Find the Unit Tangent Vector to the Curve R (T) $\left\{ e ^ { 2 t } \cos t , e ^ { 2 t } \sin t , e ^ { 2 t } \right\}$

question 46

Multiple Choice

Find the unit tangent vector to the curve r (t) = $\left\{ e ^ { 2 t } \cos t , e ^ { 2 t } \sin t , e ^ { 2 t } \right\}$ at the point where t = $\frac { \pi } { 2 }$ .

• A) \(\left( \frac { 2 } { 3 } , - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)\)
• B) \(\left( \frac { 2 } { 3 } , - \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)\)
• C) \(\left( - \frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 2 } { 3 } \right)\)
• D) \(\left( \frac { 2 } { 3 } , \frac { 2 } { 3 } , - \frac { 1 } { 3 } \right)\)
• E) \(\left( \frac { 3 } { \sqrt { 14 } } , \frac { 2 } { \sqrt { 14 } } , - \frac { 1 } { \sqrt { 14 } } \right)\)
• F) \(\left( - \frac { 2 } { \sqrt { 14 } } , \frac { 1 } { \sqrt { 14 } } , \frac { 3 } { \sqrt { 14 } } \right)\)
• G) \(\left( \frac { 1 } { \sqrt { 14 } } , - \frac { 3 } { \sqrt { 14 } } , \frac { 2 } { \sqrt { 14 } } \right)\)
• H) \(\left( \frac { 1 } { \sqrt { 6 } } , \frac { 2 } { \sqrt { 6 } } , \frac { 1 } { \sqrt { 6 } } \right)\)

---

Definitions:

Ventromedial

Relating to the ventromedial region of the brain, which is associated with regulating emotions and decision-making.

Cephalocaudal

A principle of development that describes the growth pattern of organisms from head to toe, as in human infants.

Neonates

Neonates refer to newborn infants, particularly within the first 28 days of life, highlighting a critical period for development and adjustment outside the womb.

Organic Illness

A medical condition that is caused by a detectable physical or biochemical change in the body, rather than a psychological origin.