Deck 14: Calculus of Vector-Valued Functions

Question 1

Find the point of intersection of the space curves , and .

Answer 1

Question 2

Find the curve traced by the following vector valued function and describe it with a Cartesian equation.

Answer 2

Question 3

Let be the intersection curve between the surfaces and .
If , , and is a parameterization of then:

A)
B)
C)
D)
E) cannot be parameterized by a parameterization in this form.

Answer 3

Question 4

Particle 1 and Particle 2 are flying through space. At time , the position of Particle 1 is given by and the position of Particle 2 is given by . Do the particles ever collide? If so, where?

Question 5

Parameterize the circle of radius with center located in the plane .

Question 6

Find a parameterization for the line of intersection of the planes and .

Question 7

Find the point of intersection between the curve and the plane .

Question 8

Parameterize the curve of intersection between the sphere and the cone and identify the curve.

Question 9

Parameterize the curve of intersection of the surfaces and .

Question 10

The curve intersects the plane at which of the following points?

A)
B)
C)
D)
E) The curve does not intersect the plane.

Question 11

Consider the linear paths and , described by and . Do the lines traced by and intersect? If so, where?

Question 12

Determine whether the paths and intersect.

Question 13

Consider the curve traced by .
This curve lies on both a sphere and a plane.

A) Find the equation of the sphere.
B) Find the equation of the plane.
C) Explain why the curve traced by lies on a circle.

Question 14

The curve intersects the plane at which of the following points?

A)
B) and
C)
D)
E) The curve does not intersect the - plane.

Question 15

Describe the curve traced by the following vector valued function, and sketch the graph of this curve.

Question 16

Determine the radius, center, and plane containing the circle parametrized by .

Question 17

Find where .

Question 18

Find a parametric equation for the tangent line to the path at the point where .

Question 19

Parameterize the curve of intersection of the hemisphere and the parabolic cylinder .

Question 20

Find the domain of the vector-valued function .

Question 21

Parametrize the tangent line to the curve at the point where .

Question 22

Find a point on the curve where the tangent line is parallel to the plane .

Question 23

Find the arc length of the curve .

Question 24

Find the tangent line to the curve at point .

Question 25

Find , where and :

A) by first computing the cross product and then differentiating.
B) using the cross product rule.

Question 26

Compute .

Question 27

A moving object has a position vector function

A) Find the speed of the object at time .
B) Find the distance traveled by the object between times and .
C) When does the object have minimum speed?

Question 28

Evaluate .

Question 29

Find the points on the curve where the tangent line is parallel to the plane

Question 30

Find the length of the curve described by the vector function .

Question 31

Find a vector parametrization for the tangent line to the curve at the point .

Question 32

The path of a particle satisfies At , the particle is located at:

A) .
B) .
C) .
D) .
E) The data are not sufficient.

Question 33

Find an arc length parametrization for the line .

Question 34

Find .

Question 35

Find the arc length parametrization of the curve .

Question 36

Evaluate the definite integrals.

A)
B)

Question 37

Let , , and .
A) Compute using the cross product and the dot product rules for differentiating.

A).
B) Compute the scalar triple product, differentiate it, and compare with the result in

Question 38

Let and

A) Compute and then differentiate the resulting function.
B) Find using the cross-product rule for differentiation.

Question 39

The points on the curve , where the tangent line is perpendicular to the plane , are:

A) .
B) .
C) and .
D) and .
E) There are no such points.

Question 40

Find , and for

Question 41

The path of a certain particle is parametrized by , . At what time is the minimum speed of the particle achieved?

Question 42

The path of a projectile is parametrized by . Find the projectile's speed when it reached the point .

Question 43

Find the arc length parametrization of the curve .

Question 44

Find the speed at time of the particle that is traveling along the curve .

Question 45

The curvature for is:

A) .
B) .
C) .
D) .
E) none of the above.

Question 46

Find the curvature of at .

Question 47

A particle moves along the curve The speed of the particle at is approximately:

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

Question 48

Find an arc length parametrization of the curve .

Question 49

Find the unit normal vector , the curvature , and the center of the osculating circle at , for the curve .

Question 50

The position of a particle traversing a circular path is given by . Find the speed of the particle at time

Question 51

Find the length of the curve for .

Question 52

Particle 1 follows the path parametrized by , while Particle 2 follows the path parametrized by . At what time does the speed of Particle 1 match the speed of Particle 2?

Question 53

Find the length of the curve described by the vector function .

Question 54

Find an arc length parametrization of the curve ,
and identify the curve.

Question 55

Find the length of the curve for .

Question 56

The length of the curve for is:

A) .
B) .
C) .
D) .
E) none of the above.

Question 57

The position of a particle for time is given by . Find the speed of the particle when it touches the curve parametrized by .

Question 58

The position of a particle is given by , . What is the limit of the particle's speed as

Question 59

Find the arc length of the Archimedes spiral for . Assume a is positive.

Question 60

The curvature of at the point is:

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

Question 61

A particle moves along the curve . Compute the velocity and acceleration vectors at .

Question 62

For the parametrization , find the curvature of the path at

Question 63

Find the unit normal vector to at .

Question 64

Find the curvature of the plane curve , and identify the point at which the curve has maximum curvature.

Question 65

Find the unit tangent of the curve at .

Question 66

The curvature of at is:

A) .
B) .
C) .
D) .
E) none of the above.

Question 67

Find a parametrization of the osculating circle at if .

Question 68

Let . Decompose into tangential and normal components at .

Question 69

For the parametrization , find the unit normal vector at .

Question 70

The acceleration vector of a moving particle is . Its initial position is , and its initial velocity is .
Find the vector position at time .

Question 71

The curvature of at the point where is:

A) .
B) .
C) .
D) 1.
E) .

Question 72

Consider the helix .

A) Find the unit tangent vector and the unit normal vector
B) Find the curvature

Question 73

Consider the helix .

A) Compute the center and the radius of the osculating circle at .
B) Compute the equation of the plane containing the osculating circle.

Question 74

A particle moves along the curve .
Compute the velocity and acceleration vectors at .

Question 75

Find the center and radius of the osculating circle at if .

Question 76

A particle moves so that , , and . The location of the particle at time is:

A) .
B) .
C) .
D) .
E) The data are not sufficient to give an answer.

Question 77

A particle moves along the curve .
Decompose the acceleration into tangential and normal components.

Question 78

Evaluate for the parametrization

Question 79

A particle has acceleration initial velocity , and initial position . Decompose into tangential and normal components.

Question 80

A path is parametrized by . Find the curvature of the path at the point .