Question 1

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

Accepted Answer

The answer of The path of a projectile is parametrized...

Question 2

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

Accepted Answer

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

Question 3

The aphelion (farthest distance from the Sun) of Mercury is   km, and the perihelion (closest distance to the Sun) is   km.
Find the ratio   , where   and   are the speed of Mercury at the aphelion and perihelion, respectively. Approximate your answer to the nearest hundredth.

Accepted Answer

The answer of The aphelion (farthest distance from the Sun)...

Question 4

The angular momentum in kg m2/s of Mercury when it is located at   km relative to the Sun and has a velocity   m/s and mass   kg is:

Accepted Answer

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

Question 5

Find the tangent line to the curve   at point   .

Accepted Answer

The answer of Find the tangent line to the curve...

Question 6

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

Accepted Answer

The answer of Parametrize the tangent line to the curve...

Question 7

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

Accepted Answer

The answer of Find a parametric equation for the tangent...

Question 8

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

Accepted Answer

@#IMG-DLM& This is the circle

Question 9

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?

Accepted Answer

A) @#IMG-DLM& B) @#IMG-DLM& C)

Question 10

An asteroid traverses a circular orbit of radius 9000 km about a planet of mass   kg. What is the speed of the asteroid? Recall   .

Accepted Answer

The answer of An asteroid traverses a circular orbit of...

Question 11

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

Accepted Answer

The answer of The acceleration vector of a moving particle...

Question 12

The curvature of   at the point where   is:

Accepted Answer

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

Question 13

Let   . Decompose   into tangential and normal components at   .

Accepted Answer

The answer of Let   . Decompose  ...

Question 14

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

Accepted Answer

The answer of Find the curvature of the plane curve...

Question 15

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

Accepted Answer

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

Question 16

A particle's position at time   is given by   . Find   at

Accepted Answer

The answer of A particle's position at time  ...

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

The answer of A particle moves along the curve ...

Question 19

The length of the curve   for   is:

Accepted Answer

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

Question 20

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

Accepted Answer

The answer of Find the point of intersection between the...

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

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

The aphelion (farthest distance from the Sun) of Mercury is km, and the perihelion (closest distance to the Sun) is km. Find the ratio , where and are the speed of Mercury at the aphelion and perihelion, respectively. Approximate your answer to the nearest hundredth.

The angular momentum in kg m²/s of Mercury when it is located at km relative to the Sun and has a velocity m/s and mass kg is:

Find the tangent line to the curve at point .

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

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

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

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?

An asteroid traverses a circular orbit of radius 9000 km about a planet of mass kg. What is the speed of the asteroid? Recall .

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

The curvature of at the point where is:

Let . Decompose into tangential and normal components at .

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

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

A particle's position at time is given by . Find at

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

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

The length of the curve for is:

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

Precalculus Review

Limits

Differentiation

Applications of the Derivative

The Integral

Applications of the Integral

Exponential Functions

Techniques of Integration

Further Applications of the Integral and Taylor Polynomials

Introduction to Differential Equations

Infinite Series

Parametric Equations, Polar Coordinates, and Conic Sections

Vector Geometry

Differentiation in Several Variables

Multiple Integration

Line and Surface Integrals

Fundamental Theorems of Vector Analysis

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Exam 14: Calculus of Vector-Valued Functions