Question 1

In $\mathbf { F } ( t ) = - \frac { \operatorname { GmM } \mathbf { r } ( t ) } { \| \mathbf { r } ( t ) \| ^ { 3 } }$ , m represents

Accepted Answer

A) The mass of a planet around the sun 
B) The mass of a star around the moon 
C) The gravitational constant 
D) The mass of the sun 
E) The mass of Earth 
A) The mass of a planet around the sun 
B) The mass of a star around the moon 
C) The gravitational constant 
D) The mass of the sun 
E) The mass of Earth

Question 2

The solution $\mathbf { r } ( t )$ to the differential equation $\mathbf { r } ^ { \prime } ( t ) = 6 t ^ { 2 } \mathbf { i } - e ^ { t } \mathbf { j } + e ^ { - t } \mathbf { k }$ with the condition $\mathbf { r } ( 0 ) = 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }$ is

Accepted Answer

A)  $- \left( 2 t ^ { 3 } + 3 \right) \mathbf { i } + \left( e ^ { t } - 2 \right) \mathbf { j } - \left( e ^ { - t } + 1 \right) \mathbf { k }$ 
B)  $- \left( 2 t ^ { 3 } + 3 \right) \mathbf { i } - \left( e ^ { t } - 2 \right) \mathbf { j } + \left( e ^ { - t } + 1 \right) \mathbf { k }$ 
C)  $\left( 2 t ^ { 3 } + 3 \right) \mathbf { i } + \left( e ^ { t } - 2 \right) \mathbf { j } - \left( e ^ { - t } + 1 \right) \mathbf { k }$ 
D)  $\left( 2 t ^ { 3 } + 3 \right) \mathbf { i } - \left( e ^ { t } - 2 \right) \mathbf { j } + \left( e ^ { - t } + 1 \right) \mathbf { k }$ 
E)  $\left( 2 t ^ { 3 } + 3 \right) \mathbf { i } - \left( e ^ { t } - 2 \right) \mathbf { j } - \left( e ^ { - t } + 1 \right) \mathbf { k }$ 
A)  $- \left( 2 t ^ { 3 } + 3 \right) \mathbf { i } + \left( e ^ { t } - 2 \right) \mathbf { j } - \left( e ^ { - t } + 1 \right) \mathbf { k }$ 
B)  $- \left( 2 t ^ { 3 } + 3 \right) \mathbf { i } - \left( e ^ { t } - 2 \right) \mathbf { j } + \left( e ^ { - t } + 1 \right) \mathbf { k }$ 
C)  $\left( 2 t ^ { 3 } + 3 \right) \mathbf { i } + \left( e ^ { t } - 2 \right) \mathbf { j } - \left( e ^ { - t } + 1 \right) \mathbf { k }$ 
D)  $\left( 2 t ^ { 3 } + 3 \right) \mathbf { i } - \left( e ^ { t } - 2 \right) \mathbf { j } + \left( e ^ { - t } + 1 \right) \mathbf { k }$ 
E)  $\left( 2 t ^ { 3 } + 3 \right) \mathbf { i } - \left( e ^ { t } - 2 \right) \mathbf { j } - \left( e ^ { - t } + 1 \right) \mathbf { k }$

Question 3

The speed of a particle moving along the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } - \cos ( 3 t ) \mathbf { j } + \sin ( 3 t ) \mathbf { k }$ is

Accepted Answer

A)  $\sqrt { 8 t ^ { 2 } - 1 }$ 
B)  $\sqrt { 11 }$ 
C)  $\sqrt { 8 t ^ { 2 } + 1 }$ 
D)  $\sqrt { 5 }$ 
E)  $\sqrt { 13 }$ 
A)  $\sqrt { 8 t ^ { 2 } - 1 }$ 
B)  $\sqrt { 11 }$ 
C)  $\sqrt { 8 t ^ { 2 } + 1 }$ 
D)  $\sqrt { 5 }$ 
E)  $\sqrt { 13 }$

Question 4

The radius of curvature of $\mathbf { r } ( t ) = e ^ { t } \mathbf { i } + e ^ { - t } \mathbf { j } + \mathbf { k }$ at the point corresponding to t = 0 is

Accepted Answer

A)  $4 \sqrt { 2 }$ 
B)  $\frac { 1 } { 2 \sqrt { 2 } }$ 
C) 2 
D)  $\sqrt { 2 }$ 
E)  $\frac { 7 } { 125 }$ 
A)  $4 \sqrt { 2 }$ 
B)  $\frac { 1 } { 2 \sqrt { 2 } }$ 
C) 2 
D)  $\sqrt { 2 }$ 
E)  $\frac { 7 } { 125 }$

Question 5

Let $\mathbf { a } ( t ) = - 9.5 \mathbf { k }$ be the acceleration of an object with velocity $\mathbf { v } ( t )$ . If $\mathbf { v } ( 0 ) = 2 \mathbf { i } + 3 \mathbf { j }$ then $\mathbf { v } ( t )$ is

Accepted Answer

A)  $- 2 \mathbf { i } + 3 \mathbf { j } - 9.5 t \mathbf { k }$ 
B)  $2 \mathbf { i } + 3 \mathbf { j } + 9.5 t \mathbf { k }$ 
C)  $2 \mathbf { i } - 3 \mathbf { j } + 9.5 t \mathbf { k }$ 
D)  $2 \mathbf { i } + 3 \mathbf { j } - 9.5 t \mathbf { k }$ 
E)  $2 \mathbf { i } - 3 \mathbf { j } - 9.5 t \mathbf { k }$ 
A)  $- 2 \mathbf { i } + 3 \mathbf { j } - 9.5 t \mathbf { k }$ 
B)  $2 \mathbf { i } + 3 \mathbf { j } + 9.5 t \mathbf { k }$ 
C)  $2 \mathbf { i } - 3 \mathbf { j } + 9.5 t \mathbf { k }$ 
D)  $2 \mathbf { i } + 3 \mathbf { j } - 9.5 t \mathbf { k }$ 
E)  $2 \mathbf { i } - 3 \mathbf { j } - 9.5 t \mathbf { k }$

Question 6

Let $\mathbf { r } ( t ) = 4 t \mathbf { i } - \cos t \mathbf { j } + \sin t \mathbf { k }$ Then $\mathbf { r } ^ { \prime } \left( \frac { \pi } { 6 } \right)$ is

Accepted Answer

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

Question 7

Let $y = 2 x ^ { 2 } - x ^ { 3 }$ . Then the curvature K at (-1, 3) is

Accepted Answer

A)  $\frac { 1 } { 25 \sqrt { 2 } }$ 
B)  $\frac { \sqrt { 2 } } { 11 }$ 
C)  $\frac { 2 } { 5 \sqrt { 5 } }$ 
D)  $\frac { 4 } { 17 \sqrt { 17 } }$ 
E)  $\frac { 1 } { 2 }$ 
A)  $\frac { 1 } { 25 \sqrt { 2 } }$ 
B)  $\frac { \sqrt { 2 } } { 11 }$ 
C)  $\frac { 2 } { 5 \sqrt { 5 } }$ 
D)  $\frac { 4 } { 17 \sqrt { 17 } }$ 
E)  $\frac { 1 } { 2 }$

Question 8

Let $\mathbf { r } ( t ) = \cos ( 2 t ) \mathbf { i } - \sin ( 2 t ) \mathbf { j } + 5 t \mathbf { k }$ Then $\mathbf { r } ^ { \prime } \left( \frac { \pi } { 4 } \right)$ is

Accepted Answer

A)  $- 5 \mathbf { i } + 2 \mathbf { k }$ 
B)  $2 \mathbf { i } - 5 \mathbf { k }$ 
C)  $- 2 \mathbf { i } + 5 \mathbf { k }$ 
D)  $- 2 \mathbf { i } - 5 \mathbf { k }$ 
E)  $2 \mathbf { i } + 5 \mathbf { k }$ 
A)  $- 5 \mathbf { i } + 2 \mathbf { k }$ 
B)  $2 \mathbf { i } - 5 \mathbf { k }$ 
C)  $- 2 \mathbf { i } + 5 \mathbf { k }$ 
D)  $- 2 \mathbf { i } - 5 \mathbf { k }$ 
E)  $2 \mathbf { i } + 5 \mathbf { k }$

Question 9

The normal component $a _ { \mathrm { N } } ( t )$ of the acceleration of a particle moving along the curve $\mathbf { r } ( t ) = \sin ( 2 t ) \mathbf { i } - \cos ( 2 t ) \mathbf { j } + 2 t \mathbf { k }$ is

Accepted Answer

A) 4 
B)  $\sqrt { 8 t ^ { 2 } + 1 }$ 
C)  $\sqrt { 56 }$ 
D)  $\sqrt { 65 }$ 
E)  $\frac { 8 t } { \sqrt { 8 t ^ { 2 } + 1 } }$ 
A) 4 
B)  $\sqrt { 8 t ^ { 2 } + 1 }$ 
C)  $\sqrt { 56 }$ 
D)  $\sqrt { 65 }$ 
E)  $\frac { 8 t } { \sqrt { 8 t ^ { 2 } + 1 } }$

Question 10

The magnitude of the acceleration of a particle moving along the curve $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } - t \mathbf { j } - t ^ { 2 } \mathbf { k }$ is

Accepted Answer

A)  $2 \sqrt { 2 }$ 
B)  $\sqrt { 11 }$ 
C)  $\sqrt { 8 t ^ { 2 } + 1 }$ 
D)  $\sqrt { 5 }$ 
E)  $\sqrt { 8 t ^ { 2 } - 1 }$ 
A)  $2 \sqrt { 2 }$ 
B)  $\sqrt { 11 }$ 
C)  $\sqrt { 8 t ^ { 2 } + 1 }$ 
D)  $\sqrt { 5 }$ 
E)  $\sqrt { 8 t ^ { 2 } - 1 }$

Question 11

Let $y = 2 x ^ { 2 }$ . Then the curvature K at (1, 2) is

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 13 }$ 
B)  $\frac { \sqrt { 2 } } { 11 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $\frac { 4 } { 17 \sqrt { 17 } }$ 
E)  $\frac{4}{\sqrt{17}}$ 
A)  $\frac { \sqrt { 2 } } { 13 }$ 
B)  $\frac { \sqrt { 2 } } { 11 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $\frac { 4 } { 17 \sqrt { 17 } }$ 
E)  $\frac{4}{\sqrt{17}}$

Question 12

Let $\mathbf { r } ( t ) = ( 2 - 3 t ) \mathbf { i } + 4 t \mathbf { j } - ( 1 - t ) \mathbf { k }$ Then $\mathbf { r } ^ { \prime } ( 0 )$ is

Accepted Answer

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

Question 13

Let $\mathbf { r } ( t ) = \sin ( 3 t ) \mathbf { i } + \cos ( 3 t ) \mathbf { j } - t \mathbf { k }$ Then the unit normal vector $\mathbf { n } \left( \frac { \pi } { 6 } \right)$ is

Accepted Answer

A)  $- \mathbf { i }$ 
B) k 
C) j 
D) i 
E)  $- \mathbf { j }$ 
A)  $- \mathbf { i }$ 
B) k 
C) j 
D) i 
E)  $- \mathbf { j }$

Question 14

Let $\mathbf { r } ( t ) = \cos ( 2 t ) \mathbf { i } + \sin ( 2 t ) \mathbf { j } + 4 t \mathbf { k }$ Then the unit tangent vector $\mathbf { T } \left( \frac { \pi } { 4 } \right)$ is

Accepted Answer

A)  $\frac { 1 } { \sqrt { 5 } } \mathbf { i } + \frac { 4 } { \sqrt { 5 } } \mathbf { k }$ 
B)  $- \frac { 1 } { \sqrt { 5 } } \mathbf { i } + \frac { 4 } { \sqrt { 5 } } \mathbf { k }$ 
C)  $- \frac { 1 } { \sqrt { 5 } } \mathbf { i } - \frac { 4 } { \sqrt { 5 } } \mathbf { k }$ 
D)  $- \frac { 4 } { \sqrt { 5 } } \mathbf { i } + \frac { 1 } { \sqrt { 5 } } \mathbf { k }$ 
E)  $\frac { 4 } { \sqrt { 5 } } \mathbf { i } + \frac { 1 } { \sqrt { 5 } } \mathbf { k }$ 
A)  $\frac { 1 } { \sqrt { 5 } } \mathbf { i } + \frac { 4 } { \sqrt { 5 } } \mathbf { k }$ 
B)  $- \frac { 1 } { \sqrt { 5 } } \mathbf { i } + \frac { 4 } { \sqrt { 5 } } \mathbf { k }$ 
C)  $- \frac { 1 } { \sqrt { 5 } } \mathbf { i } - \frac { 4 } { \sqrt { 5 } } \mathbf { k }$ 
D)  $- \frac { 4 } { \sqrt { 5 } } \mathbf { i } + \frac { 1 } { \sqrt { 5 } } \mathbf { k }$ 
E)  $\frac { 4 } { \sqrt { 5 } } \mathbf { i } + \frac { 1 } { \sqrt { 5 } } \mathbf { k }$

Question 15

Let $\mathbf { a } ( t ) = - 9.5 \mathbf { k }$ be the acceleration of an object with velocity $\mathbf { v } ( t )$ . If $\mathbf { v } ( 0 ) = \mathbf { i } + 2 \mathbf { k }$ then $\mathbf { v } ( t )$ is

Accepted Answer

A)  $- \mathbf { i } + 2 \mathbf { j } + 9.5 t \mathbf { k }$ 
B)  $\mathbf { i } + 2 \mathbf { j } + 9.5 t \mathbf { k }$ 
C)  $\mathbf { i } + 2 \mathbf { j } - 9.5 t \mathbf { k }$ 
D)  $\mathbf { i } - 2 \mathbf { j } - 9.5 t \mathbf { k }$ 
E)  $- \mathbf { i } + 2 \mathbf { j } - 9.5 t \mathbf { k }$ 
A)  $- \mathbf { i } + 2 \mathbf { j } + 9.5 t \mathbf { k }$ 
B)  $\mathbf { i } + 2 \mathbf { j } + 9.5 t \mathbf { k }$ 
C)  $\mathbf { i } + 2 \mathbf { j } - 9.5 t \mathbf { k }$ 
D)  $\mathbf { i } - 2 \mathbf { j } - 9.5 t \mathbf { k }$ 
E)  $- \mathbf { i } + 2 \mathbf { j } - 9.5 t \mathbf { k }$

Question 16

Which of the following is Kepler's Second Law of Motion?
​

Accepted Answer

A) The speed of a planet around the sun is a constant. 
B) The speed of a planet varies so that the line joining the sun to the planet sweeps out equal areas in equal time. 
C) The planets travel in circular orbits with the sun at the center. 
D) The planets travel in circular orbits with Earth at the center. 
E) The planets travel in circular orbits with the moon at the center. 
A) The speed of a planet around the sun is a constant. 
B) The speed of a planet varies so that the line joining the sun to the planet sweeps out equal areas in equal time. 
C) The planets travel in circular orbits with the sun at the center. 
D) The planets travel in circular orbits with Earth at the center. 
E) The planets travel in circular orbits with the moon at the center.

Question 17

Let $\mathbf { r } ( t ) = ( 2 - 3 t ) \mathbf { i } + 4 t \mathbf { j } - ( 1 - t ) \mathbf { k }$ Then the curvature K(t) is

Accepted Answer

A) 0 
B)  $\frac { 4 } { 13 }$ 
C)  $\frac { 4 } { 5 }$ 
D) 2 
E)  $\frac { 1 } { 2 }$ 
A) 0 
B)  $\frac { 4 } { 13 }$ 
C)  $\frac { 4 } { 5 }$ 
D) 2 
E)  $\frac { 1 } { 2 }$

Question 18

Let $\mathbf { a } ( t ) = - 9.5 \mathbf { k }$ be the acceleration of an object with velocity $\mathbf { v } ( t )$ . If $\mathbf { v } ( 0 ) = 2 \mathbf { i } + 3 \mathbf { j }$ then the speed $\| \mathbf { v } ( t ) \|$ is

Accepted Answer

A)  $\sqrt { 90.25 t ^ { 2 } + 15 }$ 
B)  $\sqrt { 90.25 t ^ { 2 } - 25 }$ 
C)  $\sqrt { 90.25 t ^ { 2 } + 13 }$ 
D)  $\sqrt { 90.25 t ^ { 2 } + 5 }$ 
E)  $\sqrt { 90.25 t ^ { 2 } - 13 }$ 
A)  $\sqrt { 90.25 t ^ { 2 } + 15 }$ 
B)  $\sqrt { 90.25 t ^ { 2 } - 25 }$ 
C)  $\sqrt { 90.25 t ^ { 2 } + 13 }$ 
D)  $\sqrt { 90.25 t ^ { 2 } + 5 }$ 
E)  $\sqrt { 90.25 t ^ { 2 } - 13 }$

Question 19

The solution $\mathbf { r } ( t )$ to the differential equation $\mathbf { r } ^ { \prime } ( t ) = e ^ { t } \mathbf { i } - 3 t \mathbf { j } + 3 t ^ { 2 } \mathbf { k }$ with the condition $\mathbf { r } ( 1 ) = \mathbf { j } + \mathbf { k }$ is

Accepted Answer

A)  $- \left( e ^ { t } - e \right) \mathbf { i } - \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } + t ^ { 3 } \mathbf { k }$ 
B)  $- \left( e ^ { t } - e \right) \mathbf { i } + \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } + t ^ { 3 } \mathbf { k }$ 
C)  $\left( e ^ { t } - e \right) \mathbf { i } - \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } + t ^ { 3 } \mathbf { k }$ 
D)  $\left( e ^ { t } - e \right) \mathbf { i } + \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } + t ^ { 3 } \mathbf { k }$ 
E)  $\left( e ^ { t } - e \right) \mathbf { i } - \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } - t ^ { 3 } \mathbf { k }$ 
A)  $- \left( e ^ { t } - e \right) \mathbf { i } - \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } + t ^ { 3 } \mathbf { k }$ 
B)  $- \left( e ^ { t } - e \right) \mathbf { i } + \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } + t ^ { 3 } \mathbf { k }$ 
C)  $\left( e ^ { t } - e \right) \mathbf { i } - \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } + t ^ { 3 } \mathbf { k }$ 
D)  $\left( e ^ { t } - e \right) \mathbf { i } + \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } + t ^ { 3 } \mathbf { k }$ 
E)  $\left( e ^ { t } - e \right) \mathbf { i } - \left( \frac { 3 } { 2 } t ^ { 2 } - \frac { 5 } { 2 } \right) \mathbf { j } - t ^ { 3 } \mathbf { k }$

Question 20

According to Kepler's First Law, which of the following is located at a focus of an ellipse?
​

Accepted Answer

A) The moon 
B) The sun 
C) Mars 
D) Earth 
E) Mercury 
A) The moon 
B) The sun 
C) Mars 
D) Earth 
E) Mercury

In $\mathbf { F } ( t ) = - \frac { \operatorname { GmM } \mathbf { r } ( t ) } { \| \mathbf { r } ( t ) \| ^ { 3 } }$ , m represents

The solution $\mathbf { r } ( t )$ to the differential equation $\mathbf { r } ^ { \prime } ( t ) = 6 t ^ { 2 } \mathbf { i } - e ^ { t } \mathbf { j } + e ^ { - t } \mathbf { k }$ with the condition $\mathbf { r } ( 0 ) = 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }$ is

The speed of a particle moving along the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } - \cos ( 3 t ) \mathbf { j } + \sin ( 3 t ) \mathbf { k }$ is

The radius of curvature of $\mathbf { r } ( t ) = e ^ { t } \mathbf { i } + e ^ { - t } \mathbf { j } + \mathbf { k }$ at the point corresponding to t = 0 is

Let $\mathbf { a } ( t ) = - 9.5 \mathbf { k }$ be the acceleration of an object with velocity $\mathbf { v } ( t )$ . If $\mathbf { v } ( 0 ) = 2 \mathbf { i } + 3 \mathbf { j }$ then $\mathbf { v } ( t )$ is

Let $\mathbf { r } ( t ) = 4 t \mathbf { i } - \cos t \mathbf { j } + \sin t \mathbf { k }$ Then $\mathbf { r } ^ { \prime } \left( \frac { \pi } { 6 } \right)$ is

Let $y = 2 x ^ { 2 } - x ^ { 3 }$ . Then the curvature K at (-1, 3) is

Let $\mathbf { r } ( t ) = \cos ( 2 t ) \mathbf { i } - \sin ( 2 t ) \mathbf { j } + 5 t \mathbf { k }$ Then $\mathbf { r } ^ { \prime } \left( \frac { \pi } { 4 } \right)$ is

The normal component $a _ { \mathrm { N } } ( t )$ of the acceleration of a particle moving along the curve $\mathbf { r } ( t ) = \sin ( 2 t ) \mathbf { i } - \cos ( 2 t ) \mathbf { j } + 2 t \mathbf { k }$ is

The magnitude of the acceleration of a particle moving along the curve $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } - t \mathbf { j } - t ^ { 2 } \mathbf { k }$ is

Let $y = 2 x ^ { 2 }$ . Then the curvature K at (1, 2) is

Let $\mathbf { r } ( t ) = ( 2 - 3 t ) \mathbf { i } + 4 t \mathbf { j } - ( 1 - t ) \mathbf { k }$ Then $\mathbf { r } ^ { \prime } ( 0 )$ is

Let $\mathbf { r } ( t ) = \sin ( 3 t ) \mathbf { i } + \cos ( 3 t ) \mathbf { j } - t \mathbf { k }$ Then the unit normal vector $\mathbf { n } \left( \frac { \pi } { 6 } \right)$ is

Let $\mathbf { r } ( t ) = \cos ( 2 t ) \mathbf { i } + \sin ( 2 t ) \mathbf { j } + 4 t \mathbf { k }$ Then the unit tangent vector $\mathbf { T } \left( \frac { \pi } { 4 } \right)$ is

Let $\mathbf { a } ( t ) = - 9.5 \mathbf { k }$ be the acceleration of an object with velocity $\mathbf { v } ( t )$ . If $\mathbf { v } ( 0 ) = \mathbf { i } + 2 \mathbf { k }$ then $\mathbf { v } ( t )$ is

Which of the following is Kepler's Second Law of Motion?

Let $\mathbf { r } ( t ) = ( 2 - 3 t ) \mathbf { i } + 4 t \mathbf { j } - ( 1 - t ) \mathbf { k }$ Then the curvature K(t) is

Let $\mathbf { a } ( t ) = - 9.5 \mathbf { k }$ be the acceleration of an object with velocity $\mathbf { v } ( t )$ . If $\mathbf { v } ( 0 ) = 2 \mathbf { i } + 3 \mathbf { j }$ then the speed $\| \mathbf { v } ( t ) \|$ is

The solution $\mathbf { r } ( t )$ to the differential equation $\mathbf { r } ^ { \prime } ( t ) = e ^ { t } \mathbf { i } - 3 t \mathbf { j } + 3 t ^ { 2 } \mathbf { k }$ with the condition $\mathbf { r } ( 1 ) = \mathbf { j } + \mathbf { k }$ is

According to Kepler's First Law, which of the following is located at a focus of an ellipse?

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Exam 12: Vector Functions

In F(t)=−GmM⁡r(t)∥r(t)∥3\mathbf { F } ( t ) = - \frac { \operatorname { GmM } \mathbf { r } ( t ) } { \| \mathbf { r } ( t ) \| ^ { 3 } }F(t)=−∥r(t)∥3GmMr(t)​ , m represents

The speed of a particle moving along the curve r(t)=2ti−cos⁡(3t)j+sin⁡(3t)k\mathbf { r } ( t ) = 2 t \mathbf { i } - \cos ( 3 t ) \mathbf { j } + \sin ( 3 t ) \mathbf { k }r(t)=2ti−cos(3t)j+sin(3t)k is

The radius of curvature of r(t)=eti+e−tj+k\mathbf { r } ( t ) = e ^ { t } \mathbf { i } + e ^ { - t } \mathbf { j } + \mathbf { k }r(t)=eti+e−tj+k at the point corresponding to t = 0 is

Let a(t)=−9.5k\mathbf { a } ( t ) = - 9.5 \mathbf { k }a(t)=−9.5k be the acceleration of an object with velocity v(t)\mathbf { v } ( t )v(t) . If v(0)=2i+3j\mathbf { v } ( 0 ) = 2 \mathbf { i } + 3 \mathbf { j }v(0)=2i+3j then v(t)\mathbf { v } ( t )v(t) is

Let r(t)=4ti−cos⁡tj+sin⁡tk\mathbf { r } ( t ) = 4 t \mathbf { i } - \cos t \mathbf { j } + \sin t \mathbf { k }r(t)=4ti−costj+sintk Then r′(π6)\mathbf { r } ^ { \prime } \left( \frac { \pi } { 6 } \right)r′(6π​) is

Let y=2x2−x3y = 2 x ^ { 2 } - x ^ { 3 }y=2x2−x3 . Then the curvature K at (-1, 3) is

Let r(t)=cos⁡(2t)i−sin⁡(2t)j+5tk\mathbf { r } ( t ) = \cos ( 2 t ) \mathbf { i } - \sin ( 2 t ) \mathbf { j } + 5 t \mathbf { k }r(t)=cos(2t)i−sin(2t)j+5tk Then r′(π4)\mathbf { r } ^ { \prime } \left( \frac { \pi } { 4 } \right)r′(4π​) is

The normal component aN(t)a _ { \mathrm { N } } ( t )aN​(t) of the acceleration of a particle moving along the curve r(t)=sin⁡(2t)i−cos⁡(2t)j+2tk\mathbf { r } ( t ) = \sin ( 2 t ) \mathbf { i } - \cos ( 2 t ) \mathbf { j } + 2 t \mathbf { k }r(t)=sin(2t)i−cos(2t)j+2tk is

The magnitude of the acceleration of a particle moving along the curve r(t)=t2i−tj−t2k\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } - t \mathbf { j } - t ^ { 2 } \mathbf { k }r(t)=t2i−tj−t2k is

Let y=2x2y = 2 x ^ { 2 }y=2x2 . Then the curvature K at (1, 2) is

Let r(t)=(2−3t)i+4tj−(1−t)k\mathbf { r } ( t ) = ( 2 - 3 t ) \mathbf { i } + 4 t \mathbf { j } - ( 1 - t ) \mathbf { k }r(t)=(2−3t)i+4tj−(1−t)k Then r′(0)\mathbf { r } ^ { \prime } ( 0 )r′(0) is

Let r(t)=sin⁡(3t)i+cos⁡(3t)j−tk\mathbf { r } ( t ) = \sin ( 3 t ) \mathbf { i } + \cos ( 3 t ) \mathbf { j } - t \mathbf { k }r(t)=sin(3t)i+cos(3t)j−tk Then the unit normal vector n(π6)\mathbf { n } \left( \frac { \pi } { 6 } \right)n(6π​) is

Let r(t)=cos⁡(2t)i+sin⁡(2t)j+4tk\mathbf { r } ( t ) = \cos ( 2 t ) \mathbf { i } + \sin ( 2 t ) \mathbf { j } + 4 t \mathbf { k }r(t)=cos(2t)i+sin(2t)j+4tk Then the unit tangent vector T(π4)\mathbf { T } \left( \frac { \pi } { 4 } \right)T(4π​) is

Let a(t)=−9.5k\mathbf { a } ( t ) = - 9.5 \mathbf { k }a(t)=−9.5k be the acceleration of an object with velocity v(t)\mathbf { v } ( t )v(t) . If v(0)=i+2k\mathbf { v } ( 0 ) = \mathbf { i } + 2 \mathbf { k }v(0)=i+2k then v(t)\mathbf { v } ( t )v(t) is

Which of the following is Kepler's Second Law of Motion? ​

Let r(t)=(2−3t)i+4tj−(1−t)k\mathbf { r } ( t ) = ( 2 - 3 t ) \mathbf { i } + 4 t \mathbf { j } - ( 1 - t ) \mathbf { k }r(t)=(2−3t)i+4tj−(1−t)k Then the curvature K(t) is

Let a(t)=−9.5k\mathbf { a } ( t ) = - 9.5 \mathbf { k }a(t)=−9.5k be the acceleration of an object with velocity v(t)\mathbf { v } ( t )v(t) . If v(0)=2i+3j\mathbf { v } ( 0 ) = 2 \mathbf { i } + 3 \mathbf { j }v(0)=2i+3j then the speed ∥v(t)∥\| \mathbf { v } ( t ) \|∥v(t)∥ is

According to Kepler's First Law, which of the following is located at a focus of an ellipse? ​

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

In $\mathbf { F } ( t ) = - \frac { \operatorname { GmM } \mathbf { r } ( t ) } { \| \mathbf { r } ( t ) \| ^ { 3 } }$ , m represents

The speed of a particle moving along the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } - \cos ( 3 t ) \mathbf { j } + \sin ( 3 t ) \mathbf { k }$ is

The radius of curvature of $\mathbf { r } ( t ) = e ^ { t } \mathbf { i } + e ^ { - t } \mathbf { j } + \mathbf { k }$ at the point corresponding to t = 0 is

Let $\mathbf { a } ( t ) = - 9.5 \mathbf { k }$ be the acceleration of an object with velocity $\mathbf { v } ( t )$ . If $\mathbf { v } ( 0 ) = 2 \mathbf { i } + 3 \mathbf { j }$ then $\mathbf { v } ( t )$ is

Let $\mathbf { r } ( t ) = 4 t \mathbf { i } - \cos t \mathbf { j } + \sin t \mathbf { k }$ Then $\mathbf { r } ^ { \prime } \left( \frac { \pi } { 6 } \right)$ is

Let $y = 2 x ^ { 2 } - x ^ { 3 }$ . Then the curvature K at (-1, 3) is

Let $\mathbf { r } ( t ) = \cos ( 2 t ) \mathbf { i } - \sin ( 2 t ) \mathbf { j } + 5 t \mathbf { k }$ Then $\mathbf { r } ^ { \prime } \left( \frac { \pi } { 4 } \right)$ is

The normal component $a _ { \mathrm { N } } ( t )$ of the acceleration of a particle moving along the curve $\mathbf { r } ( t ) = \sin ( 2 t ) \mathbf { i } - \cos ( 2 t ) \mathbf { j } + 2 t \mathbf { k }$ is

The magnitude of the acceleration of a particle moving along the curve $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } - t \mathbf { j } - t ^ { 2 } \mathbf { k }$ is

Let $y = 2 x ^ { 2 }$ . Then the curvature K at (1, 2) is

Let $\mathbf { r } ( t ) = ( 2 - 3 t ) \mathbf { i } + 4 t \mathbf { j } - ( 1 - t ) \mathbf { k }$ Then $\mathbf { r } ^ { \prime } ( 0 )$ is

Let $\mathbf { r } ( t ) = \sin ( 3 t ) \mathbf { i } + \cos ( 3 t ) \mathbf { j } - t \mathbf { k }$ Then the unit normal vector $\mathbf { n } \left( \frac { \pi } { 6 } \right)$ is

Let $\mathbf { r } ( t ) = \cos ( 2 t ) \mathbf { i } + \sin ( 2 t ) \mathbf { j } + 4 t \mathbf { k }$ Then the unit tangent vector $\mathbf { T } \left( \frac { \pi } { 4 } \right)$ is

Let $\mathbf { a } ( t ) = - 9.5 \mathbf { k }$ be the acceleration of an object with velocity $\mathbf { v } ( t )$ . If $\mathbf { v } ( 0 ) = \mathbf { i } + 2 \mathbf { k }$ then $\mathbf { v } ( t )$ is

Which of the following is Kepler's Second Law of Motion?

Let $\mathbf { r } ( t ) = ( 2 - 3 t ) \mathbf { i } + 4 t \mathbf { j } - ( 1 - t ) \mathbf { k }$ Then the curvature K(t) is

Let $\mathbf { a } ( t ) = - 9.5 \mathbf { k }$ be the acceleration of an object with velocity $\mathbf { v } ( t )$ . If $\mathbf { v } ( 0 ) = 2 \mathbf { i } + 3 \mathbf { j }$ then the speed $\| \mathbf { v } ( t ) \|$ is

According to Kepler's First Law, which of the following is located at a focus of an ellipse?