Q3/ 1- Let f: Zn → Z10 define by f(x) = 2x+2. Is fhomomorphism or not, explain your answer.

Q: Examine the continuity of f(z) = 2-√z+4 Z at z = 0

A: Given: fz=2-z+4z We have to find the continuity of f(z) at z=0.

Q: 7- The function f(z) = z is: a) Analytic on C b) Analytic only at zo = 0 c) Not analytic

A:

Q: The graph of f(x)= - x2-1 is concave up on interval (-1,1)^O (0,1) 0 (-1,0) U (1, 0 ).co (-∞, – 1)U…

A: Given, fx = xx2-1  We have to find the interval in which graph of  fx = xx2-1 is concave up.

Q: (a) Let f(x) = tanx- 2, g(x) = x5 + x³ and h(x) = Vcsc5x. Then find %3D the f(g(h(x))), using chain…

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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: what is the quadratic polynomial for the function f(x,y) sin(xy) + cos(x/y) at (pie/4, 1)

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Q: Prove that a decreasing function f on [a,b]is integrable.

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Q: Define the function f(x, v) = 4x³-x³y-4y5 3x5-3y5 at the origin so that it becomes continuous there.

A: A function is said to be continuous at (a,b) if the the value of function at (a,b) is equal to…

Q: du -xy du Ifu =f(x² +2yz,y² +2zx) Prove that (y- zx)-(x² - yz) -NX ду

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Q: Let F = <5x^2, 2(x+y)^2, 9(x+y+z)^2> Find divF (divergence)

A: First of all, consider the vector F and find F1, F2, and F3.

Q: Prove that f(x) = [[sin²(2log r)– 3 sin ( 2logt)+2] dt %D 1. Cannot be decreasing any where,

A: To prove that f(x) can not be decreasing anywhere, where f(x) is defined as…

Q: - Find the Linearization of f (x) = sinx at a 5. Provide your answer as L(x) =? %3D %3D

A:

Q: Q2: Let f(z) = z² is f(z) satisfy Cauchy-Riemann equation ?

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

Q: Use the comparison theorem with f(x) = = 1 √x to show that [₂²= 1 √x-1 is divergent.

A: See this solution in below

Q: Find the Linearization L(x) of y = (1+x)¬1/² at a = 8. L(x) = %3D

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Q: Decide whether or not the real function In (eon 2) I+0, = (1) 1= 0, is continuous at r = 0.

A: The given function is fx=ln cos2x2x2     if  x≠0-1                if  x=0. Compute the limit…

Q: Find the linearization L(x) of f(x) at x=a f(x)= x+1/x a=3 show all work

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Q: Let g(x, y) = x4 + y² – 4x³ + 5x – 6y+3. Show that g(x,y) continuous (actually at least aC')…

A: Recall [ Implicit Function Theorem ]

Q: Suppose that f: R-Cis a measurable function with 1+ |xf° for almost all x E R. Show that its Fourier…

A: Let, f⏜ be Fourier transform of fx. Then nth derivative of f⏜ is Fourier transform of…

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: Let f(x) =esin 3x + 2x a. find the linearization of f(x)=esin3x+2x at x=0. b. Use your linearizato…

A: a). Given f(x) =esin 3x + 2x f(x) ≈ L(x) = f(a)+f′(a)(x−a). where the function L(x) is called the…

Q: Q2. Show that F(x) = 3 – x², x € is contractive on its domain, but does not have any fixed points.…

A: To show - Show that Fx = 3 - x2, x ∈ -14, 14 is contractive on its domain, but does not have any…

Q: how (in terms of  − δ) that a function f : R 3 → R defined by f(x, y, z) = (2x + 3y + 4z) is…

A: Solution of given problem is followed by

Q: Q4. Prove that f(z) = Iz/2 never be holomorphic function at z = 0. %3D

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Q: Let fc Cx)= Vx if osxs 4 c.(x-4)+2 if x>4 denotes a real constant. - where C Show that lim fe (x) =…

A: The function fcx=x,              if 0≤x≤4cx-4+2   if x>4. Where c is a real constant. We have to…

Q: (b)Assume that f(x) is a function of the form -x-1; when -T < x <0 f(r) = when 0 <r < T Prove…

A: For the solution of the problem follow the next step.

Q: 5. Determine whether the closed interval [a,b]is homeomorphism to the closed unit…

A: This is a problem from topology.

Q: Q3/ 1- Let f: Z, → Z10 define by f(x) = 2x+2. Is fhomomorphism or not, explain your %3D answer.

A:

Q: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

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Q: Show that there does not exist a function f(x, y) such that VJ= (y2,x). Hint: Use Clairaut's Theorem…

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Q: Suppose that ƒ(x) = x2 and g(x) = | x | . Then thecompositions( ƒ ∘ g)(x) = | x | 2 = x2 and (g ∘…

A: Given: The composite of the two functions are given as,

Q: Find aland U so that the function S 7z³ – 3x² + 7, z -2 is both continuous and differentiable. al =

A: To Determine: fins a and b so that the function is continuous and differentiable given: we have a…

Q: Q, Frd the donain of H fallowing function: fa) = 5x -3x-4

A:

Q: Show that f (a) = is Libschitz function on r+ 2

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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: Q3: a) Let f : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its -…

A: the function is invertible if its one -one and onto. The function is one - one if f(x1)=f(x2)…

Q: Determine the value of m such that the function 5 + 5 y fi^ - ^ - f(z, y) = is continuous at (0,0).

A: Given that function s continuous at (0,0). We need to find the value of m.

Q: Consider the function { that is defined on R as: f(x)=-x² +4x-3. Designate by (C) its representative…

A: The given function is, fx=-x2+4x-3 The orthonormal system is, O; i→,j→ To evaluate limx→-∞fx  and…

Q: Q2 Evaluate whether the function 1-xy if xy # 1 z = x-y if xy = 1 continuous at (1,1) or not.

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Q: (3x² -1, x 1, Determine c and d so that f(x) is continuous everywhere.

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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: Find and sketch the domain of the function. f(x, y, z) = ln(64 − 4x2 − 16y2 − z2)

A: Given function is      f(x,y,z) = ln(64-4x2-16y2-z2) We have to find the domain of this function.

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 f(z) = z = is differentiable but not analytic at z = 0

A: Givenf(z)=zz

Q: yLet f(2) be analytic in {z: |2| < 1} with f(0) that |f"(0)| < 2. = 0 and |f(2)| < 1 for all z in…

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Q: Estimate the global maximum value for the function F on the interval [-2,1] if F(x)= || dt . t2 +1

A: The given function is: F(x)=∫-2x tt2+1 dt

Q: f[in (w) ² ] ^ 4 3 dw

A: ∫lnω24ωdω=∫2lnω4ωdω=∫16ln4ωωdω=16∫ln4ωωdω Substitute u=lnω⇒dudω=1ω⇒dω=ωdu; 16∫ln4ωωdω=16∫u4du…

Q: Let f(z) = 1 Find all the points where f(z) %3D Z + 4 is analytic.

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Q: Determine local extremum f(x,g) = xy(4-x-). %3D

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Q: 16. Consider the function f:(-2, 0) → [0, 0) defined by f (x) = vx +2. Prove that f is injective.

A: f(x) here is injective if and only if for all a,b∈(-2, ∞) we have a=b⇔f(a)=f(b)