Deck 11: Vector Functions

Question 1

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle 2 t , t ^ { 2 } \right\rangle$

Answer 1

$x ( t ) = 2 t , y ( t ) = t ^ { 2 }$

Question 2

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \langle \sin ( t ) , 14 \rangle$

Answer 2

$x ( t ) = \sin ( t ) , y ( t ) = 14$

Question 3

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle \cos ( t ) , t ^ { 2 } - 2 \right\rangle$

A) $x ( t ) = \cos ( t ) , y ( t ) = t ^ { 2 } - 2$
B) $x ( t ) = t ^ { 2 } - 2 , y ( t ) = \cos ( t )$
C) $x ( t ) = - \sin ( t ) , y ( t ) = 2 t$
D) $x ( t ) = 2 t , y ( t ) = - \sin ( t )$

Answer 3

$x ( t ) = \cos ( t ) , y ( t ) = t ^ { 2 } - 2$

Question 4

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle t , 2 t ^ { 2 } , 3 t ^ { 3 } \right\rangle$

Question 5

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( 2 t ) , \tan ( 3 t ) \rangle$

Question 6

Find the parametric equations for the vector-valued function $\vec { r } ( t ) = \left\langle \cos ( t ) , e ^ { 2 t } , t ^ { 3 } \right\rangle$

A) $x ( t ) = t ^ { 3 } , y ( t ) = \cos ( t ) , z ( t ) = t ^ { 3 }$
B) $x ( t ) = \cos ( t ) , y ( t ) = e ^ { 3 t } , z ( t ) = t ^ { 3 }$
C) $x ( t ) = t ^ { 3 } , y ( t ) = \cos ( t ) z ( t ) = e ^ { 2 t }$
D) $x ( t ) = e ^ { 2 t } , y ( t ) = t ^ { 3 } , z ( t ) = \cos ( t )$

Question 7

Evaluate and simplify the quantity. $$3 < 2 t , \cos ( t ) >$$

Question 8

Evaluate and simplify the quantity. $$t < \cos ( t ) , e ^ { t } >$$

Question 9

Evaluate and simplify the quantity. $$\left\langle t , e ^ { t } \right\rangle \cdot \left\langle - e ^ { t } , - t \right\rangle$$

A) $t ^ { 2 } + e ^ { 2 t }$
B) $2 t e ^ { 2 t }$
C) 0
D) $- 2 t e ^ { t }$

Question 10

Evaluate and simplify the quantity. $$t < \cos ( t ) , \sin ( t ) , e ^ { t } >$$

Question 11

Evaluate and simplify the quantity. $$\left\langle t , e ^ { t } , \cos ( t ) \right\rangle + \left\langle 4 , e ^ { t } , t ^ { 2 } \right\rangle$$

Question 12

Evaluate and simplify the quantity. $$\langle \cos ( t ) , \sin ( t ) , 3 \rangle \cdot \langle \cos ( t ) , \sin ( t ) , t \rangle$$

Question 13

Evaluate and simplify the quantity. $$\langle t , 7 t , \cos ( t ) \rangle \times \left\langle 1 , t , t ^ { 2 } \right\rangle$$

A) $\left\langle t \cos ( t ) - 7 t ^ { 2 } , t ^ { 3 } - \cos ( t ) , 7 t - t ^ { 3 } \right\rangle$
B) $\left\langle 7 t ^ { 3 } - t \cos ( t ) , \cos ( t ) - t ^ { 3 } , t ^ { 2 } - 7 t \right\rangle$
C) $\left\langle t , 7 t ^ { 3 } , t ^ { 2 } \cos ( t ) \right\rangle$
D) $t + 7 t ^ { 3 } + t ^ { 2 } \cos ( t )$

Question 14

Evaluate the limit: $$\lim _ { x \rightarrow 0 } \left\langle t ^ { 2 } , \cos ( t ) \right\rangle$$

Question 15

Evaluate the limit: $$\lim _ { x \rightarrow 1 } \left\langle t , e ^ { t } \right\rangle$$

Question 16

Evaluate the limit: $$\lim _ { x \rightarrow \frac { \pi } { 2 } } \left\langle t ^ { 2 } , \cos ( t ) \right\rangle$$

A) $< 0 , \frac { \pi ^ { 2 } } { 2 } >$
B) $\langle 0,1 \rangle$
C) $< \frac { \pi ^ { 2 } } { 4 } , - 1 >$
D) $\left. < \frac { \pi ^ { 2 } } { 4 } , 0 \right\rangle$

Question 17

Evaluate the limit: $$\lim _ { x \rightarrow 0 } \left\langle t , t ^ { 2 } , \cot ( t ) \right\rangle$$

Question 18

Evaluate the limit: $$\lim _ { x \rightarrow 1 } \left\langle \frac { t ^ { 2 } - 1 } { t - 1 } , \cos ( \pi t ) , t ^ { 2 } \right\rangle$$

Question 19

Evaluate the limit: $$\lim _ { x \rightarrow \frac { \pi } { 2 } } \left\langle \frac { t } { \pi } , t ^ { 2 } , \tan ( t ) \right\rangle$$

A) $< \frac { 1 } { 2 } , \frac { \pi ^ { 2 } } { 4 } , 1 >$
B) $< 1 , \frac { \pi } { 2 } , - 1 >$
C) DNE
D) $< \frac { 1 } { 2 } , \frac { \pi ^ { 2 } } { 4 } , \frac { \pi } { 2 } >$

Question 20

Find $\vec { r } ( t ) = \langle x ( t ) , y ( t ) \rangle$ determined by $\begin{array} { l } x ^ { \prime } ( t ) = x ( t ) , y ^ { \prime } ( t ) = t \\x ( 0 ) = 1 , y ( 0 ) = 1\end{array}$

Question 21

Find the derivative $\frac { d \vec { r } } { d s }$ $\vec { r } ( t ) = \left\langle t , \sin ( t ) , t ^ { 2 } \right\rangle , t = s ^ { 2 }$

Question 22

Find the derivative $\frac { d \vec { r } } { d s }$ $\vec { r } ( t ) = \left\langle \cos ( t ) , e ^ { t } , t \right\rangle , t = e ^ { s }$

Question 23

Find the derivative $\frac { d \vec { r } } { d s }$ $\vec { r } ( t ) = \left\langle t , t ^ { 3 } , \sqrt { t } \right\rangle , t = \left( s ^ { 2 } + 1 \right) ^ { 3 }$

A) $\vec { r } ^ { \prime } ( s ) = \left\langle 1,3 \left( s ^ { 2 } + 1 \right) ^ { 6 } , \frac { 1 } { 2 } \left( s ^ { 2 } + 1 \right) ^ { - \frac { 3 } { 2 } } \right\rangle$
B) $\vec { r } ^ { \prime } ( s ) = \left\langle 1,3 t ^ { 2 } , \frac { 1 } { 2 } t ^ { - \frac { 1 } { 2 } } \right\rangle$
C) $\vec { r } ^ { \prime } ( s ) = \left\langle 1,3 \left( s ^ { 2 } + 1 \right) ^ { 6 } , \frac { 1 } { 2 } \left( s ^ { 2 } + 1 \right) ^ { - \frac { 3 } { 2 } } > 6 s \left( s ^ { 2 } + 1 \right) ^ { 2 } \right.$
D) $\vec { r } ^ { \prime } ( s ) = \left\langle t , t ^ { 3 } , \sqrt { t } \right\rangle 6 s \left( s ^ { 2 } + 1 \right)$

Question 24

Find an equation for the line tangent to the given curve at the specified point. $$\vec { r } ( t ) = \left\langle t , t ^ { 3 } , t ^ { 5 } \right\rangle , ( 1,1,1 )$$

Question 25

Find an equation for the line tangent to the given curve at the specified point. $$\vec { r } ( t ) = \left\langle e ^ { t } , \cos ( t ) , t \right\rangle , ( 1,1,0 )$$

Question 26

Find an equation for the line tangent to the given curve at the specified point. $$\vec { r } ( t ) = \left\langle e ^ { t } , \cos ( t ) , t \right\rangle , ( 1,1,0 )$$

A) $\vec { T } ( t ) = \langle 1,1,0 \rangle + t \langle 1,0,1 \rangle$
B) $\vec { T } ( t ) = \left\langle 0 , e ^ { 2 } , 1 \right\rangle + t \left\langle 0 , e ^ { 2 } , 1 \right\rangle$
C) $\vec { T } ( t ) = \left\langle 0 , e ^ { 2 } , 1 \right\rangle + t \left\langle 1,2 e ^ { 2 } , \frac { \pi } { 2 } \right\rangle$
D) $\vec { T } ( t ) = \left\langle 1,2 e ^ { 2 } , 0 \right\rangle + t \left\langle 0 , e ^ { 2 } , 1 \right\rangle$

Question 27

Find the velocity for the given position vector, $\vec { r } ( t ) = \left\langle t , \sin ( t ) , t ^ { 2 } \right\rangle$

Question 28

Find the velocity for the given position vector, $\vec { r } ( t ) = \left\langle \ln ( t ) , e ^ { t ^ { 2 } } , \ln \left( e ^ { t } \right) \right\rangle$

Question 29

Find the velocity for the given position vector, $\vec { r } ( t ) = \left\langle 2 t , \left( 1 + t ^ { 2 } \right) ^ { 3 } , t \right\rangle$

A) $\vec { v } ( t ) = \left\langle 2 t , \left( 1 + t ^ { 2 } \right) ^ { 3 } , t \right\rangle$
B) $\vec { v } ( t ) = \left\langle 2,6 t \left( 1 + t ^ { 2 } \right) ^ { 2 } , 1 \right\rangle$
C) $\vec { v } ( t ) = \left\langle 2,3 \left( 1 + t ^ { 2 } \right) ^ { 2 } , 1 \right\rangle$
D) $\vec { v } ( t ) = \left\langle 2,3 t ^ { 2 } \left( 1 + t ^ { 2 } \right) ^ { 2 } , 1 \right\rangle$

Question 30

Find the acceleration for the given position vector, $\vec { r } ( t ) = \left\langle t , \sin ( t ) , t ^ { 2 } \right\rangle$

Question 31

Find the acceleration for the given position vector, $\vec { r } ( t ) = \left\langle \ln ( t ) , e ^ { t ^ { 2 } } , \ln \left( e ^ { t } \right) \right\rangle$

Question 32

Find the acceleration for the given position vector, $\vec { r } ( t ) = \left\langle 2 t , \left( 1 + t ^ { 2 } \right) ^ { 3 } , t \right\rangle$

A) $\vec { a } ( t ) = \left\langle 2,6 t \left( 1 + t ^ { 2 } \right) ^ { 2 } , t \right\rangle$
B) $\vec { a } ( t ) = \left\langle 0,6 \left( 1 + t ^ { 2 } \right) ^ { 2 } + 24 t ^ { 2 } \left( 1 + t ^ { 2 } \right) , 0 \right\rangle$
C) $\vec { a } ( t ) = \left\langle 0,2 t \left( 1 + t ^ { 2 } \right) , 0 \right\rangle$
D) $\vec { a } ( t ) = \left\langle 0,6 t ^ { 2 } ( 1 + 2 t ) ^ { 2 } , 0 \right\rangle$

Question 33

Evaluate the integral. $\int \left\langle t , e ^ { t } , \sin ( t ) \right\rangle d t$

Question 34

Evaluate the integral. $\int _ { 1 } ^ { 2 } < \frac { 1 } { t } , 3 t ^ { 2 } , t e ^ { t ^ { 2 } } > d t$

Question 35

Evaluate the integral. $\int < t , \cot ( t ) , e ^ { t } > d t$

A) $\left. < \frac { t ^ { 2 } } { 2 } , \sin ( t ) , e ^ { t } \right\rangle$
B) $< \frac { t ^ { 2 } } { 2 } , \sin ( t ) , e ^ { t } > + \vec { c }$
C) $\left\langle \frac { t ^ { 2 } } { 2 } , - \sin ( t ) , e ^ { t } \right\rangle$
D) $< \frac { t ^ { 2 } } { 2 } , - \sin ( t ) , e ^ { t } > + \vec { c }$

Question 36

Given the velocity vector $\vec { v } ( t ) = \left\langle t , - \sin ( t ) , t ^ { 2 } \right\rangle$ and $\vec { r } ( 0 ) = \langle 1,1,1 \rangle$ find $\vec { r } ( t )$

Question 37

Given the velocity vector $\vec { v } ( t ) = \left\langle \frac { 1 } { t + 1 } , e ^ { t } , t \right\rangle$ and $\vec { r } ( 0 ) = \langle 1,0,2 \rangle$ , find $\vec { r } ( t )$

Question 38

Given the velocity vector $\vec { v } ( t ) = \left\langle 1 , t , t ^ { 2 } \right\rangle$ and $\vec { r } ( 0 ) = \langle 1,2,3 \rangle$ , find $\vec { r } ( t )$

A) $\vec { r } ( t ) = \left\langle t , t ^ { 2 } , t ^ { 3 } \right\rangle + \langle 1,2,3 \rangle$
B) $\vec { r } ( t ) = \left\langle t , \frac { t ^ { 2 } } { 2 } , \frac { t ^ { 3 } } { 3 } \right\rangle + \left\langle 0 , \frac { 3 } { 2 } , \frac { 8 } { 3 } \right\rangle$
C) $\vec { r } ( t ) = \left\langle 1 , t ^ { 2 } , t ^ { 3 } \right\rangle + \langle 0,2,3 \rangle$
D) $\vec { r } ( t ) = \left\langle t , \frac { t ^ { 2 } } { 2 } , \frac { t ^ { 3 } } { 3 } \right\rangle + \langle 1,2,3 \rangle$

Question 39

Given the acceleration vector $\vec { a } ( t ) = \left\langle 2 t , \cos ( t ) , 12 t ^ { 2 } \right\rangle$ and initial velocity and position $\vec { v } ( 0 ) = \langle 1,2,1 \rangle$ and $\vec { r } ( 0 ) = \langle 1,1,1 \rangle$ , find $\vec { r } ( t )$

Question 40

Find the unit tangent for $\vec { r } ( t ) = \left\langle t , e ^ { t } \right\rangle$

Question 41

Find the unit tangent for $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle$

Question 42

Find the unit tangent for $r ( t ) = \left\langle e ^ { t } , t \right\rangle$

A) $T ( t ) = \left\langle 1 , e ^ { t } \right\rangle$
B) $\vec { T } ( t ) = \frac { \left\langle 1 , e ^ { t } \right\rangle } { \sqrt { 1 + e ^ { t } } }$
C) $\vec { T } ( t ) = \frac { \left\langle 1 , e ^ { t } \right\rangle } { \sqrt { 1 + e ^ { 2 t } } }$
D) $\vec { T } ( t ) = \frac { \left\langle 1 , e ^ { t } \right\rangle } { \sqrt { t ^ { 2 } + e ^ { 2 t } } }$

Question 43

Find the unit tangent for $\vec { r } ( t ) = \left\langle 1 , t ^ { 2 } , t \right\rangle$

Question 44

Find the unit tangent for $\vec { r } ( t ) = \langle \cos ( 2 t ) , \sin ( 2 t ) , 2 \rangle$

Question 45

Find the unit tangent for $\vec { r } ( t ) = \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle$

A) $\vec { T } ( t ) = \frac { \left\langle - 3 \sin ( 3 t ) , 3 \cos ( 3 t ) , e ^ { t } \right\rangle } { \sqrt { 9 + e ^ { 2 t } } }$
B) $\vec { T } ( t ) = \frac { \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle } { \sqrt { 9 + e ^ { 2 t } } }$
C) $\vec { T } ( t ) = \frac { \left\langle 3 \sin ( 3 t ) , - 3 \cos ( 3 t ) , e ^ { t } \right\rangle } { \sqrt { 9 + e ^ { 2 t } } }$
D) $\vec { T } ( t ) = \left\langle \cos ( 3 t ) , \sin ( 3 t ) , e ^ { t } \right\rangle$

Question 46

Find the principle normal unit vector for $\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$

Question 47

Find the principle binormal unit vector for $\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$

Question 48

Find the osculating plane for for $\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$ for $t = 0$

Question 49

Find the principle normal unit vector for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$

Question 50

Find the principle binormal unit vector for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$

Question 51

Find the osculating plane for $\vec { r } ( t ) = \langle \cos ( 3 t ) , \sin ( 3 t ) , 3 \rangle$ for $t = 1$

Question 52

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 2 \sin ( 5 t ) , 2 \cos ( 5 t ) \rangle$ over $[ 2,4 ]$

Question 53

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) \rangle$ over $[ 4,7 ]$

Question 54

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + 7 \cos ( 3 t ) , 1 + 7 \sin ( 3 t ) \rangle$ over $[ 2,3 ]$

Question 55

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 3 + 4 \cos ( 2 t ) , 4 + 2 \sin ( 4 t ) \rangle$ over $[ 2,4 ]$

A) 4
B) 8
C) 16
D) 32

Question 56

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 2 \rangle$ over $[ 1,5 ]$

Question 57

Find the arc length of the curve defined by $\vec { r } ( t ) = \left\langle e ^ { t } \cos ( t ) , e ^ { t } , e ^ { t } \sin ( t ) \right\rangle$ over $[ \ln ( 2 ) , \ln ( 5 ) ]$

Question 58

Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 1 + \cos ( 3 t ) , 1 + \sin ( 3 t ) , 5 \rangle$ over $[ 4,7 ]$

A) 4
B) 7
C) 9
D) 11

Question 59

Give an arc length parameterization of $\vec { r } ( t ) = \langle \sin ( 2 t ) , \cos ( 2 t ) \rangle$

Question 60

Give an arc length parameterization of $\vec { r } ( t ) = \langle 1 + \sin ( 2 t ) , 2 + \cos ( 2 t ) , 3 \rangle$

A) $\vec { r } ( s ) = \langle \cos ( s ) , \sin ( s ) , 3 \rangle$
B) $\vec { r } ( s ) = \langle 1 + \cos ( 2 s ) , 2 + \sin ( 2 s ) , 3 \rangle$
C) $\vec { r } ( s ) = \langle 1 + \cos ( s ) , 2 + \sin ( s ) , 3 \rangle$
D) $\vec { r } ( s ) = \langle \cos ( 2 s ) , \sin ( 2 s ) , 0 \rangle$

Question 61

Give an arc length parameterization of $\vec { r } ( t ) = \left\langle e ^ { t } , 2 e ^ { t } \right\rangle$

Question 62

Find the curvature of the given function at the indicated value of $X$ : $f ( x ) = x ^ { 2 } , x = 1$

Question 63

Find the curvature of the given function at the indicated value of $X$ : $f ( x ) = e ^ { x } , x = 0$

Question 64

Find the curvature of the given function at the indicated value of $X$ : $f ( x ) = \cos ( x ) , x = 0$

A) $\frac { 1 } { 2 ^ { \frac { 3 } { 2 } } }$
B) 0
C) 1
D) Undefined

Question 65

Find the curvature of the vector valued function: $$\vec { r } ( t ) = \left\langle 1 , t ^ { 2 } , t ^ { 2 } \right\rangle$$

Question 66

Find the curvature of the vector valued function: $$\vec { r } ( t ) = \langle \sin ( t ) , \cos ( t ) , t \rangle$$

Question 67

Find the curvature of the vector valued function: $$\vec { r } ( t ) = \left\langle e ^ { t } , e ^ { 2 t } , e ^ { 3 t } \right\rangle$$

A) $\kappa ( t ) = \frac { 1 } { 2 }$
B) $\kappa ( t ) = \sqrt { 36 e ^ { 10 t } + 36 e ^ { 8 t } + 4 e ^ { 6 t } }$
C) $\kappa ( t ) = \frac { 2 \sqrt { 9 e ^ { 4 t } + 9 e ^ { 2 t } + 1 } } { \left( 1 + 4 e ^ { 2 t } + 9 e ^ { 4 t } \right) ^ { \frac { 3 } { 2 } } }$
D) $\kappa ( t ) = \sqrt { 9 e ^ { 4 t } + 9 e ^ { 4 t } + 1 }$

Question 68

For $\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) \right\rangle$ find the displacement vector as $t$ goes from $t = - 1$ to $t = 1$

Question 69

Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) \right\rangle$ as $t$ goes from $t = - 1$ to $t = 1$

Question 70

For $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$

A) $\langle - 2,0 \rangle$
B) $\langle - 1,0 \rangle$
C) $\langle 1,0 \rangle$
D) $\langle 1,1 \rangle$

Question 71

Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$

A) $\pi \sqrt { 3 }$
B) $3 \pi$
C) $2$
D) $2 \pi$

Question 72

For $\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) , t \right\rangle$ find the displacement vector as $t$ goes from $t = - 1$ to $t = 1$

Question 73

Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \left\langle \sin \left( \frac { \pi } { 2 } t \right) , \cos \left( \frac { \pi } { 2 } t \right) , t \right\rangle$ as $t$ goes from $t = - 1$ to $t = 1$

Question 74

For $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) , t \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$

A) $\langle - 2,0 , \pi \rangle$
B) $\langle - 1,0 , \pi \rangle$
C) $\langle 1,0 , \pi \rangle$
D) $\langle 1,1 , \pi \rangle$

Question 75

Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) , t \rangle$ as $t$ goes from $t = 0$ to $t = \pi$

A) $3 \pi$
B) $\pi \sqrt { 10 }$
C) $\pi \sqrt { 3 }$
D) $10 \pi$

Question 76

Find the tangential and normal components for the position function $\vec { r } ( t ) = \left\langle t , t ^ { 3 } \right\rangle$

Question 77

Find the tangential and normal components for the position function $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle$

Question 78

Find the tangential and normal components for the position function $\vec { r } ( t ) = \left\langle e ^ { t } , e ^ { - t } \right\rangle$

A) $\begin{array} { l } a _ { T } = e ^ { 2 t } - e ^ { - 2 t } \\a _ { N } = 2\end{array}$
B) $\begin{array} { l } a _ { T } = \frac { 2 } { \sqrt { e ^ { 2 t } + e ^ { - 2 t } } } \\a _ { N } = \frac { e ^ { 2 t } - e ^ { - 2 t } } { \sqrt { e ^ { 2 t } + e ^ { - 2 t } } }\end{array}$
C) $\begin{array} { l } a _ { T } = \frac { 2 } { \sqrt { e ^ { 2 t } + e ^ { - 2 t } } } \\a _ { N } = \frac { e ^ { 2 t } + e ^ { - 2 t } } { \sqrt { e ^ { 2 t } + e ^ { - 2 t } } }\end{array}$
D) $\begin{array} { l } a _ { T } = \frac { e ^ { 2 t } - e ^ { - 2 t } } { \sqrt { e ^ { 2 t } + e ^ { - 2 t } } } \\a _ { N } = \frac { 2 } { \sqrt { e ^ { 2 t } + e ^ { - 2 t } } }\end{array}$

Question 79

Find the tangential and normal components for the position function $\vec { r } ( t ) = \left\langle t , e ^ { t } , t ^ { 2 } \right\rangle$

Question 80

Find the tangential and normal components of acceleration for the position function $\vec { r } ( t ) = \langle \cos ( 3 t ) , t , \sin ( 3 t ) \rangle$