Show F(x, y) = (4xy³ + 6)i + (6x²y² + 2e²¹)j LF is conservative by finding a potential function f for F, and use f to compute с for 0 ≤ t ≤ π/2. f(x, y) = [F F. dr = r(t) = 2 sinº ti + F. dr, where C is the curve given by 2t -sin³ 5tj T

Q: Show that / F dr is independent of its path, given that F(r, y, :) = + tan-:) i+tan rj+ k. IHence,…

A: Here we will find out the required value and prove the required result.

Q: 21 Let F(x, y) = (3x + 2y)i + (2x - y)j. Evaluate F. dr where C is the curve X y from (0, 0) to (1,…

A: Here we calculate the integral along the curve x = y2 from (0,0) to (1,1)

Q: Consider the function 3..2 f(x, y) = cos(rxy) – a°y?. = cos(Txy) – x°y°. a) Find the equation of the…

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Q: Find the equation of the tangent plane to z = x² + 5y³ at (2, 1, 9).

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Q: F(x, y) = (4x³,3y²) along a path from (0, 0) to (2, 6). Find S F · dř. a) using a parabolic path. b)…

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Q: f (x, y) is a differentiable function with Vf(4,3) = 7i+ 5j. 2. Suppose w = (a) Find the directional…

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Q: Show that F dr is independent of its path, given that F(r. y. :) = ( i+ tanrj+, + tan k. 1+:2 %3D…

A: Here we will find out the required value and prove the result.

Q: Find f. f(t) f"(t) = 2e + 3 sin t, f(0) = 0, f(π) = 0

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Q: Sketch the curve r(t)=f°i+tj and indicate the orientation. Find r'(t) and sketch r(1) & r°(1).

A: First we will plot the given parameters.

Q: Verify that the function u, defined by u(t, x) = t sin(x – t²), is a solution to the equation ,2 U +…

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Q: 87; Let flxiy,2) =% Cos(Tx) + kon (s2). Then find the directional derivative of f the point…

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Q: Let F = (2x³y² + 2x, 2x¹y³ + y). Given that f(x, y) = x¹y¹ + ² + y² is a potential function of F.…

A: The given function is F→=2x3y4+2x,2x4y3+y, and its potential function is f(x,y)=12x4y4+x2+12y2. To…

Q: Q: Find the condition that the line x cos a +y sin a = p may touch the curve x"y" = a™*".

A: Answer

Q: For the curve r(t), write the acceleration in the form a+T+aNN. r(t) = (6t sin t+ 6 cos t)i + (6t…

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Q: Obtain the integrating factor of the non-exact DE, (2 + ycosx)dx + xcosxdy = 0. a. secx b. cosx c…

A: Integrating factor

Q: (B): use the directional derivative to estimate how much f(x, y) = cos(xy) + xy² Will change if the…

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Q: Consider the function f(x, y, z) = 3x2 cos(y) + e 25. Calculate grad f at the point (2,0,1).

A: The gradient of a function, often denoted as grad or ∇f, is a vector that points in the direction of…

Q: Given a function (r, y, 2) = 3x2 + In (3y2+ 1) + e cos 2. i. Find the directional derivative of o at…

A: Here we will find out the required value.

Q: :N N amd 9: N-oN are functions such that f is not onto but f+9 is onto Prove that 9 is onto or give…

A: Suppose that f:ℕ→ℕ and g:ℕ→ℕ are functions such that f is not onto but f+g is onto. To prove that g…

Q: Let f(x, y) = x²y² + 3xy − 5y, find af /ər where x = t² +r², and y = 2tr . Evaluate the derivative…

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Q: Evaluate S (xy + x + 2) ds along the curve r(t) = 2ti + tj + (3 - 2t)k, Osts 1.

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