Use the properties of the derivative to find the following. r(t) = ti + 4tj + t?k, u(t) = 4ti + t²j + t³k (a) r'(t) (2r(e) – u()] (b) dt (c) (4t)u(t)] dt

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A: The given integral,          ∫-119t i+3t3 j+t3kdt

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A: To find the divergence and curl.

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Q: Evaluate JJ xy dA, where R is the region in (1, 3) (5, 3) (2, 1) (4, 1)

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A: Dear learner, As per our guidelines we can provide the solution for first one only. Kindly post…

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A: Directional derivative: D f = del f • unit vector of direction

Q: Show

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A: Let's find.

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A: According to the company rule we can answer only first question and rest can be reposted.

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A: We need to evaluate the following line integral ∫C-ydx+xdy Where,  C: y2 = 6x  from 6, 6 to 0, 0

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A: We have to find it from Trapezoidal rule.

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A: Given, r= 5ti -8j +(1+5t)k Therefore, The equation is given by, r= <0,-8,1>+t<5,0,5>

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