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, 1/3t³ - 1/2t², t = 4

Answer 1

To find the unit tangent vector t(t), calculate the derivative of r(t) to get the velocity vector, evaluate it at the given t value, and normalize the resulting vector.

Step-by-step explanation:

To find the unit tangent vector t(t) at a specific point for a given parametric equation r(t), you must follow these steps:

1. Compute the derivative of r(t) with respect to t to get the velocity vector v(t).
2. Evaluate v(t) at the given t value to find the velocity at that point.
3. Normalize the resulting velocity vector to obtain the unit tangent vector t(t).

Given r(t) = t² - 3t, ⅛ t, ⅓ t³ - ½ t², and t = 4:

1. The velocity vector v(t) is the derivative of r(t), which would be v(t) = 2t - 3, ⅛, t² - t.
2. Evaluating at t = 4 gives us v(4).
3. Normalize v(4) to find the unit tangent vector t(4).