Let A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

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: Show that each function in satisfies a Laplaceequation. ƒ(x, y, z) = 2z3 - 3(x2 + y2)z

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Q: (2) Let F(x, y) = (cos(y)+2x, -x sin y+1). Show that F is path independent by finding %3D a…

A: For the given vector function to be independent, it must be derived from a potential scalar function…

Q: Let P(z) = | tin(t) dt. Use part 1 of the fundamental theorem of calculus to determine if P(z) is…

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Q: Let f(x, y,z) = x² sin y +y e, where x t +3,y = net-1 and z = In 2t. %3D at t = 1 is: dt O -12n O…

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Q: Find the linearization L(x) at x= a. f(x) = x+ a = - 1

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Q: Let fn E D[0, 1]. Suppose that fh g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show…

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Q: Q2= A) for E= 2xĭ-3Y3+K %3D prove That divc curl È)=0 %3D

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Q: Prove that flx)= (i+x)? + X and diverge Litx)? both

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Q: Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function…

A: We have the given Cauchy problem is y'=a(x)tan-1y,  y(0)=1   …..(1) where a(x) is a continuous…

Q: Suppose that P1(t), p2(t), and p3(t) are continuous on R, and that y1, Y2, and y3 are solutions to…

A: Wronskian of the function

Q: Show f(x) = e and g(x) = xe linearly independent by finding its Wronskian.

A: Given: fx=exgx=xex The formula to obtain the Wronskian is Wx=f(x)g(x)f'(x)g'(x), where f(x) and g(x)…

Q: Lemma 4.3 Assume • T(0) = Idx; • T(t +s) = T(t)T(s) (t, s 2 0); • Vx€ X :xH T(t)x is continuous in…

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Q: 3. Show that every y E lị defines a continuous linear functional on l through the inner product,…

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Q: If z = f(u) + g(u), where u = - and v = x y. Prove that y %3D x: Zx - y· Zy = y2 · Zyy – x2 . Zxx…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Show f(x) = e2= and g(x) = x e²z linearly independent by finding its Wronskian.

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Q: * Let A E R"Xn, ĩ E R" \ {0} and A = 0. Prove that if the origin is stable if and only if ī is…

A: Given A∈ℝn×n,x¯∈ℝn\0 and Ax¯=0 To prove that the origin is stable if and only if x¯ is stable.  If…

Q: If z = f (u) + g(v), where u = = and v = x ·y. Prove that Q1: x· Zx – y· Zy = y² · Zyy – x² · Zxx…

A: We have z=fu+gv, and u=xy  and v=x.y We need to use the chain rule and prove x.zx-y.zy=y2.zyy-x2.zxx…

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: Let F(x, y, z) = (2x cos(y), e² – x² sin(y), ye²). (a) If possible, find all potential functions for…

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Q: Let z = = g(x, y) = e* cos y +y In(x – 1).

A: Let we havez=g(x,y)=excosy+yln(x-1)If x=s-t and y=s2-t2

Q: If z = f(u) + g(u), where u = = and v = x · y. Prove that x · Zx – y · Zy = y² ·Zyy – x².Zxx using…

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Q: Let F(x, y, z) = z* +x+y sin x +x In z 0, where z =f(x, y), then at (0,0,1) and %3D are equal to:

A: Given,          Fx,y,z=z4+x+ysinx+xlnz=0 , where z=fx,y

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

A: Given function is: f(x,y,z) = ln(3xy-6) We need to find the domain  for this function.This can be…

Q: Let A: C [0,1]→C [0,1]. Prove that operator continuous reversible, find A. 2.11. Ax(t)= x(1)- -…

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: A function f(x,y) is homogeneous of degree 7 in x and y if and only if, __________.

A: Given problem is solved below.

Q: If z is a function of x and y implicitly defined by əz əz find and əx ду at (x, y, z) = (0, 2, -1).…

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Q: ; Let x, y ER" be given. (a) lf ||x|| = ||y|| = 1 and x - y = 1, show that x = y. (b) If ||x|| ≤ 1…

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Q: Prove that the function f(z2) defined by x*(1+ i) – y°(1 – 1) , (z ± 0) x² + y? f(z) = [0, (z = 0)…

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Q: If f(x, y, z) = In(x7 + yº + z®) + (x° + yº + z®), then find fr, fy and fz at (1, 1, 1).

A: →fx=∂f∂x, During derivative of ∂f∂x, consider 'y' and 'z' as a constant→fy=∂f∂y, During derivative…

Q: Consider the following semilinear first-order PDE ди ди + Y = xe¯u ду (i) Find the general solution…

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Q: If z = f(u) + g), where u = = and v = x· y. Prove that y x: Zx - y Zy = y? Zyy – x2 - Zxx using the…

A: We have to prove this

Q: Consider a regularizing term defined by where k!2* and the linear differential operator D is defined…

A: We are given the following regularizing term.                    ∫ℝN0DF(x)2 dx=∑k=0∞ak∫ℝN0DkF(x)2 dx…

Q: Let F(x,y,z) = z* + x+ y sin x + x In z 0, where z fx.y), then at (0,0,1) and are dy equal to: ax

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Q: Let C[0, 1] have inner product [f(x), g(x)] = ∫ f(x)g(x) dx. Find α so that (x + αx2 ) ⊥ (x + 1)

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Q: Suppose the goind densiy Runction Xanef y i ginen by frxy) = 44 (ス-8) ○ZXC0,○ ス %3D otheraise…

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Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = x + 1/x, a = 1

A: Given fx=x+1x

Q: 2. Let z = sin(xy) + x siny,x=u² + w² and y = uw. 2a. Compute Vz= when u = 0 and w = 1.

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Q: Let f(x,y,z) = x² sin y + y e, where x = t +3,y = net-1 and z In 2t. %3D %3D at t 1 is: dt -2 O None…

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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: Let w f(t,0) where t and 0 are functions ofr and s, %3D dw is equal to Or then according to the…

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Q: Find the Wronskian for the set of functions.{x, x2, ex, e−x}

A: Given information: x, x2, ex, e-x The Wronskian is given by the following determinant.…

Q: Find the Wronskians of the given sets of functions and determine whether the functions are linearly…

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Q: Given F(x, y) = (2xe'6(y), e'x²(eY + 0(y)) with o E C'(R) then O (A) F is conservative only if o(y)…

A: (A ) is true.

Q: Given F(x, y, z) = (tan y + cosh x Joi 2xy)i + (x sec² y - x² + 1)ĵ - 2zk, evaluate F. dR where C'…

A: Given that,  F→x,y,z=tany+coshx-2xyi^+xsec2y-x2+1j^-2zk^, Therefore, set fx=tany+coshx-2xy…

Q: Let c(t) Use the chain rule to compute (foc)'(t). (cos(t), sin(t), t²) and f(x, y, z) = x² + y² +…

A: The objective is to find the derivative.