1. a) In G = (Z43, e), find [G:H] for H = (32). b) In G = (Z54, O), find [G: H] for H = (24).

Q: Let w = r = uv, s = u – 2v. Use appropriate forms of the p2+ s2" dw dw and du chain rule to find in…

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Q: For z € C, define Tz: CC by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint,…

A: Given that: For z∈C, Define Tz:C→C by Tzu=zu. The objective is to verify the given statement. Now…

Q: show that }I f,lm) Ck-m) 2.

A: For, ∑m=0kf1mf2k-m, substitute k-m=n or m=k-n: ∑m=0kf1mf2k-m=∑k-n=0kf1k-nf2k-k-n…

Q: Consider j1,ỹ2, ỹ,ER° and that ỹ,+y;+ÿ;=0. For Î=span{ÿ, ,ÿ and Y=span{ÿ2, ÿ}, show that Ý=Ỹ.

A: Given y1→, y2→, y3→∈ℝ5 such that y1→+y2→+y3→=0. Let Y^=spany1→,y2→ and Y~=spany2→,y3→. To show…

Q: let (X; Z) v it piece wise comnected ?

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Q: find the funotion with hejp of a indegral meau value betw een 4,1] of g(x)=\xl-2

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Q: Suppose that ∇fp =〈2, −4, 4〉. Is f increasing or decreasing at P in the direction v =〈2, 1, 3〉?

A: 1) v=2,1,3 v=22+12+33 = 14 u=vv=1142,1,3

Q: Suppose that f(x, y) = e-2r² – 1y² + 2x +8y then find the discriminantD= faz fyy – fay- Question…

A: Solution: The objective is to find the discriminant

Q: For the function f(x,y,z) = In (5x + 4y + 9z), find f,, fy, and f,.

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Q: Prove that the quartic function q(z) = r* - 16 has only

A: Given that: q(x)=x4-16

Q: Let D, be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

A:

Q: Suppose that f(z, y) = 10z²y² + 8z? + 2y? then find the discriminant of f.

A: We have to find the discriminant of the function of two variables fx, y. It is given that the…

Q: T: R2 R2 is defined by х + 7х, [3x, – 4x2] X1 %3D and B = {u,,u,} and B' = {v,v,}, where %3D -- --…

A: Given transformation T:R2→R2, defined by Tx1x2=x1+7x23x1-4x2 and B=u1, u2 and B'=v1,v2 Here u1=22,…

Q: for w = ze(zy²+2:). Do not include =" in your answer.

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Q: is the gropt Q whch of Hhe of followring

A: Given that: 1. y=x2ex2. Graph of f(x)

Q: Let f(x,y) = ax + by2 af where a and b are constants. If of at x = 1 and %3D ху y 2, then the…

A: Introduction :

Q: Verify that ƒxy = ƒyx, for ƒ(x, y) = xey.

A: Given function is fx,y=xey We have to verify fxy=fyx Solution: fxy means that first we find partial…

Q: f(z) =z2 is discontinuous in C

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Q: () Let z = x + iy. Find Im and Im 2 in terms of x and y. Im Im z2 ||

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Q: For the function f(x,y,z) = In (6x + 7y + 4z), find fx, fy, and f,.

A: Given: f(x,y,z)=ln(6x+7y+4z)To find the fx, fy, and fzddx (ln x)=1x

Q: Determine if the functions Y1=e^{4x}senx and Y2=e^{4x}cosx are linearly independent or linearly…

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Q: 2. Use the Squeeze Theorem to show that f(r) = 2 is in R(0, 3).

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Q: (b) Show that the Linear guen Operators. are not with T(f)= Lf6), f()+1].

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Q: Show that f: R → R defined as f(a) = 3a³- 4 is a one to one %3D iuicuon.

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Q: 3. Prove that R² is Hausdorff. Is R²T3? Is R² T4?

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Compute the subdifferential of f(x) = max(1-r,0), for r E Rn.

A: As per the question we have to find the subdiffernetial of the function f(x) = max (1-x,0) And we…

Q: Differentiate w.r.to "x" y = xsin-1x +x/(1-x3)

A:

Q: 4. If f(x, y)=: Vx² +y? 1 , find f, and f,.

A: Given data is : fx,y=1x2+y2  To find fx and fy

Q: dual of c, is isometric to l'.

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Q: For x, y E [0, 1], prove that 1 xy(1 – x – y) < 27

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Q: Given f(x, y, z) = /2x – y – 5z, find - - f-(x, Y, z) = %3D f,(x, Y, 2) = %3D f-(x, Y, z) =

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Q: (c) Show that |(−1,1)| = |R|, where (−1, 1) = {x € R : −1 < x < 1}.

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Q: Assume that ƒ′(3) = -1, g′(2) = 5, g(2) = 3, and y = ƒ(g(x)). What is y′ at x = 2?

A: Given:

Q: O show thet fu) =2a +3b is Candin Uous where a, b are Constants.

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Q: 2.) Use the chain cule to find dz and d ond guen the fllowng: Yoy must show in work that your the…

A: We have to find partial derivatives.  Concept:

Q: Consider P2 with the inner product = f (x)g(x) dr. Then d(x, 2) Jo

A: d(x,2) represents distance between the vectors x and 2 P2 is the set of all polynomials with degree…

Q: ニメ Find the eg uaton (x) 4a| %3D 3tan x when fo f the tor

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Q: For the function f(x,y,z) = In (5x + 7y+ 9z), find fx, fy, and fz.

A:

Q: Let D, be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

A: Dx(f)=8x+5 Apply integration both side  ∫Dx(f)dX=∫(8x+5)dx   f(x)=(8x2)/2 + 5x +c (here c is the…

Q: Show that the dyfferential equatian IS exact x(スt5in t) do + cast -2tu) abe +x² dt-

A: We will show that the given differential equation is exact.

Q: Compute the value of f.dr If F= -yi+xj and c is a from (-1,1Y to CI,1) %3D SemICirele

A: Proceed as shown below

Q: Show that F. dr is independent of its path, given that

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Q: Consider f (x, y) = re63ry f-(x, y) = e63ry and fy(x, y) = %3D e63ry

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Q: Given that f(r, y) = }2 - - . (a) Calculate: f,, (b) Calculate: f„, (c) Are they equal?

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Q: Ihow do you bnd the imit of as x approaches 0?

A: The limit of x+1-1x as x approaches to 0

Q: f(z) =z? is discontinuous

A: f(z)= z2 is a polynomial function in z and polynomials are continuous everywhere.

Q: Compute the convolution y(n) = x(n) * h(n) , Where x(n) = {1,1,0,1,1} and h(n) = {1,-2,-3,4} , Using…

A: Given: Given convolution expression is Y(n)=x(n)*h(n) x(n)={1,1,0,1,1} h(n)={1,-2,-3,4}