3. Use the Fun.Theorem to evaluate F dr, where F = 3xy?ī+ 3x*yj , πί and C is the curve described by r(t) = (t + sin())i + (t + cos())j, 0

Q: Evaluate Se (5xy + x² + y²) dx + (x² - y)dy by using Green's theorem, where C is a closed curve that…

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Q: find the work done by F over the curve in thedirection of increasing t. F = 6z i + y2j + 12x kr(t) =…

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Q: 2. Let y be the curve in R° given by t (t, 2 cos t, 2 sin t) for 0 < t< 47. Compute y? ds

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Q: ne if the curve (In t. t In t. 1) is tangent to the surface rz- uz? + cos Tu =1 at (0.0. 1).

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Q: 1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal…

A: TRUE Slope of the curve, when in parametric form = dy/dx = g'(t) / f'(t)

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

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Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (c) Determine the…

A: Here is the solution of the given problem.  To solve this problem we use simple concept of…

Q: Q2) find the slop for the tangent of the function f (x) sin(xcos(x)) at x = 2n. %3D

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Q: Find f F. dr for F = 2yi – (sin y)j on the curve counterclockwise around the unit circle C starting…

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Q: Let C be a portion of a curve from (2,0) to (2,2). Let F(r,y) = (tan(x²) + e²v¸ 2re²v

A: Introduction: A vector field is conservative if ∂P∂y=∂Q∂x, Where f(x,y)=Pi+Qj.

Q: Consider the curve in the xy-plane repre- sented by = e' and y = The slope of the line tangent to…

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Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

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Q: Calculate F.dr, where C is the curve given by ti + 123+ tk, with t = [0, 1] and F(x, y, z) = 4xei +…

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Q: Let C be the curve defined by T(t) = (2 + 2 sin(t), 3 – 2 cos(t)) for 0 < t < 4x=²) dx + (2xyev" +…

A: Given the integral is ∫Cy2exy2+4x-2dx+2xyexy2+2dy. Rewrite the integral as ∫CF·→dr→:…

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 the smooth parametric curve defined by the equations = t (t² − 3) = 3 (t² − 3)' X Y where…

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Q: Which one of the following equations describes the tangent plane to the curve, f(x, y) = x² + y² +…

A: Explanation of the answer is as follows

Q: Let C be the curve.

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Q: 4. Let f(z) = 1 23 1 - and find [ f(z) dz, where C a circle of radius 2 about the origin with…

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

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Q: Use Green's Theorem to evaluate F. dr, where F C consists of the arc of the curve y 1x x2 - from…

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Q: Let C be the curve parameterized as follows F(t) := (9 cos(t), 9 sin(t), 12t), 0<ts 10 and let f(x,…

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Q: Find the equation for the Tangent Plane on the given surface coS ax – x²y + e*z+ yz = 4 Po(0, 1, 2)

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Q: Use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate the given curve's length. %3D c(t)…

A: We are given the curve :c(t)=(cos t, esint) The length of the curve: s=∫02π-sin t2+(cos t esin t)2…

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

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Q: 7. Consider the function f: R → R³ given by: t+1 f(t) = sin(cos(t)), tet) t2 +4° Find f'(t) for all…

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Q: Let F = e'i+(xe' -e')j- ye'k and cbe a smooth curve defined by Then the value of the line integral…

A: The formula is  ∫F.dr=∫F(r(t)).r'(t)dt Here…

Q: Find the work done by F over the curve in the direction of increasing t. F = 3xyi + 4yj - 2yzk r(t)…

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Q: Evaluate F. dr where F = (x – 3y)i + (y – 2r)j and C is the closed curve in the xy plane, x = 2 cos…

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Q: Find an equation for the tangent plane to the graph of z = xe' cos z – 2 - at the point (2,0,0). 1.…

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Q: 2. Let F(x,y) = (cos(x – 2y), –2 cos(x – 2y)). Find curves C and C2 so | that, F. dr = 0 F. dr = 1

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Q: Find the work done by F over the curve in the direction of increasing t. F= 11zi + 11xj + 11yk, r(t)…

A: We have to find the work done by F over the curve

Q: Compute / f ds over the curve specified. (1) f(x, y, z) = z², r(t) = for 0 < t < 2

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Q: 13. Let F (x, y) = (1(²y²)x, af(x²y? , a(y²) Y) and suppose that C is a smooth curve parametrized…

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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: x² = y + 3y.

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Q: Find the Giniindex for the lorenz curve L (X)= 0.3X + O=7X 11

A: Given the Lorenz curve Lx=0.3x2+0.7x. Find the Gini index of the Lorenz curve.

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: Use Green's Theorem to evaluate / F · dĩ, where F = (VE + 3y, 2x + vũ) C consists of the arc of the…

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Q: Find the work done by F over the curve in the direction of increasing t. F=xyi + 8j + 6xk; C: r(t) =…

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Q: Sketch the curve. O 68°F r = 2 - 2 sin(0) -2 -3 1 y 1 2 1 1 XX 2 O EZ 1 2 3 3 X -2

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Q: Let C denote the curve of intersection between x = -5z° and y = 6z", starting from A(0,0,0) and…

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Q: Ci is (x(t), y(t)) = (t, 0) for - 1<t< 1 and C2 is (¤(t), y(t)) = (cos t, sin t) for 0 <t< r.…

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Q: The parametric form of the tangent line to the image of f(t) = (5t², 2,t – 2) at t = -1 is L(t) =

A: Topic = Application of derivative

Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: Find the work done by F over the curve C in the direction of increasing t,where F=-5 yi+5xj+8z°k and…

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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: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A: According to the given information, it is required to use green’s theorem to evaluate the integral.