Consider F and C below. F(x, y) = x² i + y² j C is the arc of the parabola y = 4x² from (-1, 4) to (2, 16) (a) Find a function f such that F = Vf.

Q: - Compute 5y dr a dy, where C is the arc of the parabola y = from (-2, 4) to (0, 0).

A: The line integral of the function with respect to x, y is solved using the formula…

Q: 13 - x² – y? at (3, 2, 0), then If the point (1, 1, k) lies on the tangent plane to the paraboloid z…

A: z=13-x2-y2 Partially differentiating with respect to x, we get zx=0-2x-0=-2x Partially…

Q: Consider F and C below. F(x, y) = (3 + 4xy2)i + 4x²yj, C is the arc of the hyperbola y 1/x from (1,…

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Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = 2xy3i…

A: Consider F = 2xy3i + 4x2y2j We have M = 2xy3 and N = 4x2y2

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A: Equation of given curve is y=2x32 Differentiate with respect to x. dydx=232x32-1=3x

Q: Find the function f if. Vf-(3a²y + y², a² + 2xy), Vf dr with and without the Fundamental Theorem of…

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Q: If f'(x)= cosec* x-1 then what is the arclength of the curve y = f (x) on TC %3D V3 B) 3 A) 1 C) 2…

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Q: Consider F and C below. F(x, y) = x2 i + y² j C is the arc of the parabola y = 2x2 from (-1, 2) to…

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Q: Find the point on the ellipse x = 2 cos t, y = sin t, 0 … t … 2pi closest to the point (3/4, 0).…

A: To Find: Point on the ellipse x=2cost, y=sint, 0≤t≤2π, closet to the point 34,0. Concept: Distance…

Q: (a) Find the solution of the limit using polar coordinate system. x² + y2 lim (x,y)-(0,0) x2 + y?…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let L be a tangent line to the hyperbola xy=4 at x=9 Find the area of the triangle bounded by L and…

A: Compute the tangent line as follows.

Q: Find the tangent plane to the equation 3y cos(4x – 2y) at the point (2,4,12) Z =

A: Z=3ycos(4x-2y)  Point= (2,4,12)

Q: 4. Find f such that F = Vf and use it to evaluate SF. dr. (a) F(x, y) = (3 + 2xy²) î+ 2x²y ĵ, C is…

A: (4) To find: f such that F→=∇f and To Evaluate: ∫CF→·dr→. For the following Functions: (a)…

Q: Find the arc length of y = -6x + 2 on [2,5] (use calculus). %3D L = 1 + f'(x)² dx

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Q: Use Green's Theorem to evaluate F. · dr, where F = (Va + 4y, 2x + and C consists of the arc of the…

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Q: Consider F and C below. F(x, y) = (9 + 4xy2)i + 4x2yj, C is the arc of the hyperbola y = 1/x from…

A: Consider the given information: F(x,y)=(9+4xy2)i+4x2yj And C be the arc of the hyperbola y=1/x from…

Q: 7. (~Ch16 Review #6) Compute y3 dx + x² dy where C is the arc of the parabola x = 1- y² from (0, –…

A: In this question we need to find the given line integral .

Q: If f'(x)=Vcosec* x-1 then what is the arclength of the curve y= f (x) on TC TC A) 1 B) 3 C) 2 D) V3…

A: Given that, f'x=csc4x-1

Q: Find a path c(t) that traces the parabolic arc y = x2 from (0, 0) to (3, 9) for 0 ≤ t ≤ 1.

A: Given parabolic arc y=x2 from 0,0 to 3,9 for 0≤t≤1. The path ct that traces the parabolic arc has to…

Q: Use Green's Theorem to evaluate | F. dř, where F = (Va + 4y, 2x + 9) C consists of the arc of the…

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Q: A particle moves in the xy-plane in such a way that its path is defined by x = e^t cos t and y = e^t…

A: Given that, x = e^t cos t y = e^t sin 2t Differentiating x with respect to t we get, dx/dt = e^t cos…

Q: Find the work done by F in moving a particle once counterclockwise around the given curve. F= (4x-…

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Q: Consider the curve: r(t) = In (t) i + +(/2)tk Find the length of the curve between the point Po…

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Q: Consider the tangent line to the curve r(t) = t2i −(1/(t + 1))j +(4 − t2)k at the point P(4, 1, 0).…

A: Given the tangent line to the curve r→t=t2i^−1t+1j^+4−t2k^  at the point P4,1,0. We have to…

Q: 1. Sketch each curve in the z-plane, and sketch its image under w = 2². (a) |z1|=1 (b) x = 1 (e) y²…

A: We are authorized to answer three subparts at a time since you have not mentioned which part you are…

Q: Consider F and C below. F(x, y) = x² i + y² j C is the arc of the parabola y = 2x² from (-1, 2) to…

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Q: Given F=. By using Gauss's theorem, find f F ds for a paraboloid x = 4-y² - 2² with x ≥ 0 as shown…

A: We are given  F=-3x+1+z3,y+1+x3,6z+xy So, the divergence of F is div…

Q: 2. Find the orthogonal trajectories of the family of parabolas y = cx O In(x² + y² ) = c O 2r? + y²…

A: Explanation of the answer is as follows

Q: Use Green's Theorem to evaluate | F. dr, where F = (Va + 5y, 2x + Vỹ ) and C consists of the arc of…

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Q: Find the arc length from (x1, y1) to (x2, y2) on the graph of f(x) = mx + b.

A: Given Data The first point on the graph is (x1, y1). The second point on the graph is (x2, y2). The…

Q: - Find the aera under the curve y = cosx Form x = 0 to x = -fx sec² (ax² )dx a#0

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Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: Consider the given function, xy+y2-3x-3=0 now differentiate with respect to x,…

Q: Let curve C be the arc of x2 + y2 = 16 traversed once counterclockwise from (0, 4) to (0, -4). Let…

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Q: Show that the graph of ř = (t sin t, t cos t, t²) lies on the paraboloid z = x? + y?.

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Q: Consider F and C below. F(x, y) = x² i + y² j C is the arc of the parabola y = 3x2 from (-2, 12) to…

A: Consider Fx,y=x2i+y2j. a. We have to find a function f such that F=∇f. Step 1: We have…

Q: 1. Use a change of variables to evaluate [[ y- dA where R is the trapezoid with vertices (1,0,…

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Q: Use Green's Theorem to evaluate / F · dř, where F = (Va + 6y, 3x + y9) C consists of the arc of the…

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Q: b. What is fo, f (z) dz?

A: f(z) is holomorphic except at z0   where f(z) has pole at z0 we can write  f(z)=g(z)z-z0  where g(z)…

Q: ind the equation of the tangent line to right branch of the hyperbola == sec t, y= tant, 2 - sts at…

A: Topic:- application of derivatives

Q: Consider the given curve: R(f) = (2 cos f, 12f +3,-2 sin f) Reparametrize the curve using arclength…

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Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: the equation we have  (x3-y4)6+ln(x2+y)=1

Q: Let C be the closed curve starting at (0, 1) traveling along the parabola y = (x – 1)? up to the…

A: Using Green's Theorem we solve the above question.

Q: Consider F and C below. F(x, y) = x2 i + y² j C is the arc of the parabola y = 5x2 from (1, 5) to…

A: Find the curl in step2 of part (b).

Q: Consider F and C below. F(x, y) = (6 + 4xy²)i + 4x²yj, %3D C is the arc of the hyperbola y = 1/x…

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Q: (1) Let f(x, y) = xe" + x2. (a) Find the tangent plane of f at the point (2,0). (b) Use your answer…

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Q: Q1. Find the length of the of the curve y = In(cos x) from x = 0 to x = . %3D

A: Explanation of the solution is given below...

Q: Consider F and C below. F(x, y) = (9 + 2xy²)i + 2x²yj, C is the arc of the hyperbola y = (a) Find a…

A: The function f(x,y) may contains a constant, but we can choose a f(x,y) without constant.

Q: Find eauation of ny Perbola at o Ne tne foci Fz the and crankshafto f Fi Fz ny Perbola= Fi F

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