Question 1

Find the radius of curvature of the plane curve $y = 3 x ^ { 2 } + 2 \text { at } x = - 1 \text {. Round }$ your answer to three decimal places.

Accepted Answer

A) 24.694 
B) 85.333 
C) 28.133 
D) 34.510 
E) 37.510 
A) 24.694 
B) 85.333 
C) 28.133 
D) 34.510 
E) 37.510

Question 2

Find the length of the space curve given below. $\mathbf { r } ( t ) = 2 t \mathbf { i } + 5 \cos t \mathbf { j } + 5 \sin t \mathbf { k } , [ 0,3 ]$

Accepted Answer

A)  $\sqrt { 29 }$ 
B)  3 
C)  $3 \sqrt { 29 }$ 
D)  $6 \sqrt { 29 }$ 
E)  29 
A)  $\sqrt { 29 }$ 
B)  3 
C)  $3 \sqrt { 29 }$ 
D)  $6 \sqrt { 29 }$ 
E)  29

Question 3

Find the curvature K of the curve $$\mathbf { r } ( t ) = e ^ { t } \mathbf { i } + 9 t \mathbf { j } \text { at the point } P ( 1,0 ) \text {. Round your }$$ answer to three decimal places.

Accepted Answer

A) 0.883 
B) 0.994 
C) 0.019 
D) 0.012 
E) 0.017 
A) 0.883 
B) 0.994 
C) 0.019 
D) 0.012 
E) 0.017

Question 4

$\text { Find } \mathbf { r } ^ {t } ( t ) \cdot \mathbf { r } ^ {tt } ( t ) \text { given the following vector function: }$ 
$\mathbf { r } ( t ) = - 3 t ^ { 3 } \mathbf { i } - 5 t ^ { 5 } \mathbf { j }$

Accepted Answer

A)  $162 t ^ { 7 } \mathbf { i } + 2500 t ^ { 9 } \mathbf { j }$ 
B)  0 
C)  $162 t ^ { 3 } - 2500 t ^ { 7 }$ 
D)  $162 t ^ { 3 } + 2500 t ^ { 7 }$ 
E)  $162 t ^ { 8 } + 2500 t ^ { 8 }$ 
A)  $162 t ^ { 7 } \mathbf { i } + 2500 t ^ { 9 } \mathbf { j }$ 
B)  0 
C)  $162 t ^ { 3 } - 2500 t ^ { 7 }$ 
D)  $162 t ^ { 3 } + 2500 t ^ { 7 }$ 
E)  $162 t ^ { 8 } + 2500 t ^ { 8 }$

Question 5

Use the properties of the derivative to find vector-valued functions. $D _ { t } [ \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) ] \text { given the following }$ $\begin{array} { l } 
\mathbf { r } ( t ) = t \mathbf { i } + 2 \cos 4 t \mathbf { j } + 2 \sin 4 t \mathbf { k } \
\mathbf { u } ( t ) = \frac { 5 } { t } \mathbf { i } + 2 \cos 4 t \mathbf { j } + 2 \sin 4 t \mathbf { k }
\end{array}$

Accepted Answer

A)  2 
B)  $\frac { 2 } { t }$ 
C)  $2 t$ 
D)  5 
E)  0 
A)  2 
B)  $\frac { 2 } { t }$ 
C)  $2 t$ 
D)  5 
E)  0

Question 6

Match the equation with the graph shown in red below.

Accepted Answer

A)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + t ^ { 3 } \mathbf { k } \quad 0 \leq t \leq 1$ 
B)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + t \mathbf { k } , \quad 0 \leq t \leq 1$ 
C)  $\mathbf { r } ( t ) = t \mathbf { i } + t \mathbf { j } + t ^ { 2 } \mathbf { k } , \quad 0 \leq t \leq 1$ 
D)  $ \mathbf { r } ( t ) = \mathbf { i } + t \mathbf { j } + t \mathbf { k } , \quad 0 \leq t \leq 4$ 
E)  $\mathbf { r } ( t ) = t \mathbf { i } + t \mathbf { j } + t \mathbf { k } , \quad 0 \leq t \leq 1$ 
A)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + t ^ { 3 } \mathbf { k } \quad 0 \leq t \leq 1$ 
B)  $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + t \mathbf { k } , \quad 0 \leq t \leq 1$ 
C)  $\mathbf { r } ( t ) = t \mathbf { i } + t \mathbf { j } + t ^ { 2 } \mathbf { k } , \quad 0 \leq t \leq 1$ 
D)  $ \mathbf { r } ( t ) = \mathbf { i } + t \mathbf { j } + t \mathbf { k } , \quad 0 \leq t \leq 4$ 
E)  $\mathbf { r } ( t ) = t \mathbf { i } + t \mathbf { j } + t \mathbf { k } , \quad 0 \leq t \leq 1$

Question 7

A baseball is hit 3 feet above the ground at 80 feet per second and at an angle of $45 ^ { \circ }$ with respect to the ground. Find the maximum height. Round your answer to the nearest integer.

Accepted Answer

A) 66 feet 
B) 65 feet 
C) 53 feet 
D) 56 feet 
E) 61 feet 
A) 66 feet 
B) 65 feet 
C) 53 feet 
D) 56 feet 
E) 61 feet

Question 8

Find the length of the plane curve given below. $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } , [ 0,6 ]$

Accepted Answer

A)  20 
B)  9 
C)  17 
D)  19 
E)  18 
A)  20 
B)  9 
C)  17 
D)  19 
E)  18

Question 9

$\text { Find } \mathbf { r } ^ { t} ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } + 4 t ^ { 4 } \mathbf { j } + 2 t ^ { 3 } \mathbf { k }$

Accepted Answer

A)  $\mathbf { r } ^ { t } ( t ) = 2 t \mathbf { i } + 4 t ^ { 3 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k }$ 
B)  $\mathbf { r } ^ { t } ( t ) = 4 t ^ { 2 } \mathbf { i } + 16 t ^ { 4 } \mathbf { j } + 6 t ^ { 3 } \mathbf { k }$ 
C)  $\mathbf { r } ^ { t} ( t ) = 2 t \mathbf { i } + 4 t ^ { 2 } \mathbf { j } + 2 t ^ { 3 } \mathbf { k }$ 
D)  $\mathbf { r } ^ { t } ( t ) = 4 t \mathbf { i } + 4 t ^ { 3 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k }$ 
E)  $\mathbf { r } ^ { t} ( t ) = 4 t \mathbf { i } + 16 t ^ { 3 } \mathbf { j } + 6 t ^ { 2 } \mathbf { k }$ 
A)  $\mathbf { r } ^ { t } ( t ) = 2 t \mathbf { i } + 4 t ^ { 3 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k }$ 
B)  $\mathbf { r } ^ { t } ( t ) = 4 t ^ { 2 } \mathbf { i } + 16 t ^ { 4 } \mathbf { j } + 6 t ^ { 3 } \mathbf { k }$ 
C)  $\mathbf { r } ^ { t} ( t ) = 2 t \mathbf { i } + 4 t ^ { 2 } \mathbf { j } + 2 t ^ { 3 } \mathbf { k }$ 
D)  $\mathbf { r } ^ { t } ( t ) = 4 t \mathbf { i } + 4 t ^ { 3 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k }$ 
E)  $\mathbf { r } ^ { t} ( t ) = 4 t \mathbf { i } + 16 t ^ { 3 } \mathbf { j } + 6 t ^ { 2 } \mathbf { k }$

Question 10

A particle moves in the yz-plane along the curve represented by the vector-valued function $\mathbf { r } ( t ) = ( 5 \cos t ) \mathbf { j } + ( 7 \sin t ) \mathbf { k }$. Find the maximum value of $\left\| \mathbf { r } ^ { t } \right\|$

Accepted Answer

A)  4 
B)  3 
C)  7 
D)  8 
E)  5 
A)  4 
B)  3 
C)  7 
D)  8 
E)  5

Question 11

Find the arc length for $\mathbf { r } ( t ) = \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 8 t ^ { 3 } \mathbf { k } \text { over the interval } [ 0,2 ] \text {. Round your }$ answer to two decimal places.

Accepted Answer

A) 67.63 
B) 73.37 
C) 79.40 
D) 49.75 
E) 62.97 
A) 67.63 
B) 73.37 
C) 79.40 
D) 49.75 
E) 62.97

Question 12

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

Accepted Answer

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)  $\mathbf T ( 4 ) = \frac { 4 } { \sqrt { 1060 } } \mathbf { i } + \frac { 32 } { \sqrt { 1060 } } \mathbf { j }$ 
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)  $\mathbf T ( 4 ) = \frac { 4 } { \sqrt { 1060 } } \mathbf { i } + \frac { 32 } { \sqrt { 1060 } } \mathbf { j }$

Question 13

$\text { Find } \mathbf { r } ^ { t } ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = - 5 t ^ { 4 } \mathbf { i } - 4 t ^ { 5 } \mathbf { j } + 6 t ^ { 3 } \mathbf { k }$

Accepted Answer

A)  $\mathbf { r } ^ { t } ( t ) = - 5 t ^ { 3 } \mathbf { i } - 4 t ^ { 4 } \mathbf { j } + 6 t ^ { 2 } \mathbf { k }$ 
B)  $\mathbf { r } ^ { t } ( t ) = - 5 t ^ { 3 } \mathbf { i } - 4 t ^ { 2 } \mathbf { j } + 6 t ^ { 4 } \mathbf { k }$ 
C)  $\mathbf { r } ^ { t } ( t ) = - 20 t ^ { 3 } \mathbf { i } + 4 t ^ { 4 } \mathbf { j } + 6 t ^ { 2 } \mathbf { k }$ 
D)  $\mathbf { r } ^ { t} ( t ) = - 20 t ^ { \mathbf { 4 } } \mathbf { i } - 20 t ^ { 5 } \mathbf { j } + 18 t ^ { 3 } \mathbf { k }$ 
E)  $\mathbf { r } ^ { t } ( t ) = - 20 t ^ { 3 } \mathbf { i } - 20 t ^ { 4 } \mathbf { j } + 18 t ^ { 2 } \mathbf { k }$ 
A)  $\mathbf { r } ^ { t } ( t ) = - 5 t ^ { 3 } \mathbf { i } - 4 t ^ { 4 } \mathbf { j } + 6 t ^ { 2 } \mathbf { k }$ 
B)  $\mathbf { r } ^ { t } ( t ) = - 5 t ^ { 3 } \mathbf { i } - 4 t ^ { 2 } \mathbf { j } + 6 t ^ { 4 } \mathbf { k }$ 
C)  $\mathbf { r } ^ { t } ( t ) = - 20 t ^ { 3 } \mathbf { i } + 4 t ^ { 4 } \mathbf { j } + 6 t ^ { 2 } \mathbf { k }$ 
D)  $\mathbf { r } ^ { t} ( t ) = - 20 t ^ { \mathbf { 4 } } \mathbf { i } - 20 t ^ { 5 } \mathbf { j } + 18 t ^ { 3 } \mathbf { k }$ 
E)  $\mathbf { r } ^ { t } ( t ) = - 20 t ^ { 3 } \mathbf { i } - 20 t ^ { 4 } \mathbf { j } + 18 t ^ { 2 } \mathbf { k }$

Question 14

A baseball is hit 5 feet above the ground at 140 feet per second and at an angle of $\text { of } 45 ^ { \circ }$ with respect to the ground. Find the vector-valued function for the path of the baseball.

Accepted Answer

A)  $\mathbf { r } ( t ) = ( 140 t \sqrt { 2 } ) \mathbf { i } + \left( 5 - 140 t \sqrt { 2 } - 32 t ^ { 2 } \right) \mathbf { j }$ 
B)  $\mathbf { r } ( t ) = ( 70 t \sqrt { 2 } ) \mathbf { i } + \left( 5 - 70 t \sqrt { 2 } + 32 t ^ { 2 } \right) \mathbf { j }$ 
C)  $\mathbf { r } ( t ) = ( 140 t \sqrt { 2 } ) \mathbf { i } + \left( 5 + 140 t \sqrt { 2 } + 16 t ^ { 2 } \right) \mathbf { j }$ 
D)  $\mathbf { r } ( t ) = ( 70 t \sqrt { 2 } ) \mathbf { i } + \left( 5 + 70 t \sqrt { 2 } - 16 t ^ { 2 } \right) \mathbf { j }$ 
E)  $\mathbf { r } ( t ) = ( 70 t \sqrt { 2 } ) \mathbf { i } + \left( 5 - 70 t \sqrt { 2 } + 16 t ^ { 2 } \right) \mathbf { j }$ 
A)  $\mathbf { r } ( t ) = ( 140 t \sqrt { 2 } ) \mathbf { i } + \left( 5 - 140 t \sqrt { 2 } - 32 t ^ { 2 } \right) \mathbf { j }$ 
B)  $\mathbf { r } ( t ) = ( 70 t \sqrt { 2 } ) \mathbf { i } + \left( 5 - 70 t \sqrt { 2 } + 32 t ^ { 2 } \right) \mathbf { j }$ 
C)  $\mathbf { r } ( t ) = ( 140 t \sqrt { 2 } ) \mathbf { i } + \left( 5 + 140 t \sqrt { 2 } + 16 t ^ { 2 } \right) \mathbf { j }$ 
D)  $\mathbf { r } ( t ) = ( 70 t \sqrt { 2 } ) \mathbf { i } + \left( 5 + 70 t \sqrt { 2 } - 16 t ^ { 2 } \right) \mathbf { j }$ 
E)  $\mathbf { r } ( t ) = ( 70 t \sqrt { 2 } ) \mathbf { i } + \left( 5 - 70 t \sqrt { 2 } + 16 t ^ { 2 } \right) \mathbf { j }$

Question 15

The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car Moving on the path $y = \frac { 1 } { 3 } x ^ { 3 } ( x \text { and } y \text { are measured in miles) can safely go } 25 \text { miles per hour at }$ $\left( 1 , \frac { 1 } { 3 } \right)$. How fast can it go at $\left( \frac { 5 } { 2 } , \frac { 125 } { 24 } \right)$ ? Round your answer to two decimal places.

Accepted Answer

A)  $259.31 \mathrm { mi } / \mathrm { h }$ 
B)  $149.71 \mathrm { mi } / \mathrm { h }$ 
C)  $119.77 \mathrm { mi } / \mathrm { h }$ 
D)  $75.30 \mathrm { mi } / \mathrm { h }$ 
E)  $299.42 \mathrm { mi } / \mathrm { h }$ 
A)  $259.31 \mathrm { mi } / \mathrm { h }$ 
B)  $149.71 \mathrm { mi } / \mathrm { h }$ 
C)  $119.77 \mathrm { mi } / \mathrm { h }$ 
D)  $75.30 \mathrm { mi } / \mathrm { h }$ 
E)  $299.42 \mathrm { mi } / \mathrm { h }$

Question 16

The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 42 yards directly downfield at a height of 3 feet. The pass is released at an angle of $30 ^ { \circ } \text { with the horizontal. Find the maximum height of the football. }$ Round your answer to one decimal place.

Accepted Answer

A) -295.8 feet 
B) -194.1 feet 
C) -278.9 feet 
D) -224.3 feet 
E) -197.7 feet 
A) -295.8 feet 
B) -194.1 feet 
C) -278.9 feet 
D) -224.3 feet 
E) -197.7 feet

Question 17

Use the given acceleration function and initial conditions to find the position at time t = 3. $\mathbf { a } ( t ) = 5 \cos t \mathbf { i } - 3 \sin t \mathbf { j } , \quad \mathbf { v } ( 0 ) = 8 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { r } ( 0 ) = - 5 \mathbf { i }$

Accepted Answer

A)  $\mathbf { r } ( 3 ) = - 5 \cos 3 \mathbf { i } - ( 3 \sin 3 + 15 ) \mathbf { j } + 15 \mathbf { k }$ 
B)  $\mathbf { r } ( 3 ) = - 5 \cos 3 \mathbf { i } + ( 3 \sin 3 - 15 ) \mathbf { j } + 15 \mathbf { k }$ 
C)  $\mathbf { r } ( 3 ) = - 5 \cos 3 \mathbf { i } + ( 3 \sin 3 + 15 ) \mathbf { j } + 15 \mathbf { k }$ 
D)  $\mathbf { r } ( 3 ) = 5 \cos 3 \mathbf { i } + ( 3 \sin 3 + 15 ) \mathbf { j } + 15 \mathbf { k }$ 
E)  $\mathbf { r } ( 3 ) = - 5 \cos 3 \mathbf { i } + 15 \mathbf { j } + ( 3 \sin 3 + 15 ) \mathbf { k }$ 
A)  $\mathbf { r } ( 3 ) = - 5 \cos 3 \mathbf { i } - ( 3 \sin 3 + 15 ) \mathbf { j } + 15 \mathbf { k }$ 
B)  $\mathbf { r } ( 3 ) = - 5 \cos 3 \mathbf { i } + ( 3 \sin 3 - 15 ) \mathbf { j } + 15 \mathbf { k }$ 
C)  $\mathbf { r } ( 3 ) = - 5 \cos 3 \mathbf { i } + ( 3 \sin 3 + 15 ) \mathbf { j } + 15 \mathbf { k }$ 
D)  $\mathbf { r } ( 3 ) = 5 \cos 3 \mathbf { i } + ( 3 \sin 3 + 15 ) \mathbf { j } + 15 \mathbf { k }$ 
E)  $\mathbf { r } ( 3 ) = - 5 \cos 3 \mathbf { i } + 15 \mathbf { j } + ( 3 \sin 3 + 15 ) \mathbf { k }$

Question 18

$\text { Find } a _ { \mathbf { T } } \text { at time } t = 0 \text { for the plane curve } \mathbf { r } ( t ) = e ^ { 2 t } \mathbf { i } + e ^ { - 7 t } \mathbf { j } \text {. Round your answer to }$ three decimal places.

Accepted Answer

A) -40.625 
B) -39.209 
C) -111.667 
D) -46.016 
E) -43.988 
A) -40.625 
B) -39.209 
C) -111.667 
D) -46.016 
E) -43.988

Question 19

$\text { Find } a _ { \mathrm { N } } \text { at time } t = 1 \text { for the plane curve } r ( t ) = 9 t ^ { 2 } \mathbf { i } + 5 t \mathbf { j } \text {. Round your answer to }$ three decimal places.

Accepted Answer

A) 6.581 
B) 4.818 
C) 3.407 
D) 2.154 
E) 4.962 
A) 6.581 
B) 4.818 
C) 3.407 
D) 2.154 
E) 4.962

Question 20

$\text { The position vector } \mathbf { r } ( t ) = 2 t \mathbf { i } + \left( - t ^ { 2 } + 6 \right) \mathbf { j } \text { describes the path of an object moving in }$ the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point $( 2,5 )$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Find the radius of curvature of the plane curve $y = 3 x ^ { 2 } + 2 \text { at } x = - 1 \text {. Round }$ your answer to three decimal places.

Find the length of the space curve given below. $\mathbf { r } ( t ) = 2 t \mathbf { i } + 5 \cos t \mathbf { j } + 5 \sin t \mathbf { k } , [ 0,3 ]$

Find the curvature K of the curve $\mathbf { r } ( t ) = e ^ { t } \mathbf { i } + 9 t \mathbf { j } \text { at the point } P ( 1,0 ) \text {. Round your }$ answer to three decimal places.

$\text { Find } \mathbf { r } ^ {t } ( t ) \cdot \mathbf { r } ^ {tt } ( t ) \text { given the following vector function: }$ $\mathbf { r } ( t ) = - 3 t ^ { 3 } \mathbf { i } - 5 t ^ { 5 } \mathbf { j }$

Use the properties of the derivative to find vector-valued functions. $D _ { t } [ \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) ] \text { given the following }$ (t)=t+24t+24t (t)=+24t+24t

Match the equation with the graph shown in red below.

A baseball is hit 3 feet above the ground at 80 feet per second and at an angle of $45 ^ { \circ }$ with respect to the ground. Find the maximum height. Round your answer to the nearest integer.

Find the length of the plane curve given below. $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } , [ 0,6 ]$

$\text { Find } \mathbf { r } ^ { t} ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } + 4 t ^ { 4 } \mathbf { j } + 2 t ^ { 3 } \mathbf { k }$

A particle moves in the yz-plane along the curve represented by the vector-valued function $\mathbf { r } ( t ) = ( 5 \cos t ) \mathbf { j } + ( 7 \sin t ) \mathbf { k }$ . Find the maximum value of $\left\| \mathbf { r } ^ { t } \right\|$

Find the arc length for $\mathbf { r } ( t ) = \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 8 t ^ { 3 } \mathbf { k } \text { over the interval } [ 0,2 ] \text {. Round your }$ answer to two decimal places.

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

$\text { Find } \mathbf { r } ^ { t } ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = - 5 t ^ { 4 } \mathbf { i } - 4 t ^ { 5 } \mathbf { j } + 6 t ^ { 3 } \mathbf { k }$

A baseball is hit 5 feet above the ground at 140 feet per second and at an angle of $\text { of } 45 ^ { \circ }$ with respect to the ground. Find the vector-valued function for the path of the baseball.

Use the given acceleration function and initial conditions to find the position at time t = 3. $\mathbf { a } ( t ) = 5 \cos t \mathbf { i } - 3 \sin t \mathbf { j } , \quad \mathbf { v } ( 0 ) = 8 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { r } ( 0 ) = - 5 \mathbf { i }$

$\text { Find } a _ { \mathbf { T } } \text { at time } t = 0 \text { for the plane curve } \mathbf { r } ( t ) = e ^ { 2 t } \mathbf { i } + e ^ { - 7 t } \mathbf { j } \text {. Round your answer to }$ three decimal places.

$\text { Find } a _ { \mathrm { N } } \text { at time } t = 1 \text { for the plane curve } r ( t ) = 9 t ^ { 2 } \mathbf { i } + 5 t \mathbf { j } \text {. Round your answer to }$ three decimal places.

$\text { The position vector } \mathbf { r } ( t ) = 2 t \mathbf { i } + \left( - t ^ { 2 } + 6 \right) \mathbf { j } \text { describes the path of an object moving in }$ the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point $( 2,5 )$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 12: Vector-Valued Functions

Find the radius of curvature of the plane curve y=3x2+2 at x=−1. Round y = 3 x ^ { 2 } + 2 \text { at } x = - 1 \text {. Round }y=3x2+2 at x=−1. Round your answer to three decimal places.

Find the length of the space curve given below. r(t)=2ti+5cos⁡tj+5sin⁡tk,[0,3]\mathbf { r } ( t ) = 2 t \mathbf { i } + 5 \cos t \mathbf { j } + 5 \sin t \mathbf { k } , [ 0,3 ]r(t)=2ti+5costj+5sintk,[0,3]

Find the curvature K of the curve r(t)=eti+9tj at the point P(1,0). Round your \mathbf { r } ( t ) = e ^ { t } \mathbf { i } + 9 t \mathbf { j } \text { at the point } P ( 1,0 ) \text {. Round your }r(t)=eti+9tj at the point P(1,0). Round your answer to three decimal places.

Use the properties of the derivative to find vector-valued functions. Dt[r(t)⋅u(t)] given the following D _ { t } [ \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) ] \text { given the following }Dt​[r(t)⋅u(t)] given the following (t)=t+24t+24t (t)=+24t+24t

Match the equation with the graph shown in red below.

A baseball is hit 3 feet above the ground at 80 feet per second and at an angle of 45∘45 ^ { \circ }45∘ with respect to the ground. Find the maximum height. Round your answer to the nearest integer.

Find the length of the plane curve given below. r(t)=3cos⁡ti+3sin⁡tj,[0,6]\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } , [ 0,6 ]r(t)=3costi+3sintj,[0,6]

Find the unit tangent vector to the curve given below at the specified point. r(t)=−6ti+4t2j,t=4\mathbf { r } ( t ) = - 6 t \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t = 4r(t)=−6ti+4t2j,t=4

A baseball is hit 5 feet above the ground at 140 feet per second and at an angle of of 45∘\text { of } 45 ^ { \circ } of 45∘ with respect to the ground. Find the vector-valued function for the path of the baseball.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Find the radius of curvature of the plane curve $y = 3 x ^ { 2 } + 2 \text { at } x = - 1 \text {. Round }$ your answer to three decimal places.

Find the length of the space curve given below. $\mathbf { r } ( t ) = 2 t \mathbf { i } + 5 \cos t \mathbf { j } + 5 \sin t \mathbf { k } , [ 0,3 ]$

Find the curvature K of the curve $\mathbf { r } ( t ) = e ^ { t } \mathbf { i } + 9 t \mathbf { j } \text { at the point } P ( 1,0 ) \text {. Round your }$ answer to three decimal places.

Use the properties of the derivative to find vector-valued functions. $D _ { t } [ \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) ] \text { given the following }$ (t)=t+24t+24t (t)=+24t+24t

A baseball is hit 3 feet above the ground at 80 feet per second and at an angle of $45 ^ { \circ }$ with respect to the ground. Find the maximum height. Round your answer to the nearest integer.

Find the length of the plane curve given below. $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } , [ 0,6 ]$

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 baseball is hit 5 feet above the ground at 140 feet per second and at an angle of $\text { of } 45 ^ { \circ }$ with respect to the ground. Find the vector-valued function for the path of the baseball.