Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = t2 − 3t, 1 + 4t, 1 3 t3 + 1 2 t2 , t = 4.

Question

Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = t2 − 3t, 1 + 4t, 1 3 t3 + 1 2 t2 , t = 4.

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Camila 2 weeks 2021-10-12T10:12:52+00:00 1 Answer 0

Answers ( )

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    2021-10-12T10:14:03+00:00

    Answer:

    T(t) = [1/144 , 1/180 , 1]

    Step-by-step explanation:

    if the parametric curve is

    r(t) = t² − 3t, 1 + 4t, 13*t³ + 12* t²

    then r'(t) = [ x'(t) , y'(t) , z'(t) ]

    therefore

    r'(t) = 2t – 3, 4 , 39* t² + 24t

    at t=4

    r'(4) = 2*4 -3 , 4 , 39*4² + 24*4 = 5 , 4 , 720

    also the modulus of the r'(4) vector is

    | r'(4) | = √(5²+4²+720²) ≈ 720

    thus

    T(t) = r'(4) / | r'(4) | = 5/720 , 4/720 , 720/720 = 1/144 , 1/180 , 1

    T(t) = 1/144 , 1/180 , 1

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