1. Consider the vector function (with values in R³) r(t) = ln (√2√1−t) i + (3 − √2√1 −t)j + 2 k Find a positive change of parameter from t to s where s is an arc length parameter of the curve having r(0) as its reference point.

Q: Let F(t) = (1, ²,4) and P = (2, 10, 4). Consider the curve C parametrized by F(t). Compute a tangent…

A: The objective of the question is to find the tangent vector, unit tangent vector, and the tangent…

Q: 1. Let ř(1) = . Find a vector and parametric equation of the line tangent to F(1) at the point (-…

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Q: Plot the curve c(t) = (cos t, esin t ) for 0 ≤ t ≤ 2π

A: The plot of the curve for t within 0 to 2pi is shown in figure below.

Q: Q2) consider a particle moving in a curve defined by F(t) =sin 21, cos 21, 1. Find a. The velocity…

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Q: The velocity of a particle moving in the xy-plane is given by the vector v(t) = (-5 sin(t²), 2e¬6t…

A: we now that if velocity vectoe at time t is  v(t)=vx , vywhere v1  and  v2  are integrable function…

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: Find the vector equation of the tangent line to F(t) = (2 sin 2t, cos 2t, – 6t) at the point where…

A: Consider the given vector. rt=2sint,cos2t,-6t   Find the derivative of the vector with respect to t.…

Q: 5. Find the parametric equations of the line tangent to the curve C defined by the vector equation…

A: To find the parametric equation of the given tangent to the curve. The vector equation of the curve…

Q: Find the unit tangent vector to the curve given below at the specified point. r(t) = 7ti−2t²j¸ t = 4…

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Q: Sketch the curve with vector equation r(t) = (t2 – 2, t) in the xy-plane. Indicate with arrows the…

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Q: Let r(t) = (-3t³-4, -3e¹, 5 sin( - 3t)) Find the unit tangent vector T' (t) at the point t = 0 T(0)…

A: r(t)=<-3t3-4, -3e4t, -5Sin(-3t) >

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

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Q: Let 7(t) = (– 5t4 – 4, – 2e“, 2 sin(3t)) Find the unit tangent vector T (t) at the point t = 0 T (0)…

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Q: 2. Find thé moving trihedral of the curve C traced by the vector-valued function R(t) 4 cost i+4…

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Q: Let 7 (t) = (4t³ – 4, – 3e", – 3 sin( – 4t)) - Find the unit tangent vector T(t) at the point t = 0…

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

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Q: Find the unit tangent vector to the curve defined by =(- 4t, 4t, V49 - t) at t = - 6. T(- 6)

A: