Part 3/6:
r'(t) = 3 t^2 \mathbf{i} + (1 - t) e^{-t} \mathbf{j} + 2 \cos(2t) \mathbf{k}
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This vector represents the tangent direction and rate of change of the curve at each point ( t ).
Part B: Computing the Unit Tangent Vector at a Point ( t = z )
The next step involves finding the unit tangent vector at a specific value of ( t ). The process entails:
Computing the derivative ( r'(t) ) at ( t = z ).
Calculating the magnitude (length) of this derivative vector.
Dividing ( r'(z) ) by its magnitude to normalize it.
Suppose, for instance, we evaluate at ( t=0 ). Calculating the derivative at ( t=0 ):
[