Verify if the vector field F = (x² cos(3y), -³ sin(3y)) is conservative and evaluate the line integral Je where C' is the curve parameterized by R=< t, 2t² > for 0 < t < 1. cos(6) 3 A. I = B. I O c. I - OD. I = 0 E. I sin(6) 4 23 cos(3) 2 F. dR

Q: Draw the slope field (direction field) for the following DE: y'=y-t

A: Given DE is Make table as follows:…

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A:

Q: Suppose f(x, y) = sin( (a) Vf(x, y) = = (b) Vƒ(3, π) = 2y x T 2y 2y 2y cos cos 2) Con (2) 200m (20 -…

A: Given that , fx,y=sin2yx and u is the unit vector in the direction of 1,1.

Q: 3. Consider the following helix with vector equation 7 (t) = 2 cos (t) 7 (t) 7 + tỉ + 2 sin (a)…

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Q: Consider the vector-valued function F(t) = (t², sin(t)) on the interval [-#,#]. Construct the table…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three…

Q: 1. Consider the vector-valued function F(t) = sin(2t) t (2, 1, -2), a) Find the domain of F. b) Show…

A: Disclaimer: There are two questions here and as per Bartleby policy, I am solving only the first…

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A: Given, r→t=sinti^-2costj^-tk^

Q: Let 7(t) = (t + 4, – e*, 3 sin(3t)) - Find the unit tangent vector T (t) at the point t = :0 T(0) =

A: We have to find

Q: Sketch a direction field for the equation y'= 2y(1-y) by hand. Then use that and sketch a few…

A:

Q: 2. Let T(x; y) = sin(2x – y) , P (-; ) and Q(0;0) (a) Determine the directional derivative of T at P…

A: Tx,y=sin2x-y, P=-π3,π6,𝐐=0,0                                      P1  P2                𝐐1,…

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: 6. Let f(r, y, z) = e=² cosh y + z ar ff at the point P in the direction of :

A:

Q: Compute the following definite integral of a vector-valued function: Cos t i- tj + te’k ) dt sint +1

A: Please refer the attached image for complete solution.

Q: 3. Let F = (x - y, xy) and C be arc of the circle x? + y? = 4 traversed counter-clockwise from (2,0)…

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Q: Find an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 –…

A: The complete solutions are given below

Q: Find the vector equation of the tangent line to the curve given by 7(t) = sin ti + cos (2t)j + tan…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: ● Q1/ Determine the integral of the vector function -2π [sin² (t) cos(t) i + sin(t)(cos² (t) j + 2k]…

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Q: 2. Find the length of the curve r(t) = e'i + e' sin 2tj + e' cos 2tk, 0st< 2n.

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Q: Suppose F(x, y) = −y sin(x)i + cos(x)}. (a) Find a vector parametric equation for the parabola y =…

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Q: Iterate the estimate given U(x) log²x + // (A(st) - A(s)A(1)) U () dsdt = 0(x log.x) as x → 00,…

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Q: 3. Solve xy" + y = 0 given that y(1) = 1, y(2) = 2 taking h = 0.25 using finite difference method.

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Q: Find a tangent vector of the curve7(t) = (t², 2 sin(t), 2 cos(t)) at (0,0, 2) (0,0,1) (1,0,0) None…

A: Here, Find it’s derivative This is the tangent at any point on the curve.

Q: 3. (a) Given the vector valued function r(t) = 4 cos zti + 4 sin atj + tk. Find the position vector,…

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

Q: If an object travels in the xy-plane along the curve traced out by 
the vector function r(t) = (t…

A: We have to find the total distance traveled by the object from t = 0 to t = 4.

Q: 1. Find the tangent vector to the curve 6t r(t) = [t³ – t, t+1 , (2t + 1)*] at t = 1.

A: As per our guidelines we are supposed to answer only one question. If you want solution of any…