Deck 12: Parametric Equations, Polar Coordinates, and Conic Sections

Question 1

is a parametrization of which curve?

A)
B)
C)
D)
E) None of the above

Answer 1

Question 2

Find parametric equations for the curve .

Answer 2

, where (There are other possible answers.)

Question 3

Eliminate the parameter and write an equation in and for the parametric curve

Answer 3

Question 4

The right hyperbola is parametrized by which of the following equations?

A)
B)
C)
D)
E)

Question 5

The slope of the curve at is which of the following?

A)
B)
C)
D)
E)

Question 6

Find the area of the surface generated by revolving the curve about the x-axis.

Question 7

The surface area of the surface generated by revolving the parametric curve about the x-axis for is given by the formula .
Compute the area of the surface generated by revolving the parametric curve about the x-axis.

Question 8

Find the speed and the length of the curve.

Question 9

Find the slope of the tangent line to the curve at .

Question 10

Find parametric equations for the curve .

Question 11

Find the speed and arc length of , .

Question 12

Express the parametric curve by an equation in and

Question 13

Find the equation of the tangent line to the parametric curve at .

Question 14

A moving particle is following the path . At the particle leaves the path on a straight line tangent to the path. If the particle maintains the same speed as when it left the path, what are the x- and y-coordinates of the particle's position when ?

Question 15

A particle travels along the path (where length is in feet and time is in minutes). Find the speed of the particle and compute its displacement and the distance covered during the first minute of its motion.

Question 16

Find parametric equations for the curve .

Question 17

Calculate the area under the parametrized curve given by for .

Question 18

The slope of the curve at is which of the following?

A)
B)
C)
D)

Question 19

Find the speed and the length of the curve.

Question 20

Calculate the area under the parametrized curve given by for .

Question 21

Find the arc length of the parametric curve for .

Question 22

Write the equation of in polar coordinates.

Question 23

The surface area of the surface generated by revolving a parametric curve about the x-axis for is given by the formula .
Compute the area of the surface generated by revolving the curve , about the x-axis.

Question 24

Write the equation of in polar coordinates.

Question 25

Write the equation of in Cartesian coordinates.

Question 26

The equation of in Cartesian coordinates is which of the following?

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

Question 27

Write the equation in rectangular coordinates.

Question 28

Find the arc length of the curve given by the parametric equations

Question 29

Find the arc length of the parametric curve .

Question 30

Find the polar equation of the line .

Question 31

Find the speed at and the arc length of the parametric curve between the points (corresponding to ) and (corresponding to ).

Question 32

Find the polar equation of the line through orthogonal to the line .

Question 33

The surface area of the surface generated by revolving a parametric curve about the x-axis for is given by the formula .
Compute the area of the surface generated by revolving the curve about the x-axis.

Question 34

Compute the length of the parametric curve .

Question 35

The equation of in Cartesian coordinates is which of the following?

A)
B)
C)
D)
E)

Question 36

Find the arc length of the loop of the curve given by the parametric equations .

Question 37

The length of the parametric curve is which of the following?

A)
B)
C)
D)
E)
The length integral cannot be computed analytically by the methods covered so far.

Question 38

A particle travels along a path (where time is measured in minutes and length in feet). Find the speed of the particle at , and compute the displacement and the distance traveled during the first min.

Question 39

Match the polar equations with their Cartesian form

A)
B)
C)
D)
i)
ii)
iii)
iv)

Question 40

Match the polar equations with their Cartesian form

A)
B)
C)
D)
i)
ii)
iii)
iv)

Question 41

Suppose that has polar coordinates , where x and y are not both zero.
Find the polar coordinates of the following.

A)
B)
C)
D)

Question 42

Find the area of the inner loop of .

Question 43

Give the polar coordinates of the point of intersection of the lines and .

Question 44

Convert the equation to polar coordinates.

Question 45

Consider the limaçon .

A) Express as functions of , and find and .
B) Find the slopes of the tangent lines to the inner loop of the limaçon at the origin.

Question 46

Determine the area of the inner loop of .

Question 47

The Cartesian coordinates of the point on the polar curve closest to the origin are:

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

Question 48

Find a parametrization for the curve from to .

Question 49

Find the length of the polar curve .
Use: .

Question 50

Find the polar equation of the line .

Question 51

Find the area of the inner loop of .

Question 52

Find the area enclosed by the leaf of the four-leaved rose

Question 53

The exact value of the area that lies inside and outside is approximately which of the following?

A) 36.03
B) 10.03
C) 16.3
D) 18.226
E) none of the above.

Question 54

The area inside and outside is approximately which of the following?

A) 1.571
B) 1.913
C) 1.228
D) 2.094
E) 4.188

Question 55

The area inside but outside is approximately which of the following?

A) 1.571
B) 1.913
C) 1.228
D) 1.047
E) 0.956

Question 56

The equation of in Cartesian coordinates is which of the following?

A)
B)
C)
D)
E)

Question 57

The area that lies outside and inside is approximately which of the following?

A) 36.03
B) 16.3
C) 3.57
D) 10.03
E) 9.11

Question 58

Find the area of the shaded region in the following figure.

Question 59

The area outside but inside is which of the following?

A)
B) 16
C)
D) 32
E) 8

Question 60

Find the length of the polar curve .

Question 61

The part of the parabola has a length of .
Find the focus and directrix of the parabola.
Use .

Question 62

Consider the conic section .

A) Identify the conic.
B) Find the foci, vertices, and eccentricity of the conic.

Question 63

Determine the standard equation of the parabola whose focus is the point and directrix is the line .

Question 64

Write the standard equation for the ellipse with foci at and , and vertices at and .

Question 65

Which of the following equations describes the graph of a hyperbola?

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

Question 66

Write the standard equation for the ellipse with the following characteristics: center , vertex , and focus .

Question 67

Find the area inside the cardioid and below the line

Question 68

Find the area enclosed by one turn of the logarithmic spiral .

Question 69

Identify the type of conic section whose polar equation is .

Question 70

The polar curve is the equation of:

A)
a circle.
B)
an ellipse.
C)
a parabola.
D)
a hyperbola.
E)
a line.

Question 71

Find the equation for the ellipse with foci at and and eccentricity .