Consider the function f: (-2,0) → [0, 0) defined by f (x) = Vx + 2. Prove that f is suriective.

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: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = tan x, a = π

A: The linearization of a function f(x) at x = a is, Now differentiating  given function with respect…

Q: Find the interval that the fullowing function is continous at In (9 -x) %3D A. [I,3] BL) (3,3) (1,3)…

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Q: Integrate to find F as a function of x, and demonstrate the Second Fundamental Theorem of Calculus…

A: We need to Integrate fo find F as a function of x.

Q: → [0, ∞) defined by f (x) = Vx + 2. Prove that f is 17. Consider the function f: (-2, 0) surjective.

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Q: (a) Assume that g(x, y) is a function of the form g(x, y) {} 0 if (x, y) = (0,0). (i) Show that…

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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: f(x, y) = x² + y² +1 f (x, y)

A: Let's find.

Q: af Find a d () + In (). of the function f(x,y) = arccot and ду

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Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

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Q: Let F(x) –3 cos(ť²) dt. Use the Fundamental Theorem of Calculus to find F'(x). F'(z) =

A: Given: F(x)=∫x−1(−3cos(t2))dtF'(x)=ddx∫x−1(−3cos(t2))dt=?

Q: Find a potential function for F or determine that F is not conservative.〈F = z, 1, x〉

A: F is conservative when curl F = 0 Given that f=(z, 1, x)=zi+j+xk Now evaluate curl F, curl F = ∇ × F…

Q: State what it means for a scalar function y = f (x) to beintegrable on an interval [a, b].

A: Scalar integrable  function y = f (x)

Q: What is the Laplacian of the following function? f(x,y,z)= x² - y² +z?

A: Given function is f(x,y,z)=x^2 -y^2 +z^2. We have to find Laplacian of the function f(x,y,z) .

Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A: Fx=∫1xt4dt Now consider, ∫t4 dt=∫t14 dt              =t14+114+1+C          ∫xn…

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: Let f(x) = x² Inx. Use the Three-Point central Formula with h= 0.2 to approximate f"(1.5) %3D %3D…

A: Given that, f(x)=x2lnx and h=0.2. The three point formula for second derivative is given by,…

Q: Prove that the function f : R? → R, f(x, y) = sin(/x² +|sin y|) is uniformly continuous on R?.

A: The time interval between two waves is known as a Period whereas a function that repeats its values…

Q: find a potential function for F or determine that F is not conservative. F=(z,x,y)

A: F is conservative when curl F = 0 Given that f=(z, x, y)=zi+xj+yk

Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A: To find F'x using the second fundamental theorem of calculus: Fx=∫-1xt4+1dt Solution: The second…

Q: The function f : R → R defined by f(x) = cos(x - ) is %3D

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Q: Use the Cauchy-Riemann Equations to show that f(z) = z Im(z) is only differentiable at z =0 and find…

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

Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A: Given: Fx=∫0xt21+t3dt for finding derivative of given function, we will use Leibniz theorem…

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: 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: Let f:C→C be defined by exp(-z-4) z#0 f(z) =. z = 0 Does f satisfy the Cauchy- Riemann equations? Is…

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

Q: Let f (z) = u + iv be an analytic function. If f'(z) = 0, prove using Cauchy-Riemann equations that…

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Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

A: According to second fundamental theorem of calculus, to find the derivative, one have to plug the…

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.

Q: Then, write a function f(z) such that a) it is analytic in C except its isolated singular points,…

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Q: Determine if a function f : [0,2π] →R  defined by f(x) = sin 2π is continuous

A: We need to determine if a function                f : [0,2π] →R  defined by f(x) = sin 2π is…

Q: Prove that f(x, y) = x sin y is differentiable everywhere.

A: Answer

Q: Apply the theorem of Cauchy-Riemann equations to verify that function f (z) is entire: f (z) = 3x +…

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Q: Let u be a differentiable function of x. Use the fact that ∣u∣ = √u2 to prove that

A: Given that - u = u2          ............(1)To prove - ddxu = u'uuProof - Difefrentiating equation…

Q: Consider F(x,y)=(4x^3+4y)i+(4x+6y^2)j Show that F is conservative, and find a function f such that…

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Q: Let f(2)=u+iv be an analytic function. Given that u=v² show that f(z) is a constant.

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

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Q: Let f (x) = vr +2² + 2 , ro = 1, h = 0.5, r1 using the Three-Point formulas to approximate f(1) (1);…

A: Given: The given fx=x3+x2+2, x0=1,h=0.5,x1=1.5 and x2=2

Q: Let F = (2y? + z)i+ 4ryj+ æk. (a) Show whether F is conservative.

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Q: Find all functions f(x, y, z) such that grad f = 2xi + zj + yk.

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Q: Let g(u) be a differentiable function and let f(x, y) = g(x² + y²), find the direction of maximal…

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Q: Let f: R → R defined by f(x) = sin x has 0 as a fixed point then f is a contraction. True O False

A: Contraction mapping

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

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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: For the set of functions {sin(x), sin(2x), sin(3x), ...} = {sin(nx)}, n = 1,2,3, ... on the interval…

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

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