Question 1

Let the position function of a particle be r (t) = sin 3t i+cos 3t j+sin 4t k. Find the smallest value of its speed.

Accepted Answer

A) 1 
B) 2 
C) 9 
D)  $\sqrt { 3 }$ 
E) 0
F) $\sqrt { 10 }$ 
G) $\sqrt { 2 }$  
H)3 
A) 1 
B) 2 
C) 9 
D)  $\sqrt { 3 }$ 
E) 0
F) $\sqrt { 10 }$ 
G) $\sqrt { 2 }$  
H)3

Question 2

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 1 - 2 t , t \right\rangle$ . Find the smallest value of its speed.

Accepted Answer

A) 0 
B) 1 
C)  $\sqrt { 2 }$ 
D)  $\sqrt { 3 }$ 
E) 2
F) $\sqrt { 5 }$  
G) $\sqrt { 6 }$ 
H) $\sqrt { 7 }$ 
A) 0 
B) 1 
C)  $\sqrt { 2 }$ 
D)  $\sqrt { 3 }$ 
E) 2
F) $\sqrt { 5 }$  
G) $\sqrt { 6 }$ 
H) $\sqrt { 7 }$

Question 3

Find the center of the osculating circle of the parabola $y = x ^ { 2 }$ at the origin.

Accepted Answer

The answer of Find the center of the osculating circle...

Question 4

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle 3 \sin s \cos t , 3 \sin s \sin t , 3 \cos s \rangle$ .

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 5

Find the unit tangent vector T(t) to the curve r (t) = $t \mathbf { i } + t ^ { 2 } \mathrm { j }$ when t = 0.

Accepted Answer

A) i 
B) j 
C)  $\frac { 1 } { \sqrt { 2 } } \mathbf { i } + \frac { 1 } { \sqrt { 2 } } \mathbf { j }$ 
D)  $\frac { 1 } { \sqrt { 2 } } \mathbf { i } - \frac { 1 } { \sqrt { 2 } } \mathbf { j }$ 
E)  $\frac { 2 } { \sqrt { 5 } } \mathbf { i } + \frac { 1 } { \sqrt { 5 } } \mathbf { j }$ 
F) $\frac { 1 } { \sqrt { 5 } } \mathbf { i } + \frac { 2 } { \sqrt { 5 } } \mathbf { j }$ 
G) $\frac { 2 } { \sqrt { 5 } } \mathbf { i } - \frac { 1 } { \sqrt { 5 } } \mathbf { j }$  
H) $\frac { 1 } { \sqrt { 5 } } \mathbf { i } - \frac { 2 } { \sqrt { 5 } } \mathbf { j }$ 
A) i 
B) j 
C)  $\frac { 1 } { \sqrt { 2 } } \mathbf { i } + \frac { 1 } { \sqrt { 2 } } \mathbf { j }$ 
D)  $\frac { 1 } { \sqrt { 2 } } \mathbf { i } - \frac { 1 } { \sqrt { 2 } } \mathbf { j }$ 
E)  $\frac { 2 } { \sqrt { 5 } } \mathbf { i } + \frac { 1 } { \sqrt { 5 } } \mathbf { j }$ 
F) $\frac { 1 } { \sqrt { 5 } } \mathbf { i } + \frac { 2 } { \sqrt { 5 } } \mathbf { j }$ 
G) $\frac { 2 } { \sqrt { 5 } } \mathbf { i } - \frac { 1 } { \sqrt { 5 } } \mathbf { j }$  
H) $\frac { 1 } { \sqrt { 5 } } \mathbf { i } - \frac { 2 } { \sqrt { 5 } } \mathbf { j }$

Question 6

Find the unit tangent vector T(t) to the curve r (t) = $\left( \sin t , 2 t , t ^ { 2 } \right)$ when t = 0.

Accepted Answer

A)  $\left( - \frac { 2 } { \sqrt { 5 } } , - \frac { 1 } { \sqrt { 5 } } , 0 \right)$ 
B)  $\left( \frac { 2 } { \sqrt { 5 } } , - \frac { 1 } { \sqrt { 5 } } , 0 \right)$ 
C)  $\left( - \frac { 2 } { \sqrt { 5 } } , \frac { 1 } { \sqrt { 5 } } , 0 \right)$ 
D)  $\left( \frac { 2 } { \sqrt { 5 } } , \frac { 1 } { \sqrt { 5 } } , 0 \right)$ 
E)  $\left( - \frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt { 5 } } , 0 \right)$ 
F) $\left( \frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt { 5 } } , 0 \right)$

G) $\left( - \frac { 1 } { \sqrt { 5 } } , \frac { 2 } { \sqrt { 5 } } , 0 \right)$

H) $\left( \frac { 1 } { \sqrt { 5 } } , \frac { 2 } { \sqrt { 5 } } , 0 \right)$ 
A)  $\left( - \frac { 2 } { \sqrt { 5 } } , - \frac { 1 } { \sqrt { 5 } } , 0 \right)$ 
B)  $\left( \frac { 2 } { \sqrt { 5 } } , - \frac { 1 } { \sqrt { 5 } } , 0 \right)$ 
C)  $\left( - \frac { 2 } { \sqrt { 5 } } , \frac { 1 } { \sqrt { 5 } } , 0 \right)$ 
D)  $\left( \frac { 2 } { \sqrt { 5 } } , \frac { 1 } { \sqrt { 5 } } , 0 \right)$ 
E)  $\left( - \frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt { 5 } } , 0 \right)$ 
F) $\left( \frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt { 5 } } , 0 \right)$

G) $\left( - \frac { 1 } { \sqrt { 5 } } , \frac { 2 } { \sqrt { 5 } } , 0 \right)$

H) $\left( \frac { 1 } { \sqrt { 5 } } , \frac { 2 } { \sqrt { 5 } } , 0 \right)$

Question 7

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

Accepted Answer

A)  $\sqrt { 5 }$ 
B)  $\sqrt { 6 }$ 
C)  $\sqrt { 7 }$ 
D)  $\sqrt { 8 }$ 
E) 3 
F) $\sqrt { 10 }$ 
G) $\sqrt { 11 }$ 
H) $\sqrt { 12 }$ 
A)  $\sqrt { 5 }$ 
B)  $\sqrt { 6 }$ 
C)  $\sqrt { 7 }$ 
D)  $\sqrt { 8 }$ 
E) 3 
F) $\sqrt { 10 }$ 
G) $\sqrt { 11 }$ 
H) $\sqrt { 12 }$

Question 8

Consider r (t), the vector function describing the curve shown below. Put the curvatures at A, B, and C in order from smallest to largest.

Accepted Answer

The answer of Consider r (t), the vector function describing...

Question 9

Find the equation of the osculating circle of the ellipse whose equation is given by $\left( \operatorname { acos } t , \frac { a } { 2 } \sin t , 0 \right) \text { at } t = \frac { \pi } { 2 } \text {. }$

Accepted Answer

The answer of Find the equation of the osculating circle...

Question 10

For $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + t \mathbf { j }$ , find $\mathbf { a } _ { T }$ and $\mathbf { a } _ { N }$ , the tangential and normal components of acceleration.

Accepted Answer

The answer of For \(\mathbf { r } ( t...

Question 11

Let $x = t ^ { 3 } \text { and } y = 3 t ^ { 2 } . \text { Find the point on the curve closest to } ( 0,3 ) \text {. }$

Accepted Answer

The answer of Let \(x = t ^ { 3...

Let the position function of a particle be r (t) = sin 3t i+cos 3t j+sin 4t k. Find the smallest value of its speed.

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 1 - 2 t , t \right\rangle$ . Find the smallest value of its speed.

Find the center of the osculating circle of the parabola $y = x ^ { 2 }$ at the origin.

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle 3 \sin s \cos t , 3 \sin s \sin t , 3 \cos s \rangle$ .

Find the unit tangent vector T(t) to the curve r (t) = $t \mathbf { i } + t ^ { 2 } \mathrm { j }$ when t = 0.

Find the unit tangent vector T(t) to the curve r (t) = $\left( \sin t , 2 t , t ^ { 2 } \right)$ when t = 0.

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

Consider r (t), the vector function describing the curve shown below. Put the curvatures at A, B, and C in order from smallest to largest.

Find the equation of the osculating circle of the ellipse whose equation is given by $\left( \operatorname { acos } t , \frac { a } { 2 } \sin t , 0 \right) \text { at } t = \frac { \pi } { 2 } \text {. }$

For $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + t \mathbf { j }$ , find $\mathbf { a } _ { T }$ and $\mathbf { a } _ { N }$ , the tangential and normal components of acceleration.

Let $x = t ^ { 3 } \text { and } y = 3 t ^ { 2 } . \text { Find the point on the curve closest to } ( 0,3 ) \text {. }$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Exam 10: Vector Functions

Let the position function of a particle be r (t) = sin 3t i+cos 3t j+sin 4t k. Find the smallest value of its speed.

Let the position function of a particle be r(t)=⟨t2,1−2t,t⟩\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 1 - 2 t , t \right\rangler(t)=⟨t2,1−2t,t⟩ . Find the smallest value of its speed.

Find the center of the osculating circle of the parabola y=x2y = x ^ { 2 }y=x2 at the origin.

Identify the geometric object that is represented by parametric equations r(t,s)=⟨3sin⁡scos⁡t,3sin⁡ssin⁡t,3cos⁡s⟩\mathbf { r } ( t , s ) = \langle 3 \sin s \cos t , 3 \sin s \sin t , 3 \cos s \rangler(t,s)=⟨3sinscost,3sinssint,3coss⟩ .

Find the unit tangent vector T(t) to the curve r (t) = ti+t2jt \mathbf { i } + t ^ { 2 } \mathrm { j }ti+t2j when t = 0.

Find the unit tangent vector T(t) to the curve r (t) = (sin⁡t,2t,t2)\left( \sin t , 2 t , t ^ { 2 } \right)(sint,2t,t2) when t = 0.

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

Consider r (t), the vector function describing the curve shown below. Put the curvatures at A, B, and C in order from smallest to largest.

Find the equation of the osculating circle of the ellipse whose equation is given by (acos⁡t,a2sin⁡t,0) at t=π2. \left( \operatorname { acos } t , \frac { a } { 2 } \sin t , 0 \right) \text { at } t = \frac { \pi } { 2 } \text {. }(acost,2a​sint,0) at t=2π​.

For r(t)=t2i+tj\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + t \mathbf { j }r(t)=t2i+tj , find aT\mathbf { a } _ { T }aT​ and aN\mathbf { a } _ { N }aN​ , the tangential and normal components of acceleration.

Let x=t3 and y=3t2. Find the point on the curve closest to (0,3). x = t ^ { 3 } \text { and } y = 3 t ^ { 2 } . \text { Find the point on the curve closest to } ( 0,3 ) \text {. }x=t3 and y=3t2. Find the point on the curve closest to (0,3).

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 1 - 2 t , t \right\rangle$ . Find the smallest value of its speed.

Find the center of the osculating circle of the parabola $y = x ^ { 2 }$ at the origin.

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle 3 \sin s \cos t , 3 \sin s \sin t , 3 \cos s \rangle$ .

Find the unit tangent vector T(t) to the curve r (t) = $t \mathbf { i } + t ^ { 2 } \mathrm { j }$ when t = 0.

Find the unit tangent vector T(t) to the curve r (t) = $\left( \sin t , 2 t , t ^ { 2 } \right)$ when t = 0.

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

Find the equation of the osculating circle of the ellipse whose equation is given by $\left( \operatorname { acos } t , \frac { a } { 2 } \sin t , 0 \right) \text { at } t = \frac { \pi } { 2 } \text {. }$

For $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + t \mathbf { j }$ , find $\mathbf { a } _ { T }$ and $\mathbf { a } _ { N }$ , the tangential and normal components of acceleration.

Let $x = t ^ { 3 } \text { and } y = 3 t ^ { 2 } . \text { Find the point on the curve closest to } ( 0,3 ) \text {. }$