Problem 5. Let T : P2 → R³ be defined by T(p) = (p(0), p(1), p(2)) for all p = P2. Let B = {1, x, x²] and D = {(1,0,0), (0, 1, 0), (0, 0, 1)}. Show that [1 0 0 MDB = 1 2 and conclude that T is an isomorphism.

Q: If 1 =- V(x2 + y²) [(xyz – 3xy)a, + (zy² + 3x² )a, + xy az] And B = r² a, + sin 0 aø Find JÃ x B| at…

A: At P1, 2, 3, A→=25ax+195ay+25az. As r=x2+y2+z2, θ=tan-1x2+y2z and ϕ=tan-1yx, at the given point we…

Q: Let z= y- 4a, a u'v and y-v- 3u. Compute On kum)-(1,1) -

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Q: 1 Let f (x,y) =In Then f (1, – 1) =... 3x2+4y3 XX

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Q: T(1,1)=(5,0) Let T:²²be defined as, T(1,0)=(2,6) Then, T(3,2)=(12,6) (a) T(0,1)=(-3,6) (b)…

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Q: 1. Let F(x, y) = 25x³ + 120x²y + 192xy? + 102y³ – 300x – 450y. Find the following: | ƏF (i). OF…

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Q: 11)

A: We solve by using Properties of Laplace Transform.

Q: Assume two consumers with utility functions of the form 1/2 1/2 UX (x₁, x₂) = x/²x/² and UX (y₁, 92)…

A: Given below clear explanation

Q: Find the linear approximation of the function f (x, y) x² + y² at the point (2,-1, 1)

A: Given function:- fx,y=5x2+y2

Q: Let F(z, v)-v +y+3z+1-0

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Q: 2 4 2) f(x,y)=2xy- 5x? - 2y² +4x+4y – 4 Ans. f 3'3 0 local max. %3D

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Q: (18) Find [S], where S = {(a, b, c)|a, b, c E [100], a <b, a <c}.

A: Here is how the set S is defined: S = {(a, b, c) | a, b, c ∈ [100], a &lt; b, a &lt; c}   We must…

Q: Prove that the wff is a valid argument (Va) [S(x) → (3y) (P(x, y)^T(y))]^(3x) (C(x)^S(x)) (3x) (y)…

A: We are given the following wff. ∀xS(x)→∃yP(x, y)∧T(y)∧∃xC(x)∧S(x)→∃x∃yC(x)∧T(y)∧P(x, y) Notice that…

Q: 11. Let F(x, y, z) = = (), and let P represent the path that follows the from (1, 0, 1) to (0, 1,0),…

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Q: 5. Let g(t) be a smooth function and u(x, t) be a solution of the following problem: Uų = k?u 0 0,…

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Q: xy If z show that: x- y

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Q: Work Problem 1₁ Consider the DE: (4x + 6y)dy + (-2x + 4y)dx = 0 A) Using only the notations followed…

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Q: 11. Find the linear approximation to given function at indicated point (Баллов: 2) x2 z = 3 + +, Po…

A: Topic:- linear approximation

Q: "III. Vf dr = , where f = xy²z and Cis given by r (t) = (t +1, t², t³),0<t<2

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Q: Let x1, x2, x3 be solutions of the following ODE t^x''+2tx''+x'+2022x=0 for t>0 If their Wronskian…

A: Given differential equation is t2x'''+2tx''+x'+2022x=0 and W2=20. We have to find the Wt for all…

Q: B. Let z- ( x-y)exy Show: Zxx + zyy + e x9 Cy 3- x³)- -z (xy+ 2)

A: Given that                       z=(x-y)exy

Q: Find Vf at the given point. f(x,y,z) = x3 + y2 – 3z2 +z In x, (1,2,4) Vil(1,2,4) =Di+j + ()k ( %3D

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Q: Consider the following. w = xy2 + x2z + yz?, x = t2, y = 5t, z = 5 (a) Find dw/dt using the…

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Q: with D= 0. Obtain its T.F.

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Q: Compute the derivative of g(x) = | 1 dt. 2 1+t5 2" 1+æ15 1+25z 2" In 2 1+æ15 1+25z 2" 1+a15 1+25z 2"…

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Q: Q4. Simplify the following expression: 3-5 x 39 Click on image to enlarge if needed 12 81 243 324

A: To evaluate :- 3-5 * 39

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: 8. (p.241, #4) The function f: R → (0, 0) defined by f (x) = e**+1 is bijective. Find its inverse.

A: Replace f(x) with y.

Q: Let f(u, v) = (tan(u - 1) - ev, u²-v²) and g(x, y) = (ex-y,x-y). Use the Chain Rule to find…

A: Given functions: fu,v=tanu-1-ev,u2-v2gx,y=ex-y,x-y To find: Df∘g1,1 Concept used: If the functional…

Q: Let Y₁-X₁+X2 and Y2 X1-X2. Please derive the pdf of Y₁. (Hint: The joint support of Y₁ and Y2 would…

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Q: dy a) If all of y = 4n2 + 5n + 1,x 3n + 2 find dx b) if yx2+Vx + xy? = 1 prove that = -at (1,-1)

A: Please see the answer below.

Q: 6. Use Green's Theorem to evaluate O (y* – xy?) dæ + (2 – a*) dy where C is shown below. y -2 ² +y²…

A: Green's theorem

Q: Compute F. dr where F (x, y) : (7xy + x cos x) i+ (-yev + 6x²) j and C is a piecewise closed curve…

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Q: 3. a. Assume that P(x) and Q(x) are continuous over the interval [a, b]. Use the Fundamental Theorem…

A: (a)

Q: Suppose f1 and f2 are two analytic functions so that Re(f1)=u and Re(f2)=u.  Then h(z)=f1(z)-f2(z)…

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Q: f 20f * f af If V²f(r,0) Find f(r,0) such that r dr v²S(r,0) = 0

A: Given:                          ∇2fr,θ=∂2f∂r2+2r∂f∂r+∂2f∂θ2-∂f∂θ To do:…

Q: C is the curve y = e® from (2, e²) to (0,1) and F = (x², -y), compute ſ, F-dr.

A: Given that, C is the curve y=ex from 2,e2 to 0,1 and F=x2,-y Then, F=x2,-y=x2,-ex So, F→=x2i^-exj^

Q: 1. (Theorem 2.1, Eq. 2.4) By employing change of variables r = ax +y and s=x, prove that the PDE: A…

A: Auxx+Buxy+Cuyy=0 A≠0B2-4AC=0