Use Stokes' theorem to evaluate F. dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = 7x²yi + 3x³j + 5e² tan−¹(z)k, C is the curve with parametric equations x = cos(t), y = sin(t), z = sin(t), 0 ≤ t ≤ 2π (-1, 0, 0) (1, 0, 0)

Q: Find a rectangular equation for the curve with the given parametric equations. x = 2 sin(t),y = 2…

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Q: 2. If r(t) = ti + cos(t)j + sin(t)k, evaluate (a) frit r(t) dt, (b) r' (t) (c) the parametric…

A: Note:- As per our company guidelines we are supposed to answer only first three sub-parts. Kindly…

Q: 2. For the parametric curve defined by: x(t) = t2 + 4t + 2 y(t) = t³ + 2t2 + 4t + 1 (a) determine…

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Q: 2. Consider the parametric curve r = t + 1, y = t - t. dy (a) Find d.r dy (b) Find da?

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Q: 1. Let D = the sales of DVDs in billions of dollars, S = the total sales of streaming videos in…

A: The sales of dvds given as the function D = -0.92t + 8.75 and the total sales of streaming videos…

Q: 2. Let F(x, y, z) = (6xyz+4x) i+(3x² z+ze=v)j+(3x²y-e-") k, and let C be the portion of the…

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Q: Find the slope and concavity of the parametric curve at the point t = (1/2), x(t) = (1/2)t² -1,…

A: To find the slope of the curve at t = 1/2, we need to find the derivative of y with respect to x:…

Q: Sketch the parametric curve at R* whose parameterization is given by a (t) = (cos (t), sin (1), t),…

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Q: Find the length of the curve defined by the parametric equations t, y = 5 In((t/6)? – 1) I = from t…

A: Given, x = 56t , y = 5lnt62-1 = 5lnt2-36-5ln(36)and 7≤t≤10

Q: 4. Find the parametric equation of the tangent line to r(t) = 4costi – 3tj %3D at Po(2, –n)

A: Given: To find the parametric equation of the given tangent line as follows, r(t) = 4 costi- 3tj…