How would you find the velocity vector for the problem below?

Question

Find the velocity vector, V(t), evaluated at t= 1, of a particle that moves along the curve C given by the vector-valued function r(t)= <4t, -5t^2, 3t^3> V(1) =

it could be one of the following....
<4, -6, 15>
<3, -8, 15>
<5, -6, 12>
<4, -10, 9>
<3, -10, 12>
<5, -8, 9>
or none of these

Answer 1

The velocity is, by definition, the time rate of change of the position. Or, said differently, the derivative of position wrt time.
r(t) = <4t, -5t^2, 3t^3>
Taking the derivative wrt time gives:
v(t) = <4, -10t, 9t^2>
The value of this function at t=1 is:
v(1) = <4, -10, 9>

And that does agree with your 4th choice, above.

Answer 2

take the derivative with respect to time:

r(t)= <4t, -5t^2, 3t^3> V(t) = <4.-10t , 9t^2>

Evaluate at t=1 so V(1) = <4,-10,9>