Question 1

Find the arc length of the graph of

Accepted Answer

A)    
B)    
C)  21 
D)    
E)  0 
A)    
B)    
C)  21 
D)    
E)  0

Question 2

Find the arc length parameterization of the curve given by   that has the same orientation and uses   as a reference point.

Accepted Answer

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

Question 3

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = $\langle$ 4 cos t, sin t$\rangle$at   seconds, then sketch the path of the particle together with the velocity and acceleration vectors at   seconds.

Accepted Answer

v(t) = @#LAT-DLM& -4 sin t, co

Question 4

Find the vector equation of the line which is perpendicular to the curve r(t) = $\langle$2 cos 3t, 2 sin 3t, 8t$\rangle$ at   .

Accepted Answer

The answer of Find the vector equation of the line...

Question 5

Find the direction cosines of the tangent and normal vectors to x = e t sin 2t, y = e t cos 2t, z = 2e t at t = 0.

Accepted Answer

The answer of Find the direction cosines of the tangent...

Question 6

Find the arc length of the graph of

Accepted Answer

A)  2 
B)    
C)  0 
D)    
E)  4 
A)  2 
B)    
C)  0 
D)    
E)  4

Question 7

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = 10 + e t i + e t cos t j + e t sin t k at t = 0 seconds.

Accepted Answer

v(0) = i +

Question 8

Find the arc length parameterization of the curve given by   that has the same orientation and uses   ) as a reference point.

Accepted Answer

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

Question 9

An object in orbit has rmin = 1024km and e = 0.49. Find rmax.

Accepted Answer

A)  2.92 *1026km 
B)  3.21 * 1026km 
C)  3.21 * 1025km 
D)  2.92 * 1024km 
E)  2.92 *1025km 
A)  2.92 *1026km 
B)  3.21 * 1026km 
C)  3.21 * 1025km 
D)  2.92 * 1024km 
E)  2.92 *1025km

Question 10

If, for an elliptical orbit, rmax = 1025km and e = 0.51, find a, the semimajor axis.

Accepted Answer

A)  3.25*1024km 
B)  6.62 *1025km 
C)  6.62 *1024km 
D)  5.96 * 1024km 
E)  7.28 * 1024km 
A)  3.25*1024km 
B)  6.62 *1025km 
C)  6.62 *1024km 
D)  5.96 * 1024km 
E)  7.28 * 1024km

Question 11

At t increases, the graph of   sketches

Accepted Answer

A)  Clockwise and up 
B)  Counter-clockwise and up 
C)  Clockwise and down 
D)  Counter-clockwise and down 
E)  In a circle 
A)  Clockwise and up 
B)  Counter-clockwise and up 
C)  Clockwise and down 
D)  Counter-clockwise and down 
E)  In a circle

Question 12

r(t) = 4t3 i + (17 + 2t)j is the position vector of a particle moving in a plane. Find the acceleration.

Accepted Answer

A)  12t2 i + 2j 
B)  12i 
C)  24t i + 2j 
D)  24t i 
E)  24i 
A)  12t2 i + 2j 
B)  12i 
C)  24t i + 2j 
D)  24t i 
E)  24i

Question 13

Find the unit tangent and unit normal vectors to r(t) = (e t sin t + 3)i + e t cos t j at t = 0.

Accepted Answer

Question 14

If an asteroid has an orbit with eccentricity 0.59 and semi-major axis a = 130000000, find its maximum distance from the center of the sun.

Accepted Answer

The answer of If an asteroid has an orbit with...

Question 15

If an object orbits the sun with rmax =   miles and rmin =   miles, the elliptical orbit has eccentricity

Accepted Answer

A)    
B)    
C)  40 
D)    
E)  1 
A)    
B)    
C)  40 
D)    
E)  1

Question 16

Find parametric equations for   using arc length, s, as a parameter. Use the point on the curve where t = 0 as the reference point.

Accepted Answer

The answer of Find parametric equations for   using...

Question 17

If   and   , then k(t) at t = 0 is

Accepted Answer

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

Question 18

Find the curvature   for r(t) = cos t i + sin t j - 10k.

Accepted Answer

A)    
B)  1 
C)  sin2t - cos2t 
D)  sin2t + cos2t 
E)  - sin2t + cos2t 
A)    
B)  1 
C)  sin2t - cos2t 
D)  sin2t + cos2t 
E)  - sin2t + cos2t

Question 19

If, for an elliptical orbit, rmax = 1025km and e = 0.79, find a, the semi-major axis.

Accepted Answer

A)  1.17 *1025km 
B)  4.76 *1025km 
C)  5.59 * 1025km 
D)  5.59 * 1024km 
E)  1.17 * 1024km 
A)  1.17 *1025km 
B)  4.76 *1025km 
C)  5.59 * 1025km 
D)  5.59 * 1024km 
E)  1.17 * 1024km

Question 20

If r(t) = (2t2 - 5)i + (t - 2)j + (4t + 10)k, find the curvature k(t) at t = 1.

Accepted Answer

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

Find the arc length of the graph of

Find the arc length parameterization of the curve given by that has the same orientation and uses as a reference point.

Find the vector equation of the line which is perpendicular to the curve r(t) = $\langle$ 2 cos 3t, 2 sin 3t, 8t $\rangle$ at $Find the vector equation of the line which is perpendicular to the curve r(t) = \langle 2 cos 3t, 2 sin 3t, 8t \rangle at .$ .

Find the direction cosines of the tangent and normal vectors to x = e ^t sin 2t, y = e ^t cos 2t, z = 2e ^t at t = 0.

Find the arc length of the graph of

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = 10 + e ^t i + e ^t cos t j + e ^t sin t k at t = 0 seconds.

Find the arc length parameterization of the curve given by that has the same orientation and uses ) as a reference point.

An object in orbit has r_min = 10²⁴km and e = 0.49. Find r_max.

If, for an elliptical orbit, r_max = 10²⁵km and e = 0.51, find a, the semimajor axis.

At t increases, the graph of sketches

r(t) = 4t³ i + (17 + 2t)j is the position vector of a particle moving in a plane. Find the acceleration.

Find the unit tangent and unit normal vectors to r(t) = (e ^t sin t + 3)i + e ^t cos t j at t = 0.

If an asteroid has an orbit with eccentricity 0.59 and semi-major axis a = 130000000, find its maximum distance from the center of the sun.

If an object orbits the sun with r_max = miles and r_min = miles, the elliptical orbit has eccentricity

Find parametric equations for using arc length, s, as a parameter. Use the point on the curve where t = 0 as the reference point.

If and , then k(t) at t = 0 is

Find the curvature for r(t) = cos t i + sin t j - 10k.

If, for an elliptical orbit, r_max = 10²⁵km and e = 0.79, find a, the semi-major axis.

If r(t) = (2t² - 5)i + (t - 2)j + (4t + 10)k, find the curvature k(t) at t = 1.

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Partial Derivatives

Multiple Integrals

Topics in Vector Calculus

Filters

Exam 12: Vector-Valued Functions

Find the arc length of the graph of

Find the arc length parameterization of the curve given by that has the same orientation and uses as a reference point.

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = ⟨\langle⟨ 4 cos t, sin t ⟩\rangle⟩ at seconds, then sketch the path of the particle together with the velocity and acceleration vectors at seconds.

Find the vector equation of the line which is perpendicular to the curve r(t) = ⟨\langle⟨ 2 cos 3t, 2 sin 3t, 8t ⟩\rangle⟩ at .

Find the direction cosines of the tangent and normal vectors to x = e t sin 2t, y = e t cos 2t, z = 2e t at t = 0.

Find the arc length of the graph of

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = 10 + e t i + e t cos t j + e t sin t k at t = 0 seconds.

Find the arc length parameterization of the curve given by that has the same orientation and uses ) as a reference point.

An object in orbit has rmin = 1024km and e = 0.49. Find rmax.

If, for an elliptical orbit, rmax = 1025km and e = 0.51, find a, the semimajor axis.

At t increases, the graph of sketches

r(t) = 4t3 i + (17 + 2t)j is the position vector of a particle moving in a plane. Find the acceleration.

Find the unit tangent and unit normal vectors to r(t) = (e t sin t + 3)i + e t cos t j at t = 0.

If an asteroid has an orbit with eccentricity 0.59 and semi-major axis a = 130000000, find its maximum distance from the center of the sun.

If an object orbits the sun with rmax = miles and rmin = miles, the elliptical orbit has eccentricity

Find parametric equations for using arc length, s, as a parameter. Use the point on the curve where t = 0 as the reference point.

If and , then k(t) at t = 0 is

Find the curvature for r(t) = cos t i + sin t j - 10k.

If, for an elliptical orbit, rmax = 1025km and e = 0.79, find a, the semi-major axis.

If r(t) = (2t2 - 5)i + (t - 2)j + (4t + 10)k, find the curvature k(t) at t = 1.

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Partial Derivatives

Multiple Integrals

Topics in Vector Calculus

Filters

Find the vector equation of the line which is perpendicular to the curve r(t) = $\langle$ 2 cos 3t, 2 sin 3t, 8t $\rangle$ at $Find the vector equation of the line which is perpendicular to the curve r(t) = \langle 2 cos 3t, 2 sin 3t, 8t \rangle at .$ .

Find the direction cosines of the tangent and normal vectors to x = e ^t sin 2t, y = e ^t cos 2t, z = 2e ^t at t = 0.

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = 10 + e ^t i + e ^t cos t j + e ^t sin t k at t = 0 seconds.

An object in orbit has r_min = 10²⁴km and e = 0.49. Find r_max.

If, for an elliptical orbit, r_max = 10²⁵km and e = 0.51, find a, the semimajor axis.

r(t) = 4t³ i + (17 + 2t)j is the position vector of a particle moving in a plane. Find the acceleration.

Find the unit tangent and unit normal vectors to r(t) = (e ^t sin t + 3)i + e ^t cos t j at t = 0.

If an object orbits the sun with r_max = miles and r_min = miles, the elliptical orbit has eccentricity

If, for an elliptical orbit, r_max = 10²⁵km and e = 0.79, find a, the semi-major axis.

If r(t) = (2t² - 5)i + (t - 2)j + (4t + 10)k, find the curvature k(t) at t = 1.