linearization L(x) of f(x) at x = a. f(x) = Vx +15, a =7 %3D L(x) =

A: Givne that fx=x2+15, a=7To find L(x)

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: Let g(x) V1+ 8t dt. Use the Fundamental Theorem of Calculus to find g'(x). g'(z) =

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Q: Let g(x) = kf(x) for all x E/= {x: asxsb}. Show that Jglx)dx = kJ f(x)dx if k<0.

A: To prove the above result we use the definition of lower and upper Riemann integrals. For the proof…

Q: ind the maximum value of the function f(x,y,z) = x² + y² + z² on the plane x+y+z = 1.

A: The given function is; f(x,y,z) = x2+y2+z2

Q: Let f(x, y) = x² + In(2x – y). Find the gradient Vf(x, y) at the point (1, 1). O O O O

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Q: Determine whether the function f (z)= Įz² – e3z I0

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Q: Find the linearization L(x) of f(x) = cot x at x = 4 %3D -

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Q: Consider the function f(x, y) = x2 + 2y2 – 4x – 4y, defined on R². Find and classify all the…

A: The given function is  f(x,y)=x2+2y2-4x-4y defined on R2.

Q: The domain of the function f (x, y, z) = In(3xy – 6) is %3D {(x, y) € R² : xy > 2} o {(x, y) E R? :…

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Q: Suppose that F(x,y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that дх…

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Q: Let f(x+ y) = f(x).s) Vx, ye R.Prove that f is derivable on R iff it is derivable at one point of R.

A: If the given function is continuous at x=a then it is continuous at x=b.

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: Find the linearization of the function f () = z7 at z = 1. L (z) = 固助 Using this linearlization, f…

A: Given function is

Q: Show that the function u = p(ry) + JFÿ¢(y/x) provides the quation „Pu = 0. dy

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Q: Consider the function f(x, y) = x2 + 2y2 – 4x – 4y, defined on R². Find and classify all the…

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Q: Find the domain of the piecewise function: x < 0 0<x <1 2х, y = In x, Vx-1, x 21 Select one: O a. x…

A: By finding the domain of each piece definition of functions we thereafter take the union of all to…

Q: Consider the function f(x) = 7x° + 9x° – 9 - 9. Enter the antiderivative of f(x) + C

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Q: Consider the function f(z) = 728 + 10z5 – 2r2 – 6. An antiderivative of f(x) is F(x) = Ar" + Bx™ +…

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Q: Q1. (a) Find the points (x ,y, z) in space, where the function f(x, y,z) = /x² +y² -1 Is continuous.

A: Given :- The points x,y,z in space , where the function f x,y,z = x2 +y2 -1 is continuous.

Q: xY 2. (a) Prove that the function f(x, y) is differentiable everywhere in its do- %3D x2 + y? main.

A: Solution is given below:

Q: 44. Let f(x, y) = g(u), where u = x² + y² and g(u) is differentiable. Prove that 2 4u (dg Әх du

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Q: Show that the function f(x, y) = x²y% + y° is homogeneous.

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Q: Let g(x) 8 In(t) dt. Use the Fundamental Theorem of Calculus tc g'(x) =

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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: X+Z+3Siny X +2y-2

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Q: if (x, y) # (0,0) (1 if (x,y) (0,0) x+y (a) Prove that the function f(x, y) =2x+y is not continuous…

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

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Q: Study the continuity of f(x,y) = xe-Vx at p(1,0) ey.

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