(a) Find the unit tangent vector on the curve r(t) = [cos (t),2t,3sin(4t)] when t = 0.

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A: given differential equation

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A: f(x,y)=3xy+1x−8y

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A: For a Function f( x , y ), The directional Deivative in the direction of  is given as

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A: since you have posted multiple question in this question which are not interlinked so i have solved…

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A: x=t-1y=1+2tz=3-tPoint (-2,-1,4)

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A: To find

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Q: 1) fx²e-3x dx/

A: since you have posted multiple question in this question which are not interlinked so i have solved…

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Q: A particle moves according to a law of motion s = f(t), t≥ 0, where t is measured in seconds and s…

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Q: g(t) = sin (Int) f(x) = ln(sin x)

A: g(t)=sin⁡(ln⁡t)

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Q: Q/ solve equations 1- y' = ky 2-x(1+y²) dx =y(1+x²) d;

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A: Equation of elliptic parabolid z=3+3x2+2y2Equation of planes x=2 and y=4

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A: A sequence  converges if  where L is a finite number.

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A: Concept:Conic section:A curve obtained from cone's surface interacting in plane.

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A: given graph of f(x)graph f(x - 4) - 5

Q: Find the area of the region that lies inside both curves. r = 5 + cos(0), r = 5 cos(0)

A: r=5+cos⁡(θ),r=5−cos⁡(θ)

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A: Given,

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Q: Solve the second order Differential Equations. 4y - 12y + 5y = 0

A: The given differential equation

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A: To differentiate the following function.

Q: d the derivative of :) = €5* (x² +9¹)

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Q: Find the linearization L(x) of the function at a. f(x) = cos x, a = 3π/2 L(x) =

A: Since you have asked multiple questions in a single request, we would be answering only the first…