Find the Unit Tangent Vector to the Curve Given Below $$\mathbf { r } ( t ) = - 6 t \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t = 4$$

question 72

Multiple Choice

Find the unit tangent vector to the curve given below at the specified point. $$\mathbf { r } ( t ) = - 6 t \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t = 4$$

• A) \(\mathbf T ( 4 ) = \frac { - 6 } { \sqrt { 1060 } } \mathbf { i } + \frac { 4 } { \sqrt { 1060 } } \mathbf { j }\)
• B) \(\mathbf { T } ( 4 ) = \frac { - 6 } { \sqrt { 1060 } } \mathbf { i } + \frac { 32 } { \sqrt { 1060 } } \mathbf { j }\)
• C) \(\mathbf T ( 4 ) = \frac { - 12 } { \sqrt { 1060 } } \mathbf { i } + \frac { 32 } { \sqrt { 1060 } } \mathbf { j }\)
• D) \(\mathbf { T } ( 4 ) = \frac { - 6 } { \sqrt { 1060 } } \mathbf { j } + \frac { 32 } { \sqrt { 1060 } } \mathbf { i }\)
• E) \(\mathbf T ( 4 ) = \frac { 4 } { \sqrt { 1060 } } \mathbf { i } + \frac { 32 } { \sqrt { 1060 } } \mathbf { j }\)

---

Definitions:

Behavioral Isolation

A form of reproductive isolation in which two populations develop differences in mating rituals or behaviors that prevent them from breeding with each other.

Reproductive Isolating Mechanisms

The reproductive barriers that prevent a species from interbreeding with another species; as a result, each species’ gene pool is isolated from those of other species. See prezygotic barrier and postzygotic barrier.

Sympatric Isolating Mechanisms

Sympatric isolating mechanisms are barriers to breeding that occur when two species share the same habitat but develop behaviors or characteristics that prevent them from interbreeding.

Allopatric Isolating Mechanisms

Biological barriers that prevent species from interbreeding and producing fertile offspring when they are geographically separated.