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Exam 14: Calculus of Vector-Valued Functions

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Exam 1: Precalculus Review74 Questionsadd course
Exam 2: Limits97 Questionsadd course
Exam 3: Differentiation81 Questionsadd course
Exam 4: Applications of the Derivative77 Questionsadd course
Exam 5: The Integral82 Questionsadd course
Exam 6: Applications of the Integral80 Questionsadd course
Exam 7: Exponential Functions106 Questionsadd course
Exam 8: Techniques of Integration101 Questionsadd course
Exam 9: Further Applications of the Integral and Taylor Polynomials100 Questionsadd course
Exam 10: Introduction to Differential Equations73 Questionsadd course
Exam 11: Infinite Series95 Questionsadd course
Exam 12: Parametric Equations, Polar Coordinates, and Conic Sections71 Questionsadd course
Exam 13: Vector Geometry96 Questionsadd course
Exam 14: Calculus of Vector-Valued Functions99 Questionsadd course
Exam 15: Differentiation in Several Variables95 Questionsadd course
Exam 16: Multiple Integration98 Questionsadd course
Exam 17: Line and Surface Integrals92 Questionsadd course
Exam 18: Fundamental Theorems of Vector Analysis91 Questionsadd course
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Particle 1 follows the path parametrized by 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? , while Particle 2 follows the path parametrized by 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? . At what time 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? does the speed of Particle 1 match the speed of Particle 2?

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Question 61

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

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Question 62

A satellite traverses a circular orbit about the Earth at a speed of 1100 m/s. How far removed from the Earth's surface is the satellite's orbit? Assume the Earth has a radius of 6500 km and mass of A satellite traverses a circular orbit about the Earth at a speed of 1100 m/s. How far removed from the Earth's surface is the satellite's orbit? Assume the Earth has a radius of 6500 km and mass of kg. Recall . kg. Recall A satellite traverses a circular orbit about the Earth at a speed of 1100 m/s. How far removed from the Earth's surface is the satellite's orbit? Assume the Earth has a radius of 6500 km and mass of kg. Recall . .

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Question 63

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

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Question 64

A projectile is fired form the ground at an angle of A projectile is fired form the ground at an angle of with an initial speed of 200 m/s. What is the velocity vector of the projectile after 11 seconds? Recall that is the acceleration due to gravity on the Earth's surface. Approximate the components of your answer to the nearest tenth meter per second. with an initial speed of 200 m/s. What is the velocity vector of the projectile after 11 seconds? Recall that A projectile is fired form the ground at an angle of with an initial speed of 200 m/s. What is the velocity vector of the projectile after 11 seconds? Recall that is the acceleration due to gravity on the Earth's surface. Approximate the components of your answer to the nearest tenth meter per second. is the acceleration due to gravity on the Earth's surface. Approximate the components of your answer to the nearest tenth meter per second.

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Question 65

Find Find . .

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Question 66

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

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Question 67

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

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Question 68

For the parametrization For the parametrization , find the unit normal vector at . , find the unit normal vector at For the parametrization , find the unit normal vector at . .

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Question 69

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

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Question 70

Find Find , and for , and Find , and for for Find , and for

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Question 71

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

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Question 72

Find Find and if , , and . and Find and if , , and . if Find and if , , and . , Find and if , , and . , and Find and if , , and . .

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Question 73

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

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Question 74

Consider the helix Consider the helix . A) Find the unit tangent vector and the unit normal vector B) Find the curvature . A) Find the unit tangent vector Consider the helix . A) Find the unit tangent vector and the unit normal vector B) Find the curvature and the unit normal vector Consider the helix . A) Find the unit tangent vector and the unit normal vector B) Find the curvature B) Find the curvature Consider the helix . A) Find the unit tangent vector and the unit normal vector B) Find the curvature

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Question 75

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

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Question 76

Find the arc length parametrization of the curve Find the arc length parametrization of the curve . .

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Question 77

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

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Question 78

Determine whether the paths Determine whether the paths and intersect. and Determine whether the paths and intersect. intersect. Determine whether the paths and intersect. Determine whether the paths and intersect.

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Question 79

Let 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 , 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 , and 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 . A) Compute 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 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

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