Suppose we have a surface f (x,y)=x²y+cos (4x) · Which of the following is the gradient of f(x,y) at P (ñ/2.ñ/4): < (π) ² (π) ²> 4 4 < (π) 2 +4. 4 (π)² (π)² 4 4 (π)² (π)² 4 4 O (π) ²> 2

Q: Let F(x, y) = x2 + x² + y2J- (a) Show that F. dr +| F. dr if C and C2 are the semicircular paths…

A: The function given to us is  F(x,y)=yx2+y2i-xx2+y2j

Q: Give a parametric representation for the path followed by a heat seeking particle placed at point P…

A: The given temperature field is T(x,y)=64-x2-y3------(1) P(4,2) is the given point Then the gradient…

Q: 1. The temperature at the point (r, y) on an unevenly heated plate is given by T = f(r, y) = 100+…

A: Given f=100+5x2+15x+10y2gradient of f∇f=∂f∂xi+∂f∂yj     =10x+15i+20yjAt (2, 1)∇f=10x+15i+20yj…

Q: wy ad let (s, 1-45 24. Suppose you have a function f(x, y) and a curve r(t). You know: Vf(x, y) =…

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Q: 5. In this problem, F(x, y) = (y² +2, 2xy + 3y) and C is the line segment from (4, 2) to (0,3). a)…

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Q: dz (a) Use an appropriate form of the chain rule to find dx dz in terms of y in terms of x and if z…

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Q: 2. Find the gradient of the function f(r.y) In(r y²) at the point( 1, 1).

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Q: Find a function f such that Vf = 2e²x cos(5y)i - 5e²x sin(5y)j + z³k. f(x, y, z) =

A: Given :-  = 2e2x.cos(5y)i - 5e2xsin(5y) j + z5kNow,

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Q: Compute f cos(Tæ) dæ + 10xy dy, where y is a part of parabola y = 3x? from the point (0,0) to the…

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Q: et f(z, y) = for (z, y) # (0,0). Find the level curves of 2 + y²

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Q: The gradient of f(x, y) = e. sin(1y) at (x, y) = (-3, 1) is defined as followed V f(x, y) = (fz(-3,…

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Q: Find the gradient of f(x, y) =x2-5xy+y2 at the point (0, 1)

A: Gradient of the f(x,y) at a point (a,b) is given by :∇f=dfdx(a,b)i+dfdy(a,b)j

Q: A bug crawls on the surface z = x2 - y2 directly above a path in the xy-plane given by x = ƒ(t) and…

A: We are given that the bug crawls on the surface z = x2 - y2 and x = ƒ(t) and y = g(t).  So,…

Q: Draw the level curves of f(x,y)= x² + 9y² for k=0, k=1, k=9.

A: Given curve is, fx,y=x2+9y2

Q: Find a function f such that Vf = 2e²x cos(5y)i - 5e²* sin(5y)j + z5k. f(x, y, z) =

A: The given gradient function

Q: Ev 2 x 2 [x²z dx - yx² dy + 2 dz along the curve C'shown in the figure. AZ 2

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Q: A particle moves in the xy plane so that at any time -2 ≤ t ≤ 2, x = 3 sint and y = t - 4 cost + 1.…

A: A particle moves in the xy plane so that at any time ,  and .What is the vertical component of the…

Q: The pressure P(x, y) at a point (x, y) = R² on a metal membrane is given by P(x, y) = 100e-x²-2y².…

A: In this problem, we are given a function P(x,y) that represents the pressure at a point (x,y) on a…

Q: Write the Ostrogradsky equation for the functional 2 Əz (x, y)] = [ { )* + C)²} dx dy D where the…

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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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Q: Find the directional derivative of f(x, y, z) = xy² + yz% at P(1, – 1, 1) in the direction of a = i…

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Q: Suppose at (0,1) the function f(x,y) has linearization L = 2x + 3y -3. Let v = . What are f(0,1) and…

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Q: Find the gradient of f(x, y) = 3xty3 + 4x°y® at (1, 2).

A: