Question 1

At what point does the curve $\mathbf { r } ( t ) = \left\langle t , t , 1 - a t ^ { 2 } \right\rangle , a > 0$ have maximum curvature? What is the maximum curvature?

Accepted Answer

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

Question 2

Evaluate the integral $\int \left( \sin ^ { 3 } ( 2 t ) \cos ( 2 t ) \right) \mathbf { i } + t ^ { 2 } \ln t \mathbf { j } + \frac { e ^ { \sqrt { t } } } { \sqrt { t } } \mathbf { k }$

Accepted Answer

The answer of Evaluate the integral \(\int \left( \sin ^...

Question 3

Find a parametric representation for the surface consisting of that part of the plane z = x + 3 that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .

Accepted Answer

The answer of Find a parametric representation for the surface...

Question 4

Consider the curve in the xy-plane defined parametrically by x = t3 - 3t, y = t2, z = 0. Sketch a rough graph of the curve.

Accepted Answer

Question 5

Find the unit tangent vector T(t) to the curve r (t) = $\langle \sin t , t , \cos t \rangle$ when t = 0.

Accepted Answer

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

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

H) $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$ 
A)  $( - 1,0,0 )$ 
B)  $( 0,0 , - 1 )$ 
C)  $( 1,0,0 )$ 
D)  $( 0,0,1 )$ 
E)  $\left( 0 , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$ 
F) $\left( \frac { 1 } { \sqrt { 2 } } , 0 , \frac { 1 } { \sqrt { 2 } } \right)$

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

H) $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$

Question 6

Find the tangent vector $\mathbf { r } ^ { \prime }$ (t) of the function r (t) = $\left( t , \sqrt { t } , \frac { 1 } { 2 \sqrt { t } } \right)$ when t = $\frac { 1 } { 4 }$ .

Accepted Answer

A)  $( 1,2,4)$ 
B)  $( 1,2 , - 2)$ 
C) $( 1,1,4 )$ 
D)  $( 1,1 , - 2 )$ 
E)  $( 1,2 , - 4 )$
F) $(1,2,2 )$

G) $( 1,1 , - 4 )$

H) $( 1,1,2 )$ 
A)  $( 1,2,4)$ 
B)  $( 1,2 , - 2)$ 
C) $( 1,1,4 )$ 
D)  $( 1,1 , - 2 )$ 
E)  $( 1,2 , - 4 )$
F) $(1,2,2 )$

G) $( 1,1 , - 4 )$

H) $( 1,1,2 )$

Question 7

Find the curvature $K$ of the curve $y = 2 x ^ { 2 }$ at x = 0.

Accepted Answer

A) 0 
B)  $\frac { 1 } { 8 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 2 }$ 
E) 1
F)2
G)4
H)8 
A) 0 
B)  $\frac { 1 } { 8 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 2 }$ 
E) 1
F)2
G)4
H)8

Question 8

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 2 t , e ^ { t } \right\rangle$ . Find the velocity of the particle when t = 1.

Accepted Answer

A)  $( 2,2,1 )$ 
B)  $( 2,2 , e )$ 
C)  $(2,0,1 )$ 
D)  $( 2,0 , e )$ 
E)  $( 1,1,1 )$ 
F) $( 1,1 , e )$

G) $( 1,0,1 )$

H) $( 1,0 , e )$ 
A)  $( 2,2,1 )$ 
B)  $( 2,2 , e )$ 
C)  $(2,0,1 )$ 
D)  $( 2,0 , e )$ 
E)  $( 1,1,1 )$ 
F) $( 1,1 , e )$

G) $( 1,0,1 )$

H) $( 1,0 , e )$

Question 9

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

Accepted Answer

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

Question 10

Find a parametric representation for the surface consisting of that part of the elliptic paraboloid $x + y ^ { 2 } + 2 z ^ { 2 } = 4$ that lies in front of the plane x = 0.

Accepted Answer

@#IMG-DLM& , @#IMG-DLM& , @#IMG-DLM& whe

Question 11

Find the unit tangent vector T(t) to the curve r (t) = $\left( t ^ { 2 } - 1,3 t ^ { 2 } - t ^ { 4 } , \frac { 2 } { t } \right)$ when t = 1.

Accepted Answer

A)  $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } \right)$ 
B)  $\left( \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } \right)$ 
C)  $( 0,1,0)$ 
D)  $( 0,0,1 )$ 
E)  $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$  
F) $\left( \frac { 1 } { \sqrt { 2 } } , 0 , \frac { 1 } { \sqrt { 2 } } \right)$ 
 
G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$

H) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$ 
A)  $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } \right)$ 
B)  $\left( \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } , - \frac { 1 } { \sqrt { 3 } } \right)$ 
C)  $( 0,1,0)$ 
D)  $( 0,0,1 )$ 
E)  $\left( \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right)$  
F) $\left( \frac { 1 } { \sqrt { 2 } } , 0 , \frac { 1 } { \sqrt { 2 } } \right)$ 
 
G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$

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

Question 12

Evaluate the integral $\int \frac { 1 } { t } \mathbf { i } + e ^ { - t } \mathbf { j } + t ^ { \frac { 3 } { 2 } } \mathbf { k }$

Accepted Answer

The answer of Evaluate the integral \(\int \frac { 1...

Question 13

Identify the surface with the vector equation $\mathbf { r } ( u , v ) = \cos u \sin v \mathbf { i } + \sin u \sin v \mathbf { j } + \cos v \mathbf { k } , 0 \leq u \leq 2 \pi , 0 \leq v \leq \frac { \pi } { 2 }$ . (Hint: First consider $x ^ { 2 } + y ^ { 2 }$ .)

Accepted Answer

Top half o

Question 14

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

Accepted Answer

A)  $\frac { 2 } { 27 } ( 13 \sqrt { 13 } - 8 )$ 
B)  $\frac { 13 } { 9 }$ 
C)  $\frac { 13 \sqrt { 13 } - 6 } { 27 }$ 
D)  $\frac { 16 } { 9 }$ 
E)  $\frac { 10 } { 27 } ( \sqrt { 13 } - 2 )$
F) $\frac { 19 } { 9 }$

G) $\frac { 4 } { 9 } ( 7 \sqrt { 7 } - 2 )$

H) $\frac { 22 } { 9 }$ 
A)  $\frac { 2 } { 27 } ( 13 \sqrt { 13 } - 8 )$ 
B)  $\frac { 13 } { 9 }$ 
C)  $\frac { 13 \sqrt { 13 } - 6 } { 27 }$ 
D)  $\frac { 16 } { 9 }$ 
E)  $\frac { 10 } { 27 } ( \sqrt { 13 } - 2 )$
F) $\frac { 19 } { 9 }$

G) $\frac { 4 } { 9 } ( 7 \sqrt { 7 } - 2 )$

H) $\frac { 22 } { 9 }$

Question 15

If $\mathbf { u } ( t ) = \left( - \sqrt { t } \sin t , t , t ^ { \frac { 2 } { 3 } } \right)$ and $\mathbf { v } ( t ) = \left( - \sqrt { t } \sin t , \cos ^ { 2 } t , - t ^ { \frac { 1 } { 3 } } \right)$ , compute $\frac { d } { d t } ( \mathbf { u } ( t ) \cdot \mathbf { v } ( t ) )$ and $\frac { d } { d t } ( \mathbf { u } ( t ) \cdot \mathbf { u } ( t ) )$ .

Accepted Answer

The answer of If \(\mathbf { u } ( t...

Question 16

A picture of a circular cylinder with radius a and height h is given below. Find a parametric representation of the cylinder.

Accepted Answer

@#IMG-DLM& , @#IMG-DLM& , @#IMG-DLM& Oth

Question 17

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \langle 1 + t , 3 t , 3 - 5 t \rangle$ .

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 18

A particle is traveling along a helix whose vector equation is given by $\mathbf { r } ( t ) = \langle R \cos \alpha t , R \sin \alpha t , \beta t \rangle$ . Show that its velocity and acceleration are orthogonal at all times.

Accepted Answer

To show that the velocity and accelerati

Question 19

Find $\mathbf { r } ^ { t } ( t ) \text { if } \mathbf { r } ( t ) = t \sin t + \cos t \mathbf { i } + \frac { t } { t + 1 } \mathbf { j } - e ^ { \frac { 1 } { t } } \mathbf { k }$

Accepted Answer

The answer of Find \(\mathbf { r } ^ {...

Question 20

The graphs of which three of the following four vector functions lie along the line y = 4 - x?

Accepted Answer

A) x = 6 - t2, y = t2 - 2 
B) x = cos2t, y = sin2t + 3 
C) x = 2et, y = 2et - 8 
D) x3 = ln (1/t), y = ln (e4t) 
A) x = 6 - t2, y = t2 - 2 
B) x = cos2t, y = sin2t + 3 
C) x = 2et, y = 2et - 8 
D) x3 = ln (1/t), y = ln (e4t)

At what point does the curve $\mathbf { r } ( t ) = \left\langle t , t , 1 - a t ^ { 2 } \right\rangle , a > 0$ have maximum curvature? What is the maximum curvature?

Evaluate the integral $\int \left( \sin ^ { 3 } ( 2 t ) \cos ( 2 t ) \right) \mathbf { i } + t ^ { 2 } \ln t \mathbf { j } + \frac { e ^ { \sqrt { t } } } { \sqrt { t } } \mathbf { k }$

Find a parametric representation for the surface consisting of that part of the plane z = x + 3 that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .

Consider the curve in the xy-plane defined parametrically by x = t³ - 3t, y = t², z = 0. Sketch a rough graph of the curve.

Find the unit tangent vector T(t) to the curve r (t) = $\langle \sin t , t , \cos t \rangle$ when t = 0.

Find the tangent vector $\mathbf { r } ^ { \prime }$ (t) of the function r (t) = $\left( t , \sqrt { t } , \frac { 1 } { 2 \sqrt { t } } \right)$ when t = $\frac { 1 } { 4 }$ .

Find the curvature $K$ of the curve $y = 2 x ^ { 2 }$ at x = 0.

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 2 t , e ^ { t } \right\rangle$ . Find the velocity of the particle when t = 1.

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

Find a parametric representation for the surface consisting of that part of the elliptic paraboloid $x + y ^ { 2 } + 2 z ^ { 2 } = 4$ that lies in front of the plane x = 0.

Find the unit tangent vector T(t) to the curve r (t) = $\left( t ^ { 2 } - 1,3 t ^ { 2 } - t ^ { 4 } , \frac { 2 } { t } \right)$ when t = 1.

Evaluate the integral $\int \frac { 1 } { t } \mathbf { i } + e ^ { - t } \mathbf { j } + t ^ { \frac { 3 } { 2 } } \mathbf { k }$

Identify the surface with the vector equation $\mathbf { r } ( u , v ) = \cos u \sin v \mathbf { i } + \sin u \sin v \mathbf { j } + \cos v \mathbf { k } , 0 \leq u \leq 2 \pi , 0 \leq v \leq \frac { \pi } { 2 }$ . (Hint: First consider $x ^ { 2 } + y ^ { 2 }$ .)

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

A picture of a circular cylinder with radius a and height h is given below. Find a parametric representation of the cylinder.

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \langle 1 + t , 3 t , 3 - 5 t \rangle$ .

A particle is traveling along a helix whose vector equation is given by $\mathbf { r } ( t ) = \langle R \cos \alpha t , R \sin \alpha t , \beta t \rangle$ . Show that its velocity and acceleration are orthogonal at all times.

Find $\mathbf { r } ^ { t } ( t ) \text { if } \mathbf { r } ( t ) = t \sin t + \cos t \mathbf { i } + \frac { t } { t + 1 } \mathbf { j } - e ^ { \frac { 1 } { t } } \mathbf { k }$

The graphs of which three of the following four vector functions lie along the line y = 4 - x?

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Exam 10: Vector Functions

At what point does the curve r(t)=⟨t,t,1−at2⟩,a>0\mathbf { r } ( t ) = \left\langle t , t , 1 - a t ^ { 2 } \right\rangle , a > 0r(t)=⟨t,t,1−at2⟩,a>0 have maximum curvature? What is the maximum curvature?

Evaluate the integral ∫(sin⁡3(2t)cos⁡(2t))i+t2ln⁡tj+ettk\int \left( \sin ^ { 3 } ( 2 t ) \cos ( 2 t ) \right) \mathbf { i } + t ^ { 2 } \ln t \mathbf { j } + \frac { e ^ { \sqrt { t } } } { \sqrt { t } } \mathbf { k }∫(sin3(2t)cos(2t))i+t2lntj+t​et​​k

Find a parametric representation for the surface consisting of that part of the plane z = x + 3 that lies inside the cylinder x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 .

Consider the curve in the xy-plane defined parametrically by x = t3 - 3t, y = t2, z = 0. Sketch a rough graph of the curve.

Find the unit tangent vector T(t) to the curve r (t) = ⟨sin⁡t,t,cos⁡t⟩\langle \sin t , t , \cos t \rangle⟨sint,t,cost⟩ when t = 0.

Find the tangent vector r′\mathbf { r } ^ { \prime }r′ (t) of the function r (t) = (t,t,12t)\left( t , \sqrt { t } , \frac { 1 } { 2 \sqrt { t } } \right)(t,t​,2t​1​) when t = 14\frac { 1 } { 4 }41​ .

Find the curvature KKK of the curve y=2x2y = 2 x ^ { 2 }y=2x2 at x = 0.

Let the position function of a particle be r(t)=⟨t2,2t,et⟩\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 2 t , e ^ { t } \right\rangler(t)=⟨t2,2t,et⟩ . Find the velocity of the particle when t = 1.

Find the arc length of the curve given by r(t)=⟨t2,t3},0≤t≤1\mathbf { r } ( t ) = \left\langle t ^ { 2 } , t ^ { 3 } \right\} , 0 \leq t \leq 1r(t)=⟨t2,t3},0≤t≤1

Find a parametric representation for the surface consisting of that part of the elliptic paraboloid x+y2+2z2=4x + y ^ { 2 } + 2 z ^ { 2 } = 4x+y2+2z2=4 that lies in front of the plane x = 0.

Find the unit tangent vector T(t) to the curve r (t) = (t2−1,3t2−t4,2t)\left( t ^ { 2 } - 1,3 t ^ { 2 } - t ^ { 4 } , \frac { 2 } { t } \right)(t2−1,3t2−t4,t2​) when t = 1.

Evaluate the integral ∫1ti+e−tj+t32k\int \frac { 1 } { t } \mathbf { i } + e ^ { - t } \mathbf { j } + t ^ { \frac { 3 } { 2 } } \mathbf { k }∫t1​i+e−tj+t23​k

Find the length of the curve r(t)=⟨sin⁡2t,cos⁡2t,2t32⟩0≤t≤1\mathbf { r } ( t ) = \left\langle \sin 2 t , \cos 2 t , 2 t ^ { \frac { 3 } { 2 } } \right\rangle 0 \leq t \leq 1r(t)=⟨sin2t,cos2t,2t23​⟩0≤t≤1

A picture of a circular cylinder with radius a and height h is given below. Find a parametric representation of the cylinder.

Identify the geometric object that is represented by parametric equations r(t)=⟨1+t,3t,3−5t⟩\mathbf { r } ( t ) = \langle 1 + t , 3 t , 3 - 5 t \rangler(t)=⟨1+t,3t,3−5t⟩ .

A particle is traveling along a helix whose vector equation is given by r(t)=⟨Rcos⁡αt,Rsin⁡αt,βt⟩\mathbf { r } ( t ) = \langle R \cos \alpha t , R \sin \alpha t , \beta t \rangler(t)=⟨Rcosαt,Rsinαt,βt⟩ . Show that its velocity and acceleration are orthogonal at all times.

Find rt(t) if r(t)=tsin⁡t+cos⁡ti+tt+1j−e1tk\mathbf { r } ^ { t } ( t ) \text { if } \mathbf { r } ( t ) = t \sin t + \cos t \mathbf { i } + \frac { t } { t + 1 } \mathbf { j } - e ^ { \frac { 1 } { t } } \mathbf { k }rt(t) if r(t)=tsint+costi+t+1t​j−et1​k

The graphs of which three of the following four vector functions lie along the line y = 4 - x?

Functions and Models

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Vector Calculus

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At what point does the curve $\mathbf { r } ( t ) = \left\langle t , t , 1 - a t ^ { 2 } \right\rangle , a > 0$ have maximum curvature? What is the maximum curvature?

Evaluate the integral $\int \left( \sin ^ { 3 } ( 2 t ) \cos ( 2 t ) \right) \mathbf { i } + t ^ { 2 } \ln t \mathbf { j } + \frac { e ^ { \sqrt { t } } } { \sqrt { t } } \mathbf { k }$

Find a parametric representation for the surface consisting of that part of the plane z = x + 3 that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .

Consider the curve in the xy-plane defined parametrically by x = t³ - 3t, y = t², z = 0. Sketch a rough graph of the curve.

Find the unit tangent vector T(t) to the curve r (t) = $\langle \sin t , t , \cos t \rangle$ when t = 0.

Find the tangent vector $\mathbf { r } ^ { \prime }$ (t) of the function r (t) = $\left( t , \sqrt { t } , \frac { 1 } { 2 \sqrt { t } } \right)$ when t = $\frac { 1 } { 4 }$ .

Find the curvature $K$ of the curve $y = 2 x ^ { 2 }$ at x = 0.

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 2 t , e ^ { t } \right\rangle$ . Find the velocity of the particle when t = 1.

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

Find a parametric representation for the surface consisting of that part of the elliptic paraboloid $x + y ^ { 2 } + 2 z ^ { 2 } = 4$ that lies in front of the plane x = 0.

Find the unit tangent vector T(t) to the curve r (t) = $\left( t ^ { 2 } - 1,3 t ^ { 2 } - t ^ { 4 } , \frac { 2 } { t } \right)$ when t = 1.

Evaluate the integral $\int \frac { 1 } { t } \mathbf { i } + e ^ { - t } \mathbf { j } + t ^ { \frac { 3 } { 2 } } \mathbf { k }$

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

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \langle 1 + t , 3 t , 3 - 5 t \rangle$ .

A particle is traveling along a helix whose vector equation is given by $\mathbf { r } ( t ) = \langle R \cos \alpha t , R \sin \alpha t , \beta t \rangle$ . Show that its velocity and acceleration are orthogonal at all times.

Find $\mathbf { r } ^ { t } ( t ) \text { if } \mathbf { r } ( t ) = t \sin t + \cos t \mathbf { i } + \frac { t } { t + 1 } \mathbf { j } - e ^ { \frac { 1 } { t } } \mathbf { k }$