Find det (T) for T:R- R, where T(x,,×2,*3) = (x,- x2, X2 - X3, X3-x).

Q: Let F(x, y, z) = y° z°i+ 2xyz³j+ 3xy° z²k. (a) Find the domain of F, and show that F is…

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Q: Let f(x, y) = - 4xy² – 2x² + 8y². Find and classify the critical points of f.

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Q: Let f (x, y) = 2x3 + x2y - y2. Compute d2f/dx2, d2f/dy2, and d2f/dxdy at (x, y) = (1, 2).

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

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Q: For the function f(x,y,z) = In (10x + 7y + 6z), find fy, fy, and f,.

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Q: Let f (x, y) = (x − y)e^x . Use the Chain Rule to calculate ∂f/∂u and ∂f/∂v (in terms of u and v),…

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Q: Let z = f(x, y) = 5x – x³ + 2xy – y². - Find and classify all the critical point(s) of this…

A: Differentiate the function fx,y=5x-x3+2xy-y2 partially with respect to x and y, to obtain fx and fy.…

Q: Find the linearization at x = a. f(x) = 6x®, a = 1

A: Linearization L(x) of function f(x) at a is given as L(x) = f(a) + f'(a)(x-a)

Q: Use the chain rule to evaluate fu and f, at (u,v)=(2,-2) where f(x,y,z)=y³+xz² and x=x(u,v)=u+v²,…

A: Concept: The chain rule for the function f (x,y,z) is  dfdu=∂f∂x∂x∂u+∂f∂y∂y∂u+∂f∂z∂z∂u…

Q: Evaluate f fs 2² dS, where S is the portion of x2 + y² + z² = 2 in octant 1.

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Q: Find the MRSxy of the utility function U=In(x)+y.

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Q: Let, T:R' → R³;T(x,y,z)=(2x+ y, y- z,2y+4z) 2. Find the rank and nullity of T.

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Q: Determine the linear approximation of f(x) = x² at x = 1. a. y= 2x +1 O b. y = 2x - 1 O c. y=- 2 x-1…

A: Given: f(x)=x2, x=1 To find: Linear approximation at given point.

Q: Let T : R → R³ be defined by: X1 T [3x1 – 12 – 5x12x17+ x2] X2 (a) Show that T is a linear…

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Q: find the linearization at x = a. f (x) = x4, a = 1

A: Given function: Using the formula of linearization of a function f(x) at x =a :

Q: x - y -2x Prove by definition that for f(r, y) fy(x, y) = %3D x+ y' (x+ y)2

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Q: For what value of b is

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Q: ) T:R →R (ii) T:R →R defined by T (x1» x2) = X1X2 defined by T (x1, X2) = (x1 + 1, 2x2» X1 + X2)

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Q: Find the linearization L(x) of f(x) at x = a. f(x) = Vx +15, a =7 %3D L(x) =

A: Givne that fx=x2+15, a=7To find L(x)

Q: Show that the function u(x,t) = t/2e-*/4 (t > 0) satisfies u, = uxx.

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Q: Find the linearization L(x) at x = a. 1 f(x) = - x - a = 1 %3D -

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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: Find f(t). s(s + 1).

A: L-1[F(s−a)]=eatf(t) From first shifting property, L{e^(at)*F(t)} = f(s-a)…

Q: .Determine if F(x, y) = (2xy + y-2)i + (x² – 2xy-3)j for y > 0 is conservative.

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Q: Use the Chain Rule to find- af when r = 2, t =-1 of the function ar f(x, y, z) = e3xyz cos 2xyz ; x…

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Q: For f(z) = 2z', find ip using the definition,

A: The limit definition of a derivative of a function, f(x) at a point, x=a is given by .

Q: Find (Df) & (R) for this mapping F:Z→Z, f(x)= x²

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Q: Find x' for x(t) defined implicitly by x+t°x+t -8=0 and then evaluate x' at the point (-2,2).

A: x3+t3x+t4-8=0Differentiate w.r.t…

Q: Find x' for x(t) defined implicitly by x +tx+t° - 5=0 and then evaluate x' at the point (- 1, - 3).

A: Follow the procedure given below.

Q: what is the antiderivative F(x) of f(X) = 3x1/2-x-1 that satisfies the condition  F(4)= 0? A,…

A: The given function is fx=3x12-x-1 that satisfies the condition F4=0. Find the antiderivative of Fx.

Q: Find curl F if F(x,y,z) = 16y5₂5i - 8x³₂8 j- 7xy4k. O curl F = (-28xy3 +64x³₂7)i + (7y4 + 80y5z4)j +…

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Q: Find dwdt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before…

A: (a)

Q: Compute the convolution f(t) * g(t) for f(t) = e-3, g(t) = 2t.

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Q: Q = find of るf and るこ for the Ganction f(xz)-ディズ メ+Z

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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: Let, T: R3 → R³;T(x, y, z) = (2x + y, y – z,2y + 4z) Find the rank and nullity of T.

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Q: problem in photo attached

A: The given function is,

Q: Find the linearization of L(x) of f(x) at x=a f(x)=StartRoot x2 +17, End Root a =8

A: Given function and point is

Q: Use the chain rule to find D (f o g) (–1,2) for f(u, v, w) = (v² + w², u³ – vw, u²v+ w) and g(x, y)…

A: This question is related to derivative of composite function.

Q: Determine the linear approximation of f(x) = x* at x = 1. а. у %3D 4x - 3 O b. y=- 4 x - 3 О с. у-…

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Q: Let r (t)= - The x component of r' (-1) =

A: r'(t)=<2t,3,5> r'(-1)=<-2,3,5>

Q: Find the linearization L(x)of the function at a .

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Q: Let f(x, y) = 8+ 76xy+38x2 + 240y + . Find and classify the critical points of f.

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Q: Find z'(t), where z = In (x + y), x = 2t e', and y = 5e', in the following ways. a. Replace x and y…

A: follow next step

Q: For f(x, y), find all values of x and y such that f,(x, y)= 0 and f,(x.y)= simultaneously.

A: Please refer to the image below

Q: The domain of the finction f(x, y) = In(-x² - +1) is:

A: The domain of a function is the range of the values that can be input into the function. The…

Q: Use the chain rule and compute if z = x2 + xy, x = uv? + w°y = u + ve"

A: z=x2+xy3, x=uv2+w3, y=u+vew

Q: For the function P(A, B) , where both A and B are functions of t, r and w, use the chain rule to…

A: Given function is PA,B, where both A and B are the functions of t, r and w. We have to state the…