Question

Find the unit tangent vector to the curve at the specified value of the parameter. r(t) = t³i+ 6t²j, t = 2 T(2) =

Answer 1

(√5/5)i +(2√5/5)j

Explanation:

You want the unit vector in the direction tangent to the given curve at t=2.

• r(t) = t³i +6t²j

Derivative

The derivative is ...

r'(t) = 3t²i +12tj

At t=2, this is ...

r'(2) = 3·4i +12·2j = 12i +24j

The magnitude of this vector is |12i +24j| = 12√5, so the unit vector is ...

T(2) = (1/√5)i +(2/√5)j = (√5/5)i +(2√5/5)j

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