Find the linear approximation to f(x, y, z) = at the point (2, –1, –2): f(r, y, z) z

Q: use the chain rule to find dz/dt or dw/dt w=xey/z, x=t2, y=1-t, z=1+2t

A: Given

Q: 1) Using the linear approximation of the function z = g(x, y, w) = xy³w² – 2x³ + e" + 5 around the…

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Q: Find the linearizations L(x, y, z) of the functions in Exercise atthe given points.ƒ(x, y, z) =…

A: To findLinearizations of f(x,y,z)=tan-1xyz at (1,0,0) , (1,1,0) and (1,1,1)

Q: The function f (x,y) = e* +3 tan – is a homogeneous function of degree_

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Q: B) Evaluate the line integral F-dr, where F (x, y) = r*y°i+x²y°j and C is the curve r (t) = (t³ +…

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Q: Find the domain of the vector-valued function. r(t) = n(t) 0 (-∞, +∞0) O (1, + ∞ ) O (0, +∞ ) O [1,…

A: Find the domain for all three components and then take intersection of those.

Q: Find the domain of r(t) and the interval(s) on which the vector- valued function is continuous. r…

A: We need to find the domain and the interval on which the vector-valued function is continuous.

Q: Use the local linear approximation of f(x, y, z) = In(xz + y) at (2, -1, 1) to approximate f(1.98,…

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Q: Let L(z, y) f(1, y) at (1, 1) . Then f-(1,1) = = y-2x +3 be the local linear approximation to O a. 3…

A: Topic:- partial differentiation

Q: Demonstrate that the functions Y1 = a", y2 = sin(x), Y3 = xe* are linearly independent on R. 9.

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Q: Let L(x, y) = y- x+ 3 be the local linear approximation to f(æ, y) at (1, 1) .Then f(1, 1)%3D a. 3.…

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Q: Consider F(x,y)=(4x^3+4y)i+(4x+6y^2)j Show that F is conservative, find a function f such that F=…

A: Given F(x,y)=(4x^3+4y)i+(4x+6y^2)j F is conservative as shown below

Q: If F = xi + -4yj +x2k determine the flux of F through S, where S is the portion of z2 = x2 + y2,…

A: Given - F = x i^ - 4y j^ + x2 k^  ;             S is the portion of z2 = x2 +y2 , 1 ≤ z ≤ 3 To find…

Q: Find the linearizations L(x, y, z) of the function at the given points. ƒ(x, y, z) = xy + yz + xz at…

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Q: Find a function v(x, y) such that, given u = 2x(1– y), f(z) = u + jv is analytic in z.

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Q: Compute the directional deri vative of the function f(z,y, 2) = zy² + y²9+°% at the point P(4, –2,…

A: Given function is fx,y,z=xy2+y2z3+z3x

Q: find the linearization L(x, y) of the function ateach point. ƒ(x, y) = (x + y + 2)2 at a. (0, 0), b.…

A: Given function,  f(x, y) = (x + y + 2) 2 To find linearisation L(x, y) at i) (0, 0)  ii) (1, 2)

Q: Find the directional derivative, Dvf , of (*) f(x, y, z) = 4x tan 1 at the point P = (1, 1, 1) in…

A: To find the directional derivative of f(x,y,z) at P=(1,1,1) and the Direction, v=(1,2,2).

Q: Let f(x, y) = x³ √√y. A. Find an equation for the plane tangent to the graph z = (x, y) = (1,4). B.…

A: A scalar function f(x,y) is a rule that is used to associate a number or quantity with a particular…

Q: For f(x,y, z) = x² – (1 – y)² + z?. Find directional derivative of f at P(1,1,1) in the direction of…

A: The directional derivative of a function f  in the direction of the vector u is given by…

Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 4x + 8y + 12z on the elipse…

A: Using Lagrange Multipliers with Two Constraints

Q: find the divergence of F. F(x,y) = √ +y I i+ Y Pa 꾸

A: F(x, y)=xx2+y2i+yx2+y2j

Q: Show that the function f(x, y, z) = x² + y² – 222 sat isfies the Laplace equation. - Good Luck.

A: Given function f(x,y,z)=x2+y2-2z2. We have to show function satisfies the Laplace equation. A…

Q: B) Evaluate the line integral F-dr. where F (x, y) = x*y²i+x²y°j and C is the curve r (t) = (t³ +…

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Q: Find the gradient vector, Vf, for the function, f(x). f(x, y) = x²y² + ex²y b. f(x,y,z) =x²y*…

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Q: The function f (x,y) =e/*+3tan( y/x) is homogeneous of degree A) 3 B) 2 C) 1 D

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Q: ind the maximum value of the function f(x,y,z) = x² + y² + z² on the plane x+y+z = 1.

A: The given function is; f(x,y,z) = x2+y2+z2

Q: Consider F and C below. F(x, y, z) = (y²z + 2XZ² ) Î + (2xyz)j + (xy² + 2x² Z) K₂ C² X=√E, y = t +…

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Q: Show that the function satisfies the Laplace equation f (x, y, z) = (/x2 – 2y² + z?)*

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Q: Let L(r, y) = 2y – x + 3 be the local linear approximation to f(r,y) at (1,1) . Then f-(1,1) = %3D O…

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Q: If L(x, y) is the local linear approximation of f(x, y)= x²+2y3+ 9:" at (2, 1) where g(2, 1) = 4,…

A: Given: f(x,y)=x2+2y3+g(x,y)x g(2,1)=4 gx(2,1)=4 gy(2,1)=6 At (2,1).…

Q: Find the linearization of the function at the given point. f(x y, z) = e5x - 3y - 6z at (0, 0, 0)…

A: Linear approximation in three Variables

Q: Let F(x,y, z) = rye" – 2² Jī and P(1,0, –1).

A: Given Fx,y,z=xyey-z2x and P1,0,-1Vector u=⟨2,-3,6⟩

Q: Consider the function: f(z,y) = 1 – xy coS TY (a) Find the linearization L(x, y) of ƒ at (1, 1) (b)…

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Q: Find the work done by the force F = xy i + ( y - x)j over the straight line from (1, 1) to (2, 3).

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Q: Let z = g(x, y) = e* cos y + y ln(x – 1). Use local linear approximation to estimate g(1.99,0.01).…

A: Given Z(x,y)=g(x,y)=excosy+ylnx-1

Q: se the chain rule to compute and at (u, v) = (1, I) where f(x, y, z) = x²y + y²z + z²x and r(u, v) =…

A: Given: fx,y,z=x2y+y2z+z2xr→u,v=eucosv,eusinv,u2u,v=1,π4 To find: ∂f∘r→∂u=?∂f∘r→∂v=? Concept used:…

Q: Consider z = Ax + By + D be the tangent plane to f(x, y) = 2x² + 5y³ + 9xy + 9x + 3y at (6,9).What…

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Q: (a) Let f(r, y) = x - 3ry + 18y, where 3r?y- y- 6x = 0. Prove that f(r, y) has extrema at the points…

A: Calculate partial derivative of f(x,y) with respect to x and y ,

Q: Let z=xy ,  where x=t2 , y=sint.  compute z'(t).

A:

Q: Let f(x,y)=y2 +sin(y+T)+x +cos(xe). At the point (TT,0) compute the unit vector in the direction of…

A: We have to find the unit vector in the direction of the maximum increase of the function f. This…

Q: Evaluate § F · ds, where F = 2x(x+y)i + (x² + xy + y?)j and the positively-oriented curve C encloses…

A: Here we will use greens theorem to find line integral of vector field For a vector field F→= Pi^+Qj^…

Q: Find the divergence of F. F(x, y, z) = 8xi + 8y²j + 8z²k

A: Given,           Fx,y,z=8x2i+8y2j+8z2k

Q: Use the divergence theorem to find the flux of F = [x²yz, –xy² z, 3 + z²] on the hemisphere S = {(x,…

A: Find the flux of the vector field F→=x2yz, -xy2z, 3+z2 on the hemisphere S=x, y, z : x2+y2+z2=4, z≥0…

Q: For a scalar function f(x, y, z) = x? + 3y2 + 2z?, the gradient at the point P(1, 2, - 1) is

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Q: Let f(x, y, z) = x² – y? + 2z. Compute the rate of change of f in the direction of the vector (v2,…

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Q: (b) Use the corresponding formula to evaluate the line integral F. ds where F(x, y, z) = (xz, -yz,…

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Q: Use the Chain Rule to find ôz/ðs and dz/dt. z = tan(u/v), u = 3s + 6t, v = 6s – 3t dz %3D as Ot

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Q: Let f(x, y) = x /G. At the point (1, 4), find a vector (a) In the direction of maximum rate of…

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.