Question 1

Find the curvature of the curve r(t).
-$$r ( t ) = ( 2 t + 6 ) i - 4 j + \left( 8 - 1 t ^ { 2 } \right) \mathbf { k }$$

Accepted Answer

A)  $k = \frac { 1 } { 2 \sqrt { 1 + t ^ { 2 } } }$ 
B)  $\kappa = 2 \sqrt { 1 + t ^ { 2 } }$ 
C)  $\kappa = 2 \left( 1 + t ^ { 2 } \right) ^ { 3 / 2 }$ 
D)  $\kappa = \frac { 1 } { 2 \left( 1 + t ^ { 2 } \right) ^ { 3 / 2 } }$ 
A)  $k = \frac { 1 } { 2 \sqrt { 1 + t ^ { 2 } } }$ 
B)  $\kappa = 2 \sqrt { 1 + t ^ { 2 } }$ 
C)  $\kappa = 2 \left( 1 + t ^ { 2 } \right) ^ { 3 / 2 }$ 
D)  $\kappa = \frac { 1 } { 2 \left( 1 + t ^ { 2 } \right) ^ { 3 / 2 } }$

Question 2

Solve the problem.
-The orbit of a satellite had a semimajor axis of a $= 8872 \mathrm {~km}$. Calculate the period of the satellite. (Earth's mass $= 5.975 \times 10 ^ { 24 } \mathrm {~kg}$ and $\mathrm { G } = 6.6720 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } \mathrm {~kg} - 2$ ).

Accepted Answer

A)  78.2 min 
B)  1.98 hr 
C)  12 hr 
D)  138.6 min 
A)  78.2 min 
B)  1.98 hr 
C)  12 hr 
D)  138.6 min

Question 3

Find the curvature of the space curve.
-r(t) = ti + (sinh t)j + (cosh t)k

Accepted Answer

A)  $\kappa = \sinh ^ { 2 } t$ 
B)  $\kappa = \cosh ^ { 2 } t$ 
C)  $\kappa = \frac { \tanh ^ { 2 } \mathrm { t } } { 2 }$ 
D)  $\kappa = \frac { \operatorname { sech } ^ { 2 } \mathrm { t } } { 2 }$ 
A)  $\kappa = \sinh ^ { 2 } t$ 
B)  $\kappa = \cosh ^ { 2 } t$ 
C)  $\kappa = \frac { \tanh ^ { 2 } \mathrm { t } } { 2 }$ 
D)  $\kappa = \frac { \operatorname { sech } ^ { 2 } \mathrm { t } } { 2 }$

Question 4

Find the curvature of the space curve.
-r(t) = -3i + (t + 4)j +(ln(cos t) + 9)k

Accepted Answer

A)  $\kappa = \csc t$ 
B)  $\kappa = \sin t$ 
C)  $\kappa = \sec t$ 
D)  $\kappa = \cos t$ 
A)  $\kappa = \csc t$ 
B)  $\kappa = \sin t$ 
C)  $\kappa = \sec t$ 
D)  $\kappa = \cos t$

Question 5

Provide an appropriate response.
-The position of a particle is given by r(t) = sin 8t i + cos 3t j. Find the velocity vector for the particle.

Accepted Answer

A)  cos 8t i - sin 3t j 
B)  8 i + 3 j 
C)  8 cos 8t i - 3 sin 3t j 
D)  8 sin 8t i + 3 cos 3t j 
A)  cos 8t i - sin 3t j 
B)  8 i + 3 j 
C)  8 cos 8t i - 3 sin 3t j 
D)  8 sin 8t i + 3 cos 3t j

Question 6

Evaluate the integral.
-$$\int _ { 0 } ^ { 4 } \left[ - \mathbf { i } - \frac { 2 t } { \left( 7 + t ^ { 2 } \right) ^ { 2 } } j + \frac { 15 } { 2 } \sqrt { t k } \right] d t$$

Accepted Answer

A)  $+ 4 i + \frac { 16 } { 161 } j + 40 k$ 
B)  $- 4 i - \frac { 16 } { 161 } j - 40 k$ 
C)  $- 4 i - \frac { 16 } { 161 } j + 40 k$ 
D)  $- 4 i + \frac { 4 } { 77 } j + 40 k$ 
A)  $+ 4 i + \frac { 16 } { 161 } j + 40 k$ 
B)  $- 4 i - \frac { 16 } { 161 } j - 40 k$ 
C)  $- 4 i - \frac { 16 } { 161 } j + 40 k$ 
D)  $- 4 i + \frac { 4 } { 77 } j + 40 k$

Question 7

The position vector of a particle is r(t). Find the requested vector.
-$\text { The velocity at } t = 3 \text { for } \mathbf { r } ( t ) = \left( 7 t ^ { 2 } + 4 t + 3 \right) \mathbf { i } - 3 t ^ { 3 } \mathbf { j } + \left( 8 - t ^ { 2 } \right) \mathbf { k }$

Accepted Answer

A)  v(3) = 46i + 81j + 6k 
B)  v(3) = 25i - 27j - 3k 
C)  v(3) = 46i - 81j - 6k 
D)  v(3) = 38i - 81j - 6k 
A)  v(3) = 46i + 81j + 6k 
B)  v(3) = 25i - 27j - 3k 
C)  v(3) = 46i - 81j - 6k 
D)  v(3) = 38i - 81j - 6k

Question 8

Find the unit tangent vector of the given curve.
-r(t) = (10 - 2t)i + (2t - 3)j + (10 + t)k

Accepted Answer

A)  $T = \frac { 2 } { 3 } \mathbf { i } - \frac { 2 } { 3 } j - \frac { 1 } { 3 } \mathbf { k }$ 
B)  $\mathrm { T } = - \frac { 2 } { 3 } \mathrm { i } + \frac { 2 } { 3 } \mathrm { j } + \frac { 1 } { 3 } \mathbf { k }$ 
C)  $T = \frac { 2 } { 9 } \mathbf { i } - \frac { 2 } { 9 } \mathbf { j } - \frac { 1 } { 9 } \mathbf { k }$ 
D)  $T = - \frac { 2 } { 9 } i + \frac { 2 } { 9 } j + \frac { 1 } { 9 } k$ 
A)  $T = \frac { 2 } { 3 } \mathbf { i } - \frac { 2 } { 3 } j - \frac { 1 } { 3 } \mathbf { k }$ 
B)  $\mathrm { T } = - \frac { 2 } { 3 } \mathrm { i } + \frac { 2 } { 3 } \mathrm { j } + \frac { 1 } { 3 } \mathbf { k }$ 
C)  $T = \frac { 2 } { 9 } \mathbf { i } - \frac { 2 } { 9 } \mathbf { j } - \frac { 1 } { 9 } \mathbf { k }$ 
D)  $T = - \frac { 2 } { 9 } i + \frac { 2 } { 9 } j + \frac { 1 } { 9 } k$

Question 9

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface
-An ideal projectile is launched from the origin at an angle of α radians to the horizontal and an initial speed of 175 ft/sec. Find the position function r(t) for this projectile.

Accepted Answer

A)  $\mathbf { r } ( \mathrm { t } ) = ( 175 \mathrm { t } \sin \alpha ) \mathbf { i } + \left( 175 \mathrm { t } \cos \alpha - 16 \mathrm { t } ^ { 2 } \right) \mathbf { j }$ 
B)  $\mathbf { r } ( \mathrm { t } ) = ( 175 \mathrm { t } \cos \alpha ) \mathbf { i } + \left( 175 \mathrm { t } \sin \alpha - 16 \mathrm { t } ^ { 2 } \right) \mathbf { j }$ 
C)  $\mathbf { r } ( \mathrm { t } ) = \left( 175 \mathrm { t } \sin \alpha - 16 \mathrm { t } ^ { 2 } \right) \mathbf { i } + ( 175 \mathrm { t } \cos \alpha ) \mathbf { j }$ 
D)  $\mathbf { r } ( \mathrm { t } ) = \left( 175 \mathrm { t } \cos \alpha - 32 \mathrm { t } ^ { 2 } \right) \mathbf { i } + ( 175 \mathrm { t } \sin \alpha ) \mathbf { j }$ 
A)  $\mathbf { r } ( \mathrm { t } ) = ( 175 \mathrm { t } \sin \alpha ) \mathbf { i } + \left( 175 \mathrm { t } \cos \alpha - 16 \mathrm { t } ^ { 2 } \right) \mathbf { j }$ 
B)  $\mathbf { r } ( \mathrm { t } ) = ( 175 \mathrm { t } \cos \alpha ) \mathbf { i } + \left( 175 \mathrm { t } \sin \alpha - 16 \mathrm { t } ^ { 2 } \right) \mathbf { j }$ 
C)  $\mathbf { r } ( \mathrm { t } ) = \left( 175 \mathrm { t } \sin \alpha - 16 \mathrm { t } ^ { 2 } \right) \mathbf { i } + ( 175 \mathrm { t } \cos \alpha ) \mathbf { j }$ 
D)  $\mathbf { r } ( \mathrm { t } ) = \left( 175 \mathrm { t } \cos \alpha - 32 \mathrm { t } ^ { 2 } \right) \mathbf { i } + ( 175 \mathrm { t } \sin \alpha ) \mathbf { j }$

Question 10

Find the length of the indicated portion of the curve.
-$\mathbf { r } ( t ) = ( 4 \cos t ) \mathbf { i } + ( 4 \sin t ) \mathbf { j } + 5 t \mathbf { k } , 0 \leq t \leq \pi / 2$

Accepted Answer

A)  $\frac { 3 t } { 2 } \pi$ 
B)  $\frac { \sqrt { 41 } t } { 2 } \pi$ 
C)  $\frac { \sqrt { 57 } \mathrm { t } } { 2 } \pi$ 
D)  $\frac { \sqrt { 41 } t } { 4 } \pi$ 
A)  $\frac { 3 t } { 2 } \pi$ 
B)  $\frac { \sqrt { 41 } t } { 2 } \pi$ 
C)  $\frac { \sqrt { 57 } \mathrm { t } } { 2 } \pi$ 
D)  $\frac { \sqrt { 41 } t } { 4 } \pi$

Question 11

Find the curvature of the space curve.
-$$r ( t ) = - 5 i + ( 4 + 2 t ) j + \left( t ^ { 2 } + 8 \right) \mathbf { k }$$

Accepted Answer

A)  $\kappa = \frac { 1 } { 2 \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$ 
B)  $\kappa = - \frac { 1 } { 2 \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$ 
C)  $\kappa = \frac { 1 } { 2 \sqrt { t ^ { 2 } + 1 } }$ 
D)  $\kappa = \frac { 1 } { \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$ 
A)  $\kappa = \frac { 1 } { 2 \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$ 
B)  $\kappa = - \frac { 1 } { 2 \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$ 
C)  $\kappa = \frac { 1 } { 2 \sqrt { t ^ { 2 } + 1 } }$ 
D)  $\kappa = \frac { 1 } { \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$

Question 12

Solve the problem.
-The orbit of a satellite has a period of $T = 1436.8$ minutes. Calculate the semimajor axis of the satellite. (Earth's mass $= 5.975 \times 10 ^ { 24 } \mathrm {~kg}$ and $\mathrm { G } = 6.6720 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 }$ ).

Accepted Answer

A)  $2.751 \times 10 ^ { 6 } \mathrm {~km}$ 
B)  $2751 \mathrm {~km}$ 
C)  $4.217 \times 10 ^ { 7 } \mathrm {~km}$ 
D)  $42,170 \mathrm {~km}$ 
A)  $2.751 \times 10 ^ { 6 } \mathrm {~km}$ 
B)  $2751 \mathrm {~km}$ 
C)  $4.217 \times 10 ^ { 7 } \mathrm {~km}$ 
D)  $42,170 \mathrm {~km}$

Question 13

Evaluate the integral.
-$$\int _ { 0 } ^ { 1 } \left[ \left( 27 t ^ { 2 } - 2 \right) \mathbf { i } + \frac { 20 t } { t ^ { 2 } + 1 } \mathbf { j } - \frac { t } { \sqrt { t ^ { 2 } + 1 } } \mathbf { k } \right] d t$$

Accepted Answer

A)  $11 \mathbf { i } + 10 \ln 2 \mathbf { j } + ( 1 + \sqrt { 2 } ) \mathbf { k }$ 
B)  $11 \mathbf { i } + 10 \ln 2 \mathbf { j } + ( 1 - \sqrt { 2 } ) \mathbf { k }$ 
C)  $7 \mathbf { i } + 10 \ln 2 \mathbf { j } + ( 1 - \sqrt { 2 } ) \mathbf { k }$ 
D)  $11 \mathbf { i } + 10 \ln 2 \mathbf { j } + \frac { 1 - \sqrt { 2 } } { 2 } \mathbf { k }$ 
A)  $11 \mathbf { i } + 10 \ln 2 \mathbf { j } + ( 1 + \sqrt { 2 } ) \mathbf { k }$ 
B)  $11 \mathbf { i } + 10 \ln 2 \mathbf { j } + ( 1 - \sqrt { 2 } ) \mathbf { k }$ 
C)  $7 \mathbf { i } + 10 \ln 2 \mathbf { j } + ( 1 - \sqrt { 2 } ) \mathbf { k }$ 
D)  $11 \mathbf { i } + 10 \ln 2 \mathbf { j } + \frac { 1 - \sqrt { 2 } } { 2 } \mathbf { k }$

Question 14

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0.
-$$\mathbf { r } ( \mathrm { t } ) = \sin ^ { - 1 } ( 7 \mathrm { t } ) \mathbf { i } + \ln \left( 8 \mathrm { t } ^ { 2 } + 1 \right) \mathbf { j } + \sqrt { 7 \mathrm { t } ^ { 2 } + 1 } \mathbf { k }$$

Accepted Answer

A)  $\frac { \pi } { 4 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  0 
A)  $\frac { \pi } { 4 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  0

Question 15

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
-Find the acceleration vector.
$\mathbf { r } ( t ) = ( 8 \ln ( 9 \mathrm { t } ) ) \mathbf { i } + \left( 3 \mathrm { t } ^ { 3 } \right) \mathbf { j }$

Accepted Answer

A)  $a = - \frac { 8 } { t ^ { 2 } } i + 18 t j$ 
B)  $a = \frac { 8 } { t ^ { 2 } } t ^ { - 2 } i + 18 t j$ 
C)  $a = \frac { 8 } { t } i + 9 t j$ 
D)  $\mathbf { a } = - \frac { 8 } { 9 } \mathrm { t } ^ { - 2 } \mathbf { i } + 18 \mathrm { t }$ 
A)  $a = - \frac { 8 } { t ^ { 2 } } i + 18 t j$ 
B)  $a = \frac { 8 } { t ^ { 2 } } t ^ { - 2 } i + 18 t j$ 
C)  $a = \frac { 8 } { t } i + 9 t j$ 
D)  $\mathbf { a } = - \frac { 8 } { 9 } \mathrm { t } ^ { - 2 } \mathbf { i } + 18 \mathrm { t }$

Question 16

Find T, N, and B for the given space curve.
-$$r ( t ) = ( \ln ( \cos t ) + 1 ) \mathbf { i } + 1 \mathbf { j } + ( 10 + t ) \mathbf { k } , - \pi / 2 < t < \pi / 2$$

Accepted Answer

A)  T = (sin t)i + (cos t)k; N = (cos t)i - (sin t)k; B = -j 
B)  T = (-sin t)i + (cos t)k; N = (-cos t)i - (sin t)k; B = -j 
C)  T = (-sin t)i + (cos t)k; N = (-cos t)i - (sin t)k; B = j 
D)  T = (sin t)i + (cos t)k; N = (cos t)i - (sin t)k; B = j 
A)  T = (sin t)i + (cos t)k; N = (cos t)i - (sin t)k; B = -j 
B)  T = (-sin t)i + (cos t)k; N = (-cos t)i - (sin t)k; B = -j 
C)  T = (-sin t)i + (cos t)k; N = (-cos t)i - (sin t)k; B = j 
D)  T = (sin t)i + (cos t)k; N = (cos t)i - (sin t)k; B = j

Question 17

$\text { Find the velocity vector in terms of } u _ { r } \text { and } u _ { \theta } \text {. }$
-$\mathrm { r } = \mathrm { a } ( 2 - \cos \theta )$ and $\frac { \mathrm { d } \theta } { \mathrm { dt } } = 7$

Accepted Answer

A)  $\mathbf { v } = ( 7 a \sin \theta ) \mathbf { u } _ { \mathrm { r } } + 7 \mathrm { a } ( 2 - \cos \theta ) \mathbf { u } _ { \theta }$ 
B)  $\mathbf { v } = ( 7 a \cos \theta ) \mathbf { u } _ { \mathrm { r } } + 7 \mathrm { a } ( 2 - \sin \theta ) \mathbf { u } _ { \theta }$ 
C)  $\mathbf { v } = 7 a ( 2 - \cos \theta ) \mathbf { u } _ { \mathbf { r } } + ( 7 \mathrm { a } \sin \theta ) \mathbf { u } _ { \theta }$ 
D)  $\mathbf { v } = ( 2 a \sin \theta ) \mathbf { u } _ { \mathrm { r } } + 2 \mathrm { a } ( 7 - \cos \theta ) \mathbf { u } _ { \theta }$ 
A)  $\mathbf { v } = ( 7 a \sin \theta ) \mathbf { u } _ { \mathrm { r } } + 7 \mathrm { a } ( 2 - \cos \theta ) \mathbf { u } _ { \theta }$ 
B)  $\mathbf { v } = ( 7 a \cos \theta ) \mathbf { u } _ { \mathrm { r } } + 7 \mathrm { a } ( 2 - \sin \theta ) \mathbf { u } _ { \theta }$ 
C)  $\mathbf { v } = 7 a ( 2 - \cos \theta ) \mathbf { u } _ { \mathbf { r } } + ( 7 \mathrm { a } \sin \theta ) \mathbf { u } _ { \theta }$ 
D)  $\mathbf { v } = ( 2 a \sin \theta ) \mathbf { u } _ { \mathrm { r } } + 2 \mathrm { a } ( 7 - \cos \theta ) \mathbf { u } _ { \theta }$

Question 18

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0.
-$\mathbf{r}(\mathrm{t})=\mathrm{e}^{6 \mathrm{t}_{\mathrm{i}}+\left(4+\mathrm{e}^{-6 \mathrm{t}}\right) \mathrm{j}+(7 \cos 8 \mathrm{t}) \mathbf{k}}$

Accepted Answer

A)  0 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 4 }$ 
A)  0 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 4 }$

Question 19

Find the length of the indicated portion of the curve.
-$$r ( t ) = ( 1 + 2 t ) \mathbf { i } + ( 1 + 4 t ) \mathbf { j } + ( 5 - 5 t ) \mathbf { k } , - 1 \leq t \leq 0$$

Accepted Answer

A)  $2 \sqrt { 5 }$ 
B)  $\sqrt { 29 }$ 
C)  $3 \sqrt { 5 }$ 
D)  $\sqrt { 41 }$ 
A)  $2 \sqrt { 5 }$ 
B)  $\sqrt { 29 }$ 
C)  $3 \sqrt { 5 }$ 
D)  $\sqrt { 41 }$

Question 20

Find the curvature of the space curve.
-r(t) = (t + 4)i + 9j + (ln(sec t) + 3)k

Accepted Answer

A)  $\kappa = \sin t$ 
B)  $\kappa = \csc t$ 
C)  $\kappa = \cos t$ 
D)  $\kappa = \sec t$ 
A)  $\kappa = \sin t$ 
B)  $\kappa = \csc t$ 
C)  $\kappa = \cos t$ 
D)  $\kappa = \sec t$

Find the curvature of the curve r(t). - $r ( t ) = ( 2 t + 6 ) i - 4 j + \left( 8 - 1 t ^ { 2 } \right) \mathbf { k }$

Solve the problem. -The orbit of a satellite had a semimajor axis of a $= 8872 \mathrm {~km}$ . Calculate the period of the satellite. (Earth's mass $= 5.975 \times 10 ^ { 24 } \mathrm {~kg}$ and $\mathrm { G } = 6.6720 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } \mathrm {~kg} - 2$ ).

Find the curvature of the space curve. -r(t) = ti + (sinh t)j + (cosh t)k

Find the curvature of the space curve. -r(t) = -3i + (t + 4)j +(ln(cos t) + 9)k

Provide an appropriate response. -The position of a particle is given by r(t) = sin 8t i + cos 3t j. Find the velocity vector for the particle.

Evaluate the integral. - $\int _ { 0 } ^ { 4 } \left[ - \mathbf { i } - \frac { 2 t } { \left( 7 + t ^ { 2 } \right) ^ { 2 } } j + \frac { 15 } { 2 } \sqrt { t k } \right] d t$

The position vector of a particle is r(t). Find the requested vector. - $\text { The velocity at } t = 3 \text { for } \mathbf { r } ( t ) = \left( 7 t ^ { 2 } + 4 t + 3 \right) \mathbf { i } - 3 t ^ { 3 } \mathbf { j } + \left( 8 - t ^ { 2 } \right) \mathbf { k }$

Find the unit tangent vector of the given curve. -r(t) = (10 - 2t)i + (2t - 3)j + (10 + t)k

Find the length of the indicated portion of the curve. - $\mathbf { r } ( t ) = ( 4 \cos t ) \mathbf { i } + ( 4 \sin t ) \mathbf { j } + 5 t \mathbf { k } , 0 \leq t \leq \pi / 2$

Find the curvature of the space curve. - $r ( t ) = - 5 i + ( 4 + 2 t ) j + \left( t ^ { 2 } + 8 \right) \mathbf { k }$

Solve the problem. -The orbit of a satellite has a period of $T = 1436.8$ minutes. Calculate the semimajor axis of the satellite. (Earth's mass $= 5.975 \times 10 ^ { 24 } \mathrm {~kg}$ and $\mathrm { G } = 6.6720 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 }$ ).

Evaluate the integral. - $\int _ { 0 } ^ { 1 } \left[ \left( 27 t ^ { 2 } - 2 \right) \mathbf { i } + \frac { 20 t } { t ^ { 2 } + 1 } \mathbf { j } - \frac { t } { \sqrt { t ^ { 2 } + 1 } } \mathbf { k } \right] d t$

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. -Find the acceleration vector. $\mathbf { r } ( t ) = ( 8 \ln ( 9 \mathrm { t } ) ) \mathbf { i } + \left( 3 \mathrm { t } ^ { 3 } \right) \mathbf { j }$

Find T, N, and B for the given space curve. - $r ( t ) = ( \ln ( \cos t ) + 1 ) \mathbf { i } + 1 \mathbf { j } + ( 10 + t ) \mathbf { k } , - \pi / 2 < t < \pi / 2$

$\text { Find the velocity vector in terms of } u _ { r } \text { and } u _ { \theta } \text {. }$ - $\mathrm { r } = \mathrm { a } ( 2 - \cos \theta )$ and $\frac { \mathrm { d } \theta } { \mathrm { dt } } = 7$

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. - $\mathbf{r}(\mathrm{t})=\mathrm{e}^{6 \mathrm{t}_{\mathrm{i}}+\left(4+\mathrm{e}^{-6 \mathrm{t}}\right) \mathrm{j}+(7 \cos 8 \mathrm{t}) \mathbf{k}}$

Find the length of the indicated portion of the curve. - $r ( t ) = ( 1 + 2 t ) \mathbf { i } + ( 1 + 4 t ) \mathbf { j } + ( 5 - 5 t ) \mathbf { k } , - 1 \leq t \leq 0$

Find the curvature of the space curve. -r(t) = (t + 4)i + 9j + (ln(sec t) + 3)k

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Exam 14: Vector-Valued Functions and Motion in Space

Find the curvature of the curve r(t). - r(t)=(2t+6)i−4j+(8−1t2)kr ( t ) = ( 2 t + 6 ) i - 4 j + \left( 8 - 1 t ^ { 2 } \right) \mathbf { k }r(t)=(2t+6)i−4j+(8−1t2)k