2. Prove that the function f : R? → IR defined by f(x, y) = 1- a2 – 4y? is neither injective nor surjective.

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

A: Given, f(x,y)=g(u)=x2+y2 To prove that, ∂f∂x2+∂f∂y2=4udgdu2

Q: 2. Show that the function u = e(xy) + VrY 4(y/x) provides the equation Pu = 0. dy? %3D -

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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: 1. Prove that the following functions are quasi-concave: (i) f(x, y) = ryt, r > 0, y > 0; e-24, x >…

A: We need to show that the given functions are quasi-concave. For checking quasi - concavity use…

Q: Let f be a differentiable function of one variable, and let f(p), where p = (x² + y² + z²)'/2. Show…

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Q: 2- Obtain the function x to minimize the following functional. = [ c«f + *)dt J = St: x1 = -X2 ,…

A: J=∫01x12+x22dt-------(1) x1˙=-x2-----(2) x(0)=c, x(1)=d here c,d are fixed number

Q: 4. Integrate the function f(z) = z² + 2z from z = -i to z = 1 in two ways, once along a straight…

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

A: Given : f(x) = x2   x0 =  9

Q: Suppose that the only Införmation we have ab8ut a fuRcEiOH FIS y= f'(x) (a) Use a linear…

A: Given: f(1)=-7

Q: Suppose that f(x, y) : at which {(x, y) | 0 < x < 4, – x < ys va}. 1+ x D

A: Consider a given function:  The objective is to evaluate the following double integral of fx, y…

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: 3) Let F = (y+ev=,8x+ cos(y*)) Use a theorem to compute ſ F •dT where c is the boundary of the…

A: Given that  F→=y+ex,8x+cosy3 To compute ∫cF→⋅dr→ where c is the boundary of the planar region…

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A: Cauchy-Riemann Equations: Let f=u+iv be a complex valued function , then the Cauchy-Riemann…

Q: 2. Show that the function u = p(xy) + Vry ý(y/x) provides the equation n, = 0. dy? U -

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Q: Find the domain of the following multivariable functions b. f(x,y,z) = x/y+ In (z – 1) c. f(x,y,z) =…

A: Solution:Domain: The domain of a function is the set into whcih all of the input of the function is…

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

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

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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: 2. Let f(r) = v (2 - r)2. Use the local linear approximation of f (x) at ro = V0.99 (1.01)². 1 to…

A: Given Curve is f(x) =x2-x2  find linear approximation at x0=1  when x =0.99 value of f(x) at  x0=1…

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A: It is given that fz=e-1z4     z≠00           z=0 where f:ℂ→ℂ. We have to check whether the function…

Q: Show that (d/dz)ž does not exist anywhere, i.e., f (z) =z is non-analytic anywhere.

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Q: Compute the discriminant D(x, y) of the function. S(x, y) = x' + y* – 6x – 2y + 1

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Q: c) Let f(r, y) = Vlry| Prove that f(r, y) is not differentiable at point (0, 0).

A: Given a function : f(x,y)=xy We need to show that f(x ,y) is not differentiable at (0 ,0). Let's…

Q: What is the domain of the function w = ƒ(x, y, z) = 1/xyz?

A: What is the domain of the function w = ƒ(x, y, z) = 1/xyz?

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: Find the best linear approximation, g, to f on the interval [0, 1]. (x) = 2/x g(x) %3D

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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.

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Q: Let f(x, y) = xcosy- y'e*y. fxy at (1,0)is equal to: %3D -2/3 -2 1 0.

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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: (a) Let f(r, y) = x - 3ry + 18y, where 3r?y- y- 6x = 0. Prove that f(r, y) has extrema at the points…

A: Calculate partial derivative of f(x,y) with respect to x and y ,

Q: (b) If F = (5xy- 6x )i + (2y– 4x)j, Evaluate [F.dr along the curve C in the xy-plane, y =x' from the…

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Q: Show that F = (2x - 3)i - z j + (cos z)k is not conservative.

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Q: Define T: (C¹ [0, 1], || · ||…) → (℃ [0, 1], || ||) by Tf = f'. Show that T is not continuous. Is…

A: T:C'[0,1], .∞→C'[0,1], .∞ T(f)=f' Let f(x)=xn then T(x)=nxn-1 T(x)=nxn-1 as n→∞,   T(x)→∞ So T is…

Q: Evaluate F.dr, where F(x,y)=[2y-3cos(x)]i + [4x +6sin(y)]j and C goes counterclockwise around the…

A: The objective is to evaluate ∫CF·dr where Fx,y=2y-3cosxi+4x+6sinyj and C goes counterclockwise…

Q: (b) Show that the function f(2) =u +iv, where S(2) = **(l+1) – y'(1-1) x² + y² = 0, z = 0 Satisfies…

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Q: Let f(x,y) = xcosy – ye*y. fry at (1,0)is equal to: -1 -5

A: fx is partial derivative with respect to x fy is partial derivative with respect to y

Q: Suppose that z = x + iy ∈ C and consider the function f(z) = y^2  + ix^2 Determine where f(z) is…

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Q: If f(z) = u + iv is entire and u(x, y) is bounded above, apply Liouville's Theorem to exp(f(z)) to…

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Q: 3. Verify that u(x, y) = x² + 4x – y² + 2y is harmonic and find b) the corresponding analytic…

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Q: iIm(2) Is the function f(2) = |z² e r analytic at zo = 0 ? %3D

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

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Q: Given the interval [a,b] that saitisFies the hypothesis of Rolle's Theorem For the Furnction f(x) =…

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Q: Let Juce) 2ntv () %3D A=O n!F (I+u+n) Bessel function Of the first Kind be a Of Order V Show that…

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Q: Consider the function f defined on [0, 1] such that f(0) = 1, f(1) 3, and f) = 2, f(1/5) = f(2/5) =…

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