Question 1

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } ( \cos 8 t , \sin 8 t , 8 )$

Accepted Answer

A)  $a _ { \mathbf { T } } = \sqrt { 65 } e ^ { t }$ , $a _ { \mathrm { N } } = 8 \sqrt { 65 } e ^ { t }$ 
B)  $a _ { \mathbf { T } } = 8 \sqrt { 65 } e ^ { t }$ , $a _ { \mathrm { N } } = \sqrt { 65 } e ^ { t }$ 
C)  $a _ { \mathbf { T } } = 65 e ^ { t }$ , $a _ { \mathrm { N } } ( t ) = 8 e ^ { t }$ 
D)  $a _ { \mathbf { T } } = 8 e ^ { t }$ , $a _ { \mathrm { N } } = 65 e ^ { t }$ 
A)  $a _ { \mathbf { T } } = \sqrt { 65 } e ^ { t }$ , $a _ { \mathrm { N } } = 8 \sqrt { 65 } e ^ { t }$ 
B)  $a _ { \mathbf { T } } = 8 \sqrt { 65 } e ^ { t }$ , $a _ { \mathrm { N } } = \sqrt { 65 } e ^ { t }$ 
C)  $a _ { \mathbf { T } } = 65 e ^ { t }$ , $a _ { \mathrm { N } } ( t ) = 8 e ^ { t }$ 
D)  $a _ { \mathbf { T } } = 8 e ^ { t }$ , $a _ { \mathrm { N } } = 65 e ^ { t }$

Question 2

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = \sin t \mathbf { i } + 6 \cos t \mathbf { j }$ for $t = \frac { \pi } { 4 }$ Sketch the path of the object and its velocity and acceleration vectors.

Accepted Answer

\[\begin{array} { l } 
\mathrm { v } \le

Question 3

Find the unit tangent and unit normal vectors $\mathbf { T } ( t )$ and $\mathbf { N } ( t )$ for the curve C defined by $\mathbf { r } ( t ) = t \mathbf { i } + 2 t ^ { 2 } \mathbf { j }$ Sketch the graph of C, and show $\mathbf { T } ( t )$ and $\mathbf { N } ( t )$ for $t = 1$

Accepted Answer

\[\begin{array} { l } 
\mathbf { T } ( t

Question 4

Given $\mathbf { r } ( t ) = 3 t ^ { 2 } \mathbf { i } + 4 t ^ { 3 } \mathbf { j }$ 
a. Find $\mathbf { r } ( - 1 )$ and $\mathbf { r } ^ { \prime } ( - 1 )$ . 
b. Sketch the curve defined by r and the vectors $\mathbf { r } ( - 1 )$ and $\mathbf { r } ^ { \prime } ( - 1 )$ on the same set of axes.

Accepted Answer

\[\begin{array} { l } 
\mathbf

Question 5

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 4 \sin t \mathbf { i } + \cos t \mathbf { j }$ for $t = \frac { \pi } { 4 }$ Sketch the path of the object and its velocity and acceleration vectors.

Accepted Answer

None

Question 6

A particle moves with position function $\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j } + 4 t \mathbf { k }$ .
Find the normal component of the acceleration vector.

Accepted Answer

The answer of A particle moves with position function \(\mathbf...

Question 7

The following table gives coordinates of a particle moving through space along a smooth curve. $$\begin{array} { | c | c | c | c | } 
\hline \mathbf { t } & x & y & z \
\hline 0.5 & 5.8 & 9.1 & 4.3 \
\hline 1 & 12.6 & 14.9 & 16.8 \
\hline 1.5 & 25.6 & 21.2 & 29.4 \
\hline 2 & 39.2 & 39.5 & 37.9 \
\hline 2.5 & 42.4 & 42.4 & 43 \
\hline
\end{array}$$ Find the average velocity over the time interval $[ 2.5,1.5 ]$ .

Accepted Answer

A)  $\mathbf { v } = 13.6 \mathbf { i } + 21.12 \mathbf { j } + 16.8 \mathbf { k }$ 
B)  $v = 13.6 \mathbf { i } + 16.8 \mathbf { j } + 16.8 \mathbf { k }$ 
C)  $$\mathbf { v } = 13.6 \mathbf { i } + 16.8 \mathbf { j } + 13.6 \mathbf { k }$$ 
D)  $\mathbf { v } = 16.8 \mathbf { i } + 21.12 \mathbf { j } + 13.6 \mathbf { k }$ 
E)  $\mathbf { v } = 21.12 \mathbf { i } + 21.12 \mathbf { j } + 13.6 \mathbf { k }$ 
A)  $\mathbf { v } = 13.6 \mathbf { i } + 21.12 \mathbf { j } + 16.8 \mathbf { k }$ 
B)  $v = 13.6 \mathbf { i } + 16.8 \mathbf { j } + 16.8 \mathbf { k }$ 
C)  $$\mathbf { v } = 13.6 \mathbf { i } + 16.8 \mathbf { j } + 13.6 \mathbf { k }$$ 
D)  $\mathbf { v } = 16.8 \mathbf { i } + 21.12 \mathbf { j } + 13.6 \mathbf { k }$ 
E)  $\mathbf { v } = 21.12 \mathbf { i } + 21.12 \mathbf { j } + 13.6 \mathbf { k }$

Question 8

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 3 t + 8 ) \mathbf { j }$ $- 1 \leq t \leq 2$ and indicate the orientation of the curve.

Accepted Answer

The answer of Sketch the curve of the vector function...

Question 9

Find the velocity, acceleration, and speed of an object with position vector $\mathbf { r } ( t ) = \langle \sqrt { t } , 8 + \sqrt { t } , 8 t \}$ .

Accepted Answer

\[\begin{array} { l } 
\mathbf { v } ( t

Question 10

Find the limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]$ .

Accepted Answer

A)  $3 \mathbf { i } - 6 \mathbf { k }$ 
B)  $4 \mathbf { i } + \mathbf { j }$ 
C)  $3 \mathbf { i } + \mathbf { j } - 6 \mathbf { k }$ 
D)  $4 i$ 
A)  $3 \mathbf { i } - 6 \mathbf { k }$ 
B)  $4 \mathbf { i } + \mathbf { j }$ 
C)  $3 \mathbf { i } + \mathbf { j } - 6 \mathbf { k }$ 
D)  $4 i$

Question 11

Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$ . $\mathbf { r } ( t ) = ( 5 + 3 t ) \mathbf { i } + ( 8 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

Accepted Answer

A)  $\mathbf { r } ( t ( s ) ) = \left( 5 - \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$ 
B)  $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 - \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$ 
C)  $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } + ( 6 s ) \mathbf { k }$ 
D)  $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$ 
E)  $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - ( 6 s ) \mathbf { k }$ 
A)  $\mathbf { r } ( t ( s ) ) = \left( 5 - \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$ 
B)  $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 - \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$ 
C)  $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } + ( 6 s ) \mathbf { k }$ 
D)  $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }$ 
E)  $\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - ( 6 s ) \mathbf { k }$

Question 12

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = \cos t , y = 4 e ^ { 6 t } , z = 4 e ^ { - 6 t } ; ( 1,5,5 )$

Accepted Answer

A)  $x = 0 , y = 5 - 24 t , z = 5 + 24 t$ 
B)  $x = 1 , y = 5 + 24 t , z = 4 - 24 t$ 
C)  $x = t , y = 5 + 24 t , z = 5 - 24 t$ 
D)  $x = t , y = 5 - 24 t , z = 5 + 24 t$ 
E)  $x = 1 , y = 5 + 24 t , z = 5 - 24 t$ 
A)  $x = 0 , y = 5 - 24 t , z = 5 + 24 t$ 
B)  $x = 1 , y = 5 + 24 t , z = 4 - 24 t$ 
C)  $x = t , y = 5 + 24 t , z = 5 - 24 t$ 
D)  $x = t , y = 5 - 24 t , z = 5 + 24 t$ 
E)  $x = 1 , y = 5 + 24 t , z = 5 - 24 t$

Question 13

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 2 t + 3 ) \mathbf { j }$ $- 1 \leq t \leq 2$ and indicate the orientation of the curve.

Accepted Answer

The answer of Sketch the curve of the vector function...

Question 14

Sketch the curve of the vector function $\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 6 \cos t \mathbf { j }$ , and indicate the orientation of the curve.

Accepted Answer

The answer of Sketch the curve of the vector function...

Question 15

Find $\mathbf { r } ^ { \prime \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 4 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

Accepted Answer

A)  $r ^ { \prime \prime } ( t ) = \sin t j - \cos t \mathrm { k }$ 
B)  $r ^ { \prime \prime } ( t ) = - 4 \sin t \mathrm { j } - \cos t \mathrm { k }$ 
C)  $r ^ { \prime \prime } ( t ) = - 4 \cos t j - \sin t \mathrm { k }$ 
D)  $r ^ { \prime \prime } ( t ) = \cos t j - \sin t \mathrm { k }$ 
E)  $r ^ { \prime \prime } ( t ) = - \sin t \mathrm { j } + 4 \cos t \mathrm { k }$ 
A)  $r ^ { \prime \prime } ( t ) = \sin t j - \cos t \mathrm { k }$ 
B)  $r ^ { \prime \prime } ( t ) = - 4 \sin t \mathrm { j } - \cos t \mathrm { k }$ 
C)  $r ^ { \prime \prime } ( t ) = - 4 \cos t j - \sin t \mathrm { k }$ 
D)  $r ^ { \prime \prime } ( t ) = \cos t j - \sin t \mathrm { k }$ 
E)  $r ^ { \prime \prime } ( t ) = - \sin t \mathrm { j } + 4 \cos t \mathrm { k }$

Question 16

A mortar shell is fired with a muzzle speed of 325 ft/sec. Find the angle of elevation of the mortar if the shell strikes a target located 1500 ft away. Round your answer to 2 decimal places.

Accepted Answer

A)  $12.22 ^ { \circ }$ 
B)  $0.64 ^ { \circ }$ 
C)  $13.51 ^ { \circ }$ 
D)  $0.24 ^ { \circ }$ 
A)  $12.22 ^ { \circ }$ 
B)  $0.64 ^ { \circ }$ 
C)  $13.51 ^ { \circ }$ 
D)  $0.24 ^ { \circ }$

Question 17

The torsion of a curve defined by $\mathbf { r } ( t )$ is given by $\tau = \frac { \left( \mathbf { r } ^ { t } \times \mathbf { r } ^ { \prime \prime } \right) \cdot \mathbf { r } ^ { m \prime } } { \left| \mathbf { r } ^ { t } \times \mathbf { r } ^ { \prime t } \right| ^ { 2 } }$ Find the torsion of the curve defined by $\mathbf { r } ( t ) = \cos 2 t \mathbf { i } + \sin 2 t \mathbf { j } + 5 t \mathbf { k }$ .

Accepted Answer

A)  $\frac { 27 } { 29 }$ 
B)  $\frac { 10 } { 29 }$ 
C)  $\frac { 50 } { 29 }$ 
D)  $\frac { 7 } { 29 }$ 
A)  $\frac { 27 } { 29 }$ 
B)  $\frac { 10 } { 29 }$ 
C)  $\frac { 50 } { 29 }$ 
D)  $\frac { 7 } { 29 }$

Question 18

Find parametric equations for the tangent line to the curve with parametric equations $x = 3 t$ $y = 7 t ^ { 2 }$ $z = 8 t ^ { 3 }$ at the point with $t = 1$

Accepted Answer

A)  $x = 3 + 3 t$ $y = 7 + 14 t$ $z = 8 + 24 t$ 
B)  $x = 1 + 3 t$ $y = 1 + 14 t$ $z = 1 + 24 t$ 
C)  $x = 1 + 3 t$ $y = 1 + 7 t$ $z = 1 + 8 t$ 
D)  $x = 3 + 3 t$ $y = 7 + 7 t$ $z = 8 + 8 t$ 
A)  $x = 3 + 3 t$ $y = 7 + 14 t$ $z = 8 + 24 t$ 
B)  $x = 1 + 3 t$ $y = 1 + 14 t$ $z = 1 + 24 t$ 
C)  $x = 1 + 3 t$ $y = 1 + 7 t$ $z = 1 + 8 t$ 
D)  $x = 3 + 3 t$ $y = 7 + 7 t$ $z = 8 + 8 t$

Question 19

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 6 t \mathbf { i } + \left( 5 - t ^ { 2 } \right) \mathbf { j }$ for $t = 1$ Sketch the path of the object and its velocity and acceleration vectors.

Accepted Answer

\[\begin{array} { l } 
\mathbf

Question 20

Find the derivative of the vector function. $\mathbf { r } ( t ) = a + t b + t ^ { 5 } c$

Accepted Answer

The answer of Find the derivative of the vector function....

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } ( \cos 8 t , \sin 8 t , 8 )$

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = \sin t \mathbf { i } + 6 \cos t \mathbf { j }$ for $t = \frac { \pi } { 4 }$ Sketch the path of the object and its velocity and acceleration vectors.

Find the unit tangent and unit normal vectors $\mathbf { T } ( t )$ and $\mathbf { N } ( t )$ for the curve C defined by $\mathbf { r } ( t ) = t \mathbf { i } + 2 t ^ { 2 } \mathbf { j }$ Sketch the graph of C, and show $\mathbf { T } ( t )$ and $\mathbf { N } ( t )$ for $t = 1$

Given $\mathbf { r } ( t ) = 3 t ^ { 2 } \mathbf { i } + 4 t ^ { 3 } \mathbf { j }$ a. Find $\mathbf { r } ( - 1 )$ and $\mathbf { r } ^ { \prime } ( - 1 )$ . b. Sketch the curve defined by r and the vectors $\mathbf { r } ( - 1 )$ and $\mathbf { r } ^ { \prime } ( - 1 )$ on the same set of axes.

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 4 \sin t \mathbf { i } + \cos t \mathbf { j }$ for $t = \frac { \pi } { 4 }$ Sketch the path of the object and its velocity and acceleration vectors.

A particle moves with position function $\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j } + 4 t \mathbf { k }$ . Find the normal component of the acceleration vector.

The following table gives coordinates of a particle moving through space along a smooth curve. x y z 0.5 5.8 9.1 4.3 1 12.6 14.9 16.8 1.5 25.6 21.2 29.4 2 39.2 39.5 37.9 2.5 42.4 42.4 43 Find the average velocity over the time interval $[ 2.5,1.5 ]$ .

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 3 t + 8 ) \mathbf { j }$ $- 1 \leq t \leq 2$ and indicate the orientation of the curve.

Find the velocity, acceleration, and speed of an object with position vector $\mathbf { r } ( t ) = \langle \sqrt { t } , 8 + \sqrt { t } , 8 t \}$ .

Find the limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]$ .

Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$ . $\mathbf { r } ( t ) = ( 5 + 3 t ) \mathbf { i } + ( 8 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = \cos t , y = 4 e ^ { 6 t } , z = 4 e ^ { - 6 t } ; ( 1,5,5 )$

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 2 t + 3 ) \mathbf { j }$ $- 1 \leq t \leq 2$ and indicate the orientation of the curve.

Sketch the curve of the vector function $\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 6 \cos t \mathbf { j }$ , and indicate the orientation of the curve.

Find $\mathbf { r } ^ { \prime \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 4 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

A mortar shell is fired with a muzzle speed of 325 ft/sec. Find the angle of elevation of the mortar if the shell strikes a target located 1500 ft away. Round your answer to 2 decimal places.

Find parametric equations for the tangent line to the curve with parametric equations $x = 3 t$ $y = 7 t ^ { 2 }$ $z = 8 t ^ { 3 }$ at the point with $t = 1$

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 6 t \mathbf { i } + \left( 5 - t ^ { 2 } \right) \mathbf { j }$ for $t = 1$ Sketch the path of the object and its velocity and acceleration vectors.

Find the derivative of the vector function. $\mathbf { r } ( t ) = a + t b + t ^ { 5 } c$

Functions and Limits

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Applications of Differentiation

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Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

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Final Exam

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Exam 13: Vector Functions

Find the scalar tangential and normal components of acceleration of a particle with position vector r(t)=et(cos⁡8t,sin⁡8t,8)\mathbf { r } ( t ) = e ^ { t } ( \cos 8 t , \sin 8 t , 8 )r(t)=et(cos8t,sin8t,8)

A particle moves with position function r(t)=4cos⁡ti+4sin⁡tj+4tk\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j } + 4 t \mathbf { k }r(t)=4costi+4sintj+4tk . Find the normal component of the acceleration vector.

The following table gives coordinates of a particle moving through space along a smooth curve. x y z 0.5 5.8 9.1 4.3 1 12.6 14.9 16.8 1.5 25.6 21.2 29.4 2 39.2 39.5 37.9 2.5 42.4 42.4 43 Find the average velocity over the time interval [2.5,1.5][ 2.5,1.5 ][2.5,1.5] .

Sketch the curve of the vector function r(t)=3ti+(3t+8)j\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 3 t + 8 ) \mathbf { j }r(t)=3ti+(3t+8)j −1≤t≤2- 1 \leq t \leq 2−1≤t≤2 and indicate the orientation of the curve.

Find the velocity, acceleration, and speed of an object with position vector r(t)=⟨t,8+t,8t}\mathbf { r } ( t ) = \langle \sqrt { t } , 8 + \sqrt { t } , 8 t \}r(t)=⟨t​,8+t​,8t} .

Find the limit lim⁡t→0[(t2+3)i+cos⁡5tj−6k]\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]limt→0​[(t2+3)i+cos5tj−6k] .

Reparametrize the curve with respect to arc length measured from the point where t=0t = 0t=0 in the direction of increasing ttt . r(t)=(5+3t)i+(8+9t)j−(6t)k\mathbf { r } ( t ) = ( 5 + 3 t ) \mathbf { i } + ( 8 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }r(t)=(5+3t)i+(8+9t)j−(6t)k

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=cos⁡t,y=4e6t,z=4e−6t;(1,5,5)x = \cos t , y = 4 e ^ { 6 t } , z = 4 e ^ { - 6 t } ; ( 1,5,5 )x=cost,y=4e6t,z=4e−6t;(1,5,5)

Sketch the curve of the vector function r(t)=3ti+(2t+3)j\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 2 t + 3 ) \mathbf { j }r(t)=3ti+(2t+3)j −1≤t≤2- 1 \leq t \leq 2−1≤t≤2 and indicate the orientation of the curve.

Sketch the curve of the vector function r(t)=5sin⁡ti+6cos⁡tj\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 6 \cos t \mathbf { j }r(t)=5sinti+6costj , and indicate the orientation of the curve.

Find r′′(t)\mathbf { r } ^ { \prime \prime } ( t )r′′(t) for the function given. r(t)=4i+sin⁡tj+cos⁡tk\mathbf { r } ( t ) = 4 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }r(t)=4i+sintj+costk

A mortar shell is fired with a muzzle speed of 325 ft/sec. Find the angle of elevation of the mortar if the shell strikes a target located 1500 ft away. Round your answer to 2 decimal places.

Find parametric equations for the tangent line to the curve with parametric equations x=3tx = 3 tx=3t y=7t2y = 7 t ^ { 2 }y=7t2 z=8t3z = 8 t ^ { 3 }z=8t3 at the point with t=1t = 1t=1

Find the velocity, acceleration, and speed of an object with position function r(t)=6ti+(5−t2)j\mathbf { r } ( t ) = 6 t \mathbf { i } + \left( 5 - t ^ { 2 } \right) \mathbf { j }r(t)=6ti+(5−t2)j for t=1t = 1t=1 Sketch the path of the object and its velocity and acceleration vectors.

Find the derivative of the vector function. r(t)=a+tb+t5c\mathbf { r } ( t ) = a + t b + t ^ { 5 } cr(t)=a+tb+t5c

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

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Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } ( \cos 8 t , \sin 8 t , 8 )$

A particle moves with position function $\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j } + 4 t \mathbf { k }$ . Find the normal component of the acceleration vector.

The following table gives coordinates of a particle moving through space along a smooth curve. x y z 0.5 5.8 9.1 4.3 1 12.6 14.9 16.8 1.5 25.6 21.2 29.4 2 39.2 39.5 37.9 2.5 42.4 42.4 43 Find the average velocity over the time interval $[ 2.5,1.5 ]$ .

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 3 t + 8 ) \mathbf { j }$ $- 1 \leq t \leq 2$ and indicate the orientation of the curve.

Find the velocity, acceleration, and speed of an object with position vector $\mathbf { r } ( t ) = \langle \sqrt { t } , 8 + \sqrt { t } , 8 t \}$ .

Find the limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]$ .

Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$ . $\mathbf { r } ( t ) = ( 5 + 3 t ) \mathbf { i } + ( 8 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = \cos t , y = 4 e ^ { 6 t } , z = 4 e ^ { - 6 t } ; ( 1,5,5 )$

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 2 t + 3 ) \mathbf { j }$ $- 1 \leq t \leq 2$ and indicate the orientation of the curve.

Sketch the curve of the vector function $\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 6 \cos t \mathbf { j }$ , and indicate the orientation of the curve.

Find $\mathbf { r } ^ { \prime \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 4 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

Find parametric equations for the tangent line to the curve with parametric equations $x = 3 t$ $y = 7 t ^ { 2 }$ $z = 8 t ^ { 3 }$ at the point with $t = 1$

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 6 t \mathbf { i } + \left( 5 - t ^ { 2 } \right) \mathbf { j }$ for $t = 1$ Sketch the path of the object and its velocity and acceleration vectors.

Find the derivative of the vector function. $\mathbf { r } ( t ) = a + t b + t ^ { 5 } c$