For ORSTV, suppose that RT = VS. State an equation that relates s, t, and v. V (t, v) T (s + t, v) R (0, 0) S (s, 0)

Q: Find the exact value of Duf(-2,-1,-3) in the direction of v = 3i+ 4j + 5k for the function f(x,y, z)…

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Q: 2. Find t+2 dt check your result by differentiating

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Q: Find fæ, fy, and fz. 4,5 f(x, y, 2) = 4z In (a*y° cos(5z))

A: We can answer all the three question as below.

Q: 2. Find a function f(r, y, 2) which makes (yz?, rz?, f(r, y, z)) conservative. Show that your answer…

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Q: For the function ex, the linear approximation around x = 2 is %3D

A: Introduction :

Q: the linearization of (fi,f2) = (x²y+ 3xy², 3x³ + 2xy) at (1,0) to approximate the value at…

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Q: Find the linearization of f(x, y) = V1+x - y? at (4, 1), and then use it o approximate f(3.96,…

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Q: Given : 6 (x, y, z) Compute the gradient 3 6y³z³ - 5 ex at = PA (2,2,1.5) of G (x, y, z)

A: Given  G(x,y,z)=6y2z3-5e-x We have to find the gradient at the point PA(2,2,1.5)

Q: 16 -Σ(Σ») f(x, y, z) = Σ · (x + y + zn?) 2

A: Given: fx,y,z = ∑n=116∑k=1nk-x+yn+zn22

Q: y z = x + show that : 0.

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Q: 2. If F(x, y, z) = 2z3 – 3(x² + y²)z, then verify that Frx + Fyy + F2 = 0. %3D

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Q: Given f(x, y, z) = x³yz². Find f.(1,1,1).

A: The equation we have , f(x,y,z) = x3yz2 We have to find out fz, Now we differentiate f(x,y,z) with…

Q: Use chain rule to find express answer in terms of x only. dx y = Vu² – 1 u = 4Vx

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Q: In Exercise, verify that S and T are inverses.

A: GivenS:R2→R2 by Sxy=4x+y3x+yT:R2→R2 by Txy=x-y-3x+4yTo check S  and T are inverse of each other.

Q: Find f, and f, for f(x, y) = 12(6x-3y +8)°. %3D 1,(x.y) = %3D

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Q: 4) Suppose that T : R³ → R² is defined by T((x, y, z)) = (1+2x – z , 3 – x + 6y) Is T linear?…

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Q: A particle is moved along the x-axis by a force F(x) = x² + 5x – 6 . How much work is done in moving…

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Q: Given : 6 (x, y, z) Compute the gradient = 3 6y³ z³ - 5 ex at PA (2,2,1-5) of G (x, y, z)

A: Given  G(x,y,z)=6y2z3-5e-x We have to find the gradient at the point PA(2,2,1.5)

Q: 2+i Evaluate (2x+1+iy)dz along (1-i) to (2+i) 1-i

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Q: 7. Find function q(z) = x³ − 3xy² + iv(x, y) is analytic, find v(x,y).

A: if a function is analytic then C-R equations satisfy.

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: a) Find as when s = 1,t = 1 for f (x, y) = x³ – 3xy - 6x + 6y,x = 2t + s, %3D %3D y = 2s - t.

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Q: For æ > 0, 1 dt 1+t2 A 2/7(1+z) B 2/7(1+JE) c) 1+1 D 1+z E VE

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Q: Making use of the fact that ˆα = y − βˆx and βˆ = SxySxx, show thatni=1[yi − (αˆ + βˆxi)] 2 = Syy −…

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Q: Find f, and f, for f(x,y)=y In (2x+7y).

A: F(x,y) = y ln(2x + 7y) To find : fx and fy

Q: (a) x² + zsen xyz = 0

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Q: (a) The function z depends on u and v so that z = (2u – 1)(1 –v)? , u = x + 2y and v = x – 2y az…

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Q: Find Sea)ax Jf(x)dx if = 56 foR X<to (x fok xZ6 flx): %3D

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Q: Find the zeYo k its ovdev for Sunction f2)=1-Ti + Z+ e.

A: Given the function, f(z)=1-πi+z+ez. Now, in order to find the zero of the function f(z), we will…

Q: Find the linear approximation to h(x) = Vx at x = 2. Use the linear approximation to approximate the…

A: Given: hx=x4 To approximate the values of 34 and 44 using linear approximation of hx. Note: the…

Q: Find h'(x) if rsin 2x t2 cost dt h(x) = J3 %3D

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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: Given f(æ, y, z) = 6x² + 3y² + 4z², find f-(x, y, z) = | fy(x, y, 2) = | f:(x, y, z) =

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Q: lf f(z, y) = 5ery + 1 sín (inr + y) + 2r + 3y, find-a-,-,-and

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Q: 9. Explain why dt r1/I dt 1+t2 1+t? is valid for any x > 0.

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Q: Q, Frd the donain of H fallowing function: fa) = 5x -3x-4

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Q: 2. For what value(s) of h, y belongs to Span {v1, v2, v3}? Where 5 -3 V2 = -4 V3 ,and y = 3 h

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Q: e-", if 0 3,

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Q: 3. Find and at (0,0) where if (r, y) # (0,0) if (z, y) = (0,0) f(1, y) = Are they equal?

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Q: Find f,, fy, and f, for f(x,y,z)= 5xy+ 3xz + yz.

A: It is partial derivative

Q: 8. xyzx +2y³ (z, = 1

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Q: z=f(x,y)=x^4-3xy+1 and x=2t, y=t^3 find dzdt using the chain rule

A: Given query is to find the derivative using chain rule.

Q: af when s = 1,t = 1 for f (x, y) = x³ – 3xy - 6x + 6y ,x = 2t + s, y = 2s – t. a) Find as

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Q: 2R then the constants R TCX If f(x) Rsin .2 S,f' and f(x)dx and S are respectively.

A: Introduction :Here we have asked to find the value of R and S

Q: Q,3 Evaluate [II, xyz dxdydz, 3 < x S-2,0 < y S 1,0 S zs-

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Q: 3. If V =(3r°y-z,x²+y",-2r:'). Find grad div, at the points (2,-1,0).

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