Question 1

Find the arc length of the curve given by $\mathbf { r } ( t ) = \left( t , \frac { \sqrt { 6 } } { 2 } t ^ { 2 } , t ^ { 3 } \right) 0 \leq t \leq 1$

Accepted Answer

The answer of Find the arc length of the curve...

Question 2

If a particle moves in a plane with constant acceleration, show that its path is a straight line or a parabola.

Accepted Answer

When a particle moves in a plane with co

Question 3

Let the velocity of a particle be $\mathbf { v } ( t ) = \mathbf { i } + t \mathbf { j }$ , and let its position when t = 0 be $\mathbf { r } ( 0 ) = \mathbf { j } + 2 \mathbf { k }$ . Find its position when t = 2.

Accepted Answer

A)  $3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$ 
B)  $2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$ 
C)  $3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$ 
D)  $3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$ 
E)  $3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$ 
F) $2 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$

G) $4 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$

H) $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$ 
A)  $3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$ 
B)  $2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$ 
C)  $3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$ 
D)  $3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$ 
E)  $3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$ 
F) $2 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$

G) $4 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$

H) $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$

Question 4

Find the equation of the plane normal to $\mathbf { r } ( t ) = \left\langle e ^ { t } \sin \frac { \pi } { 2 } t , e ^ { t } \cos \frac { \pi } { 2 } t , t ^ { 2 } \right\rangle \text { when } t = 1$

Accepted Answer

The answer of Find the equation of the plane normal...

Question 5

Find the point(s) on the curve described by the vector function $\mathbf { r } ( t ) = \left\langle t ^ { 2 } + 2 t , t ^ { 3 } - 12 t \right\rangle$ where the tangent vector is horizontal or vertical.

Accepted Answer

Horizontal at (0; 16

Question 6

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 = $\frac { \pi } { 2 }$ .

Accepted Answer

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) $\left( \frac { 1 } { \sqrt { 6 } } , \frac { 2 } { \sqrt { 6 } } , \frac { 1 } { \sqrt { 6 } } \right)$ 
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) $\left( \frac { 1 } { \sqrt { 6 } } , \frac { 2 } { \sqrt { 6 } } , \frac { 1 } { \sqrt { 6 } } \right)$

Question 7

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle s \cos t , s \sin t , t \rangle$ .

Accepted Answer

A) A plane 
B) A cone 
C) A straight line 
D) A circular cylinder 
E) A circle
F)A circular disk
G)A helix
H)A sphere 
A) A plane 
B) A cone 
C) A straight line 
D) A circular cylinder 
E) A circle
F)A circular disk
G)A helix
H)A sphere

Question 8

Consider the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ . Which of the following space curves have as their range points lying on this paraboloid?

Accepted Answer

A) r (t) = $\langle t \sin t , t \cos t , t \rangle , t \in \mathbb { R }$ 
B) r (t) = $\left\langle t ^ { 2 } , t \sin t , t \cos t \right\rangle , t \in \mathbb { R }$ 
C) r (t) = $\left\langle t \cos t , t \sin t , t ^ { 2 } \right\rangle , t \in \mathbb { R }$ 
D) r (t) = $( \sqrt { t } \sin t , t , \sqrt { t } \cos t ) , t > 0$ 
E) r (t) = $( \sqrt { t } \sin t , \sqrt { t } \cos t , t \} , t > 0$ 
A) r (t) = $\langle t \sin t , t \cos t , t \rangle , t \in \mathbb { R }$ 
B) r (t) = $\left\langle t ^ { 2 } , t \sin t , t \cos t \right\rangle , t \in \mathbb { R }$ 
C) r (t) = $\left\langle t \cos t , t \sin t , t ^ { 2 } \right\rangle , t \in \mathbb { R }$ 
D) r (t) = $( \sqrt { t } \sin t , t , \sqrt { t } \cos t ) , t > 0$ 
E) r (t) = $( \sqrt { t } \sin t , \sqrt { t } \cos t , t \} , t > 0$

Question 9

Evaluate the integral $\int _ { 0 } ^ { 1 } \left( t \mathbf { i } - 2 t ^ { 2 } \mathbf { j } + e ^ { - t } \mathbf { k } \right) d t$ .

Accepted Answer

A) i 
B) j 
C)  $\mathbf { i } - \mathbf { j } + \left( \frac { 1 } { e } - 1 \right) \mathbf { k }$ 
D)  $\mathbf { i } - \frac { 1 } { 2 } \mathbf { j } + \left( 1 - \frac { 1 } { e } \right) \mathbf { k }$ 
E)  $\frac { 1 } { 2 } \mathbf { i } - \mathbf { j }$
F) $\frac { 1 } { 2 } \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$  
G) $\frac { 1 } { 2 } \mathbf { i } - \frac { 2 } { 3 } \mathbf { j } + \left( \frac { 1 } { e } - 1 \right) \mathbf { k }$  
H) $\frac { 1 } { 2 } \mathbf { i } - \frac { 2 } { 3 } \mathbf { j } + \left( 1 - \frac { 1 } { e } \right) \mathbf { k }$ 
A) i 
B) j 
C)  $\mathbf { i } - \mathbf { j } + \left( \frac { 1 } { e } - 1 \right) \mathbf { k }$ 
D)  $\mathbf { i } - \frac { 1 } { 2 } \mathbf { j } + \left( 1 - \frac { 1 } { e } \right) \mathbf { k }$ 
E)  $\frac { 1 } { 2 } \mathbf { i } - \mathbf { j }$
F) $\frac { 1 } { 2 } \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$  
G) $\frac { 1 } { 2 } \mathbf { i } - \frac { 2 } { 3 } \mathbf { j } + \left( \frac { 1 } { e } - 1 \right) \mathbf { k }$  
H) $\frac { 1 } { 2 } \mathbf { i } - \frac { 2 } { 3 } \mathbf { j } + \left( 1 - \frac { 1 } { e } \right) \mathbf { k }$

Question 10

Find parametric equations of the tangent line to the curve $\mathbf { r } ( t ) = \langle t , \sqrt { 2 } \cos t , \sqrt { 2 } \sin t \rangle$ at $\left( \frac { \pi } { 4 } , 1,1 \right)$ . Then sketch the curve and its tangent line.

Accepted Answer

The answer of Find parametric equations of the tangent line...

Question 11

Find r (1) if $\mathbf { r } ^ { \prime }$ (t) = $t ^ { 2 } \mathbf { i } + t ^ { 3 } \mathbf { j }$ and r (0) = i.

Accepted Answer

A)  $\frac { 1 } { 3 } \mathbf { i } + \frac { 1 } { 4 } \mathbf { j }$ 
B)  $\frac { 4 } { 3 } \mathbf { i } + \frac { 1 } { 4 } \mathbf { j }$ 
C)  $\frac { 2 } { 3 } \mathbf { i } + \frac { 1 } { 4 } \mathbf { j }$ 
D)  $i + \frac { 1 } { 4 } j$ 
E)  $\frac { 1 } { 3 } \mathbf { i } + \frac { 3 } { 4 } \mathbf { j }$ 
F) $\frac { 4 } { 3 } i + \frac { 3 } { 4 } j$

G) $\frac { 2 } { 3 } \mathbf { i } + \frac { 3 } { 4 } \mathbf { j }$

H) $i + \frac { 3 } { 4 } j$ 
A)  $\frac { 1 } { 3 } \mathbf { i } + \frac { 1 } { 4 } \mathbf { j }$ 
B)  $\frac { 4 } { 3 } \mathbf { i } + \frac { 1 } { 4 } \mathbf { j }$ 
C)  $\frac { 2 } { 3 } \mathbf { i } + \frac { 1 } { 4 } \mathbf { j }$ 
D)  $i + \frac { 1 } { 4 } j$ 
E)  $\frac { 1 } { 3 } \mathbf { i } + \frac { 3 } { 4 } \mathbf { j }$ 
F) $\frac { 4 } { 3 } i + \frac { 3 } { 4 } j$

G) $\frac { 2 } { 3 } \mathbf { i } + \frac { 3 } { 4 } \mathbf { j }$

H) $i + \frac { 3 } { 4 } j$

Question 12

A person is standing 80 feet from a tall cliff. She throws a rock at 80 feet per second at an angle of 45° from the horizontal. Neglecting air resistance and discounting the height of the person, how far up the cliff does it hit?

Accepted Answer

The answer of A person is standing 80 feet from...

Question 13

Find the arc length of the curve given by $\mathbf { r } ( t ) = \left\langle 2 t , t ^ { 2 } , \ln t \right\rangle , 1 \leq t \leq 3$

Accepted Answer

The answer of Find the arc length of the curve...

Question 14

Find the domain of vector function $\mathbf { r } = \left( \sqrt { t - 2 } , \ln ( 4 - t ) , \frac { t } { \sqrt { t - 3 } } \right)$

Accepted Answer

A) [3,4] 
B) (3,4) 
C) (3,4] 
D) (2,$\infty$) 
E) [3,4)
F)(-$\infty$,4)
G)(3,$\infty$)
H)[2,$\infty$) 
A) [3,4] 
B) (3,4) 
C) (3,4] 
D) (2,$\infty$) 
E) [3,4)
F)(-$\infty$,4)
G)(3,$\infty$)
H)[2,$\infty$)

Question 15

Find the length of the curve $\mathbf { r } ( t ) = \left\langle 2 t ^ { \frac { 3 } { 2 } } , 2 t + 1 , \sqrt { 5 } t \right\rangle 0 \leq t \leq 3$

Accepted Answer

A)  $3 \sqrt { 21 }$ 
B) 6 
C) 8 
D) 10 
E) 189
F)14
G)16
H)18 
A)  $3 \sqrt { 21 }$ 
B) 6 
C) 8 
D) 10 
E) 189
F)14
G)16
H)18

Question 16

Let the position function of a particle be $\mathbf { r } ( t ) = \langle 3 \sin 2 t , 2 \cos 2 t , - \sin 4 t \rangle$ . Find the speed of the particle when $t = \frac { \pi } { 4 }$ .

Accepted Answer

A) 1 
B) 4 
C) 3 
D)  $4 \sqrt { 2 }$ 
E)  $\sqrt { 8 }$ 
F) $\sqrt { 10 }$
G) $\sqrt { 13 }$  
H) $\sqrt { 14 }$ 
A) 1 
B) 4 
C) 3 
D)  $4 \sqrt { 2 }$ 
E)  $\sqrt { 8 }$ 
F) $\sqrt { 10 }$
G) $\sqrt { 13 }$  
H) $\sqrt { 14 }$

Question 17

Let the position function of a particle be $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j }$ . Find the normal component of the acceleration vector when t = 1.

Accepted Answer

A)  $\frac { 1 } { \sqrt { 5 } }$ 
B)  $\frac { 2 } { \sqrt { 5 } }$ 
C)  $\frac { 3 } { \sqrt { 5 } }$ 
D)  $\frac { 4 } { \sqrt { 5 } }$ 
E)  $\sqrt { 5 }$
F) $\frac { 6 } { \sqrt { 5 } }$

G) $\frac { 7 } { \sqrt { 5 } }$

H) $\frac { 8 } { \sqrt { 5 } }$ 
A)  $\frac { 1 } { \sqrt { 5 } }$ 
B)  $\frac { 2 } { \sqrt { 5 } }$ 
C)  $\frac { 3 } { \sqrt { 5 } }$ 
D)  $\frac { 4 } { \sqrt { 5 } }$ 
E)  $\sqrt { 5 }$
F) $\frac { 6 } { \sqrt { 5 } }$

G) $\frac { 7 } { \sqrt { 5 } }$

H) $\frac { 8 } { \sqrt { 5 } }$

Question 18

Suppose C is the curve given by the vector function $\mathbf { r } ( t ) = \left\langle t , t ^ { 2 } , 1 - t ^ { 2 } \right\rangle$ . Find the unit tangent vector, the unit normal vector, and the curvature of C at the point where t = 1.

Accepted Answer

The answer of Suppose C is the curve given by...

Question 19

Find a parametric representation for the surface consisting of the upper half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + z ^ { 2 } = 1$ .

Accepted Answer

A)  $x = x , y = y , z = \sqrt { 1 + x ^ { 2 } + 5 y ^ { 2 } }$ 
B)  $x = x , y = y , z = \sqrt { 1 + x ^ { 2 } - 5 y ^ { 2 } }$ 
C)  $x = x , y = y , z = \sqrt { 1 - x ^ { 2 } + 5 y ^ { 2 } }$ 
D)  $x = x , y = y , z = \sqrt { 1 - x ^ { 2 } - 5 y ^ { 2 } }$ 
E)  $x = x , y = y , z = \sqrt { x ^ { 2 } + 5 y ^ { 2 } - 1 }$ 
F) $x = x , y = y , z = \sqrt { x ^ { 2 } - 5 y ^ { 2 } - 1 }$

G) $x = x , y = y , z = \sqrt { - x ^ { 2 } + 5 y ^ { 2 } - 1 }$ 
 
H) $x = x , y = y , z = \sqrt { - x ^ { 2 } - 5 y ^ { 2 } - 1 }$ 
A)  $x = x , y = y , z = \sqrt { 1 + x ^ { 2 } + 5 y ^ { 2 } }$ 
B)  $x = x , y = y , z = \sqrt { 1 + x ^ { 2 } - 5 y ^ { 2 } }$ 
C)  $x = x , y = y , z = \sqrt { 1 - x ^ { 2 } + 5 y ^ { 2 } }$ 
D)  $x = x , y = y , z = \sqrt { 1 - x ^ { 2 } - 5 y ^ { 2 } }$ 
E)  $x = x , y = y , z = \sqrt { x ^ { 2 } + 5 y ^ { 2 } - 1 }$ 
F) $x = x , y = y , z = \sqrt { x ^ { 2 } - 5 y ^ { 2 } - 1 }$

G) $x = x , y = y , z = \sqrt { - x ^ { 2 } + 5 y ^ { 2 } - 1 }$ 
 
H) $x = x , y = y , z = \sqrt { - x ^ { 2 } - 5 y ^ { 2 } - 1 }$

Question 20

At what point does the curve $\mathbf { r } ( t ) = \left( \sqrt { 2 } t , e ^ { t } , e ^ { - t } \right)$ have minimum curvature? What is the minimum curvature?

Accepted Answer

The answer of At what point does the curve \(\mathbf...

Find the arc length of the curve given by $\mathbf { r } ( t ) = \left( t , \frac { \sqrt { 6 } } { 2 } t ^ { 2 } , t ^ { 3 } \right) 0 \leq t \leq 1$

If a particle moves in a plane with constant acceleration, show that its path is a straight line or a parabola.

Let the velocity of a particle be $\mathbf { v } ( t ) = \mathbf { i } + t \mathbf { j }$ , and let its position when t = 0 be $\mathbf { r } ( 0 ) = \mathbf { j } + 2 \mathbf { k }$ . Find its position when t = 2.

Find the equation of the plane normal to $\mathbf { r } ( t ) = \left\langle e ^ { t } \sin \frac { \pi } { 2 } t , e ^ { t } \cos \frac { \pi } { 2 } t , t ^ { 2 } \right\rangle \text { when } t = 1$

Find the point(s) on the curve described by the vector function $\mathbf { r } ( t ) = \left\langle t ^ { 2 } + 2 t , t ^ { 3 } - 12 t \right\rangle$ where the tangent vector is horizontal or vertical.

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 = $\frac { \pi } { 2 }$ .

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle s \cos t , s \sin t , t \rangle$ .

Consider the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ . Which of the following space curves have as their range points lying on this paraboloid?

Evaluate the integral $\int _ { 0 } ^ { 1 } \left( t \mathbf { i } - 2 t ^ { 2 } \mathbf { j } + e ^ { - t } \mathbf { k } \right) d t$ .

Find parametric equations of the tangent line to the curve $\mathbf { r } ( t ) = \langle t , \sqrt { 2 } \cos t , \sqrt { 2 } \sin t \rangle$ at $\left( \frac { \pi } { 4 } , 1,1 \right)$ . Then sketch the curve and its tangent line.

Find r (1) if $\mathbf { r } ^ { \prime }$ (t) = $t ^ { 2 } \mathbf { i } + t ^ { 3 } \mathbf { j }$ and r (0) = i.

A person is standing 80 feet from a tall cliff. She throws a rock at 80 feet per second at an angle of 45° from the horizontal. Neglecting air resistance and discounting the height of the person, how far up the cliff does it hit?

Find the arc length of the curve given by $\mathbf { r } ( t ) = \left\langle 2 t , t ^ { 2 } , \ln t \right\rangle , 1 \leq t \leq 3$

Find the domain of vector function $\mathbf { r } = \left( \sqrt { t - 2 } , \ln ( 4 - t ) , \frac { t } { \sqrt { t - 3 } } \right)$

Find the length of the curve $\mathbf { r } ( t ) = \left\langle 2 t ^ { \frac { 3 } { 2 } } , 2 t + 1 , \sqrt { 5 } t \right\rangle 0 \leq t \leq 3$

Let the position function of a particle be $\mathbf { r } ( t ) = \langle 3 \sin 2 t , 2 \cos 2 t , - \sin 4 t \rangle$ . Find the speed of the particle when $t = \frac { \pi } { 4 }$ .

Let the position function of a particle be $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j }$ . Find the normal component of the acceleration vector when t = 1.

Suppose C is the curve given by the vector function $\mathbf { r } ( t ) = \left\langle t , t ^ { 2 } , 1 - t ^ { 2 } \right\rangle$ . Find the unit tangent vector, the unit normal vector, and the curvature of C at the point where t = 1.

Find a parametric representation for the surface consisting of the upper half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + z ^ { 2 } = 1$ .

At what point does the curve $\mathbf { r } ( t ) = \left( \sqrt { 2 } t , e ^ { t } , e ^ { - t } \right)$ have minimum curvature? What is the minimum curvature?

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Exam 10: Vector Functions

Find the arc length of the curve given by r(t)=(t,62t2,t3)0≤t≤1\mathbf { r } ( t ) = \left( t , \frac { \sqrt { 6 } } { 2 } t ^ { 2 } , t ^ { 3 } \right) 0 \leq t \leq 1r(t)=(t,26​​t2,t3)0≤t≤1

If a particle moves in a plane with constant acceleration, show that its path is a straight line or a parabola.

Let the velocity of a particle be v(t)=i+tj\mathbf { v } ( t ) = \mathbf { i } + t \mathbf { j }v(t)=i+tj , and let its position when t = 0 be r(0)=j+2k\mathbf { r } ( 0 ) = \mathbf { j } + 2 \mathbf { k }r(0)=j+2k . Find its position when t = 2.

Find the equation of the plane normal to r(t)=⟨etsin⁡π2t,etcos⁡π2t,t2⟩ when t=1\mathbf { r } ( t ) = \left\langle e ^ { t } \sin \frac { \pi } { 2 } t , e ^ { t } \cos \frac { \pi } { 2 } t , t ^ { 2 } \right\rangle \text { when } t = 1r(t)=⟨etsin2π​t,etcos2π​t,t2⟩ when t=1

Find the point(s) on the curve described by the vector function r(t)=⟨t2+2t,t3−12t⟩\mathbf { r } ( t ) = \left\langle t ^ { 2 } + 2 t , t ^ { 3 } - 12 t \right\rangler(t)=⟨t2+2t,t3−12t⟩ where the tangent vector is horizontal or vertical.

Find the unit tangent vector to the curve r (t) = {e2tcos⁡t,e2tsin⁡t,e2t}\left\{ e ^ { 2 t } \cos t , e ^ { 2 t } \sin t , e ^ { 2 t } \right\}{e2tcost,e2tsint,e2t} at the point where t = π2\frac { \pi } { 2 }2π​ .

Identify the geometric object that is represented by parametric equations r(t,s)=⟨scos⁡t,ssin⁡t,t⟩\mathbf { r } ( t , s ) = \langle s \cos t , s \sin t , t \rangler(t,s)=⟨scost,ssint,t⟩ .

Consider the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 . Which of the following space curves have as their range points lying on this paraboloid?

Evaluate the integral ∫01(ti−2t2j+e−tk)dt\int _ { 0 } ^ { 1 } \left( t \mathbf { i } - 2 t ^ { 2 } \mathbf { j } + e ^ { - t } \mathbf { k } \right) d t∫01​(ti−2t2j+e−tk)dt .

Find r (1) if r′\mathbf { r } ^ { \prime }r′ (t) = t2i+t3jt ^ { 2 } \mathbf { i } + t ^ { 3 } \mathbf { j }t2i+t3j and r (0) = i.

A person is standing 80 feet from a tall cliff. She throws a rock at 80 feet per second at an angle of 45° from the horizontal. Neglecting air resistance and discounting the height of the person, how far up the cliff does it hit?

Find the arc length of the curve given by r(t)=⟨2t,t2,ln⁡t⟩,1≤t≤3\mathbf { r } ( t ) = \left\langle 2 t , t ^ { 2 } , \ln t \right\rangle , 1 \leq t \leq 3r(t)=⟨2t,t2,lnt⟩,1≤t≤3

Find the domain of vector function r=(t−2,ln⁡(4−t),tt−3)\mathbf { r } = \left( \sqrt { t - 2 } , \ln ( 4 - t ) , \frac { t } { \sqrt { t - 3 } } \right)r=(t−2​,ln(4−t),t−3​t​)

Find the length of the curve r(t)=⟨2t32,2t+1,5t⟩0≤t≤3\mathbf { r } ( t ) = \left\langle 2 t ^ { \frac { 3 } { 2 } } , 2 t + 1 , \sqrt { 5 } t \right\rangle 0 \leq t \leq 3r(t)=⟨2t23​,2t+1,5​t⟩0≤t≤3

Let the position function of a particle be r(t)=⟨3sin⁡2t,2cos⁡2t,−sin⁡4t⟩\mathbf { r } ( t ) = \langle 3 \sin 2 t , 2 \cos 2 t , - \sin 4 t \rangler(t)=⟨3sin2t,2cos2t,−sin4t⟩ . Find the speed of the particle when t=π4t = \frac { \pi } { 4 }t=4π​ .

Let the position function of a particle be r(t)=ti+t2j\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j }r(t)=ti+t2j . Find the normal component of the acceleration vector when t = 1.

Suppose C is the curve given by the vector function r(t)=⟨t,t2,1−t2⟩\mathbf { r } ( t ) = \left\langle t , t ^ { 2 } , 1 - t ^ { 2 } \right\rangler(t)=⟨t,t2,1−t2⟩ . Find the unit tangent vector, the unit normal vector, and the curvature of C at the point where t = 1.

Find a parametric representation for the surface consisting of the upper half of the ellipsoid x2+5y2+z2=1x ^ { 2 } + 5 y ^ { 2 } + z ^ { 2 } = 1x2+5y2+z2=1 .

At what point does the curve r(t)=(2t,et,e−t)\mathbf { r } ( t ) = \left( \sqrt { 2 } t , e ^ { t } , e ^ { - t } \right)r(t)=(2​t,et,e−t) have minimum curvature? What is the minimum curvature?

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Find the arc length of the curve given by $\mathbf { r } ( t ) = \left( t , \frac { \sqrt { 6 } } { 2 } t ^ { 2 } , t ^ { 3 } \right) 0 \leq t \leq 1$

Let the velocity of a particle be $\mathbf { v } ( t ) = \mathbf { i } + t \mathbf { j }$ , and let its position when t = 0 be $\mathbf { r } ( 0 ) = \mathbf { j } + 2 \mathbf { k }$ . Find its position when t = 2.

Find the equation of the plane normal to $\mathbf { r } ( t ) = \left\langle e ^ { t } \sin \frac { \pi } { 2 } t , e ^ { t } \cos \frac { \pi } { 2 } t , t ^ { 2 } \right\rangle \text { when } t = 1$

Find the point(s) on the curve described by the vector function $\mathbf { r } ( t ) = \left\langle t ^ { 2 } + 2 t , t ^ { 3 } - 12 t \right\rangle$ where the tangent vector is horizontal or vertical.

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 = $\frac { \pi } { 2 }$ .

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle s \cos t , s \sin t , t \rangle$ .

Consider the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ . Which of the following space curves have as their range points lying on this paraboloid?

Evaluate the integral $\int _ { 0 } ^ { 1 } \left( t \mathbf { i } - 2 t ^ { 2 } \mathbf { j } + e ^ { - t } \mathbf { k } \right) d t$ .

Find r (1) if $\mathbf { r } ^ { \prime }$ (t) = $t ^ { 2 } \mathbf { i } + t ^ { 3 } \mathbf { j }$ and r (0) = i.

Find the arc length of the curve given by $\mathbf { r } ( t ) = \left\langle 2 t , t ^ { 2 } , \ln t \right\rangle , 1 \leq t \leq 3$

Find the domain of vector function $\mathbf { r } = \left( \sqrt { t - 2 } , \ln ( 4 - t ) , \frac { t } { \sqrt { t - 3 } } \right)$

Find the length of the curve $\mathbf { r } ( t ) = \left\langle 2 t ^ { \frac { 3 } { 2 } } , 2 t + 1 , \sqrt { 5 } t \right\rangle 0 \leq t \leq 3$

Let the position function of a particle be $\mathbf { r } ( t ) = \langle 3 \sin 2 t , 2 \cos 2 t , - \sin 4 t \rangle$ . Find the speed of the particle when $t = \frac { \pi } { 4 }$ .

Let the position function of a particle be $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j }$ . Find the normal component of the acceleration vector when t = 1.

Suppose C is the curve given by the vector function $\mathbf { r } ( t ) = \left\langle t , t ^ { 2 } , 1 - t ^ { 2 } \right\rangle$ . Find the unit tangent vector, the unit normal vector, and the curvature of C at the point where t = 1.

Find a parametric representation for the surface consisting of the upper half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + z ^ { 2 } = 1$ .

At what point does the curve $\mathbf { r } ( t ) = \left( \sqrt { 2 } t , e ^ { t } , e ^ { - t } \right)$ have minimum curvature? What is the minimum curvature?