Question 1

Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $\frac { 6 } { 7 }$ and directrix $x = - 13$.

Accepted Answer

The answer of Write a polar equation in $r$ and...

Question 2

Write a polar equation in $r$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$.

Accepted Answer

The answer of Write a polar equation in $r$ and...

Question 3

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.
$$x = \cos \theta + \sin 2 \theta + 8 , y = \sin \theta + \cos 2 \theta + 8 , \theta = \pi$$

Accepted Answer

The answer of Find an equation of the tangent to...

Question 4

$$\text { Find the surface area generated by rotating the lemniscate } r ^ { 2 } = 10 \cos 2 \theta \text { about the line } \theta = \pi \text {. }$$

Accepted Answer

The answer of \[\text { Find the surface area generated...

Question 5

The curve $x = 5 - 10 \cos ^ { 2 } t , y = \tan t \left( 1 - 2 \cos ^ { 2 } t \right)$ cross itself at some point $\left( x _ { 0 } , y _ { 0 } \right)$. Find the equations of both tangent lines at that point.

Accepted Answer

A)  $y = \frac { x } { 5 } , y = - \frac { x } { 10 }$ 
B) 
$y = \frac { x } { 2 } - 10 , y = - \frac { x } { 2 } - 2$ 
C)  $y = x + 5 , y = - x + 5$ 
D) 
$y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ 
E) 
$y = \frac { x } { 2 } + 8 , y = - \frac { x } { 2 } + 2$ 
A)  $y = \frac { x } { 5 } , y = - \frac { x } { 10 }$ 
B) 
$y = \frac { x } { 2 } - 10 , y = - \frac { x } { 2 } - 2$ 
C)  $y = x + 5 , y = - x + 5$ 
D) 
$y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ 
E) 
$y = \frac { x } { 2 } + 8 , y = - \frac { x } { 2 } + 2$

Question 6

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude $106 \mathrm {~km}$ and apolune altitude $318 \mathrm {~km}$ (above the moon). Find an equation of this ellipse if the radius of the moon is $1728 \mathrm {~km}$ and the center of the moon is at one focus.

Accepted Answer

The answer of The point in a lunar orbit nearest...

Question 7

Find the eccentricity of the conic. Select the correct answer.
$$r = \frac { 5 } { 8 - 5 \sin \theta }$$

Accepted Answer

A)  $e = \frac { 8 } { 5 }$ 
B)  $e = 8$ 
C)  $e = \frac { 5 } { 8 }$ 
D)  $e = - 5$ 
E)  $e = 5$ 
A)  $e = \frac { 8 } { 5 }$ 
B)  $e = 8$ 
C)  $e = \frac { 5 } { 8 }$ 
D)  $e = - 5$ 
E)  $e = 5$

Question 8

Find an equation of the conic satisfying the given conditions.
Hyperbola, foci $( 5,6 )$ and $( 5 , - 4 )$, asymptotes $x = 2 y + 3$ and $x = - 2 y + 7$

Accepted Answer

The answer of Find an equation of the conic satisfying...

Question 9

Find the length of the curve. Select the correct answer.
$$x = 3 t ^ { 2 } + 8 , y = 2 t ^ { 3 } + 8,0 \leq t \leq 1$$

Accepted Answer

A)  $2 \sqrt { 2 } - 2$ 
B)  $4 \sqrt { 2 }$ 
C)  $4 \sqrt { 2 } - 1$ 
D)  $4 \sqrt { 2 } - 2$ 
E)  None of these 
A)  $2 \sqrt { 2 } - 2$ 
B)  $4 \sqrt { 2 }$ 
C)  $4 \sqrt { 2 } - 1$ 
D)  $4 \sqrt { 2 } - 2$ 
E)  None of these

Question 10

Find an equation of the hyperbola centered at the origin that satisfies the given condition.
Vertices: $( \pm 4,0 )$, asymptotes: $y = \pm \frac { 7 } { 4 } x$

Accepted Answer

The answer of Find an equation of the hyperbola centered...

Question 11

Describe the motion of a particle with position $( x , y )$ as $t$ varies in the given interval $0 \leq t \leq 2 \pi$.
$$x = 8 \sin t , y = 5 \cos t$$

Accepted Answer

A)  Moves once counterclockwise along the circle $x ^ { 2 } + y ^ { 2 } = 1$ starting and ending at $( 0 , - 5 )$ 
B) 
Moves once counterclockwise along the ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 25 } = 1$ starting and ending at $( 0,5 )$. 
C) 
Moves once counterclockwise along the ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 8 } = 1$ starting and ending at $( - 5,0 )$ 
D) 
Moves once clockwise along the ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 25 } = 1$ starting and ending at $( 0,5 )$. 
E)  Moves once clockwise along the circle $( 8 x ) ^ { 2 } + ( 5 y ) ^ { 2 } = 1$ starting and ending at $( 0,5 )$. 
A)  Moves once counterclockwise along the circle $x ^ { 2 } + y ^ { 2 } = 1$ starting and ending at $( 0 , - 5 )$ 
B) 
Moves once counterclockwise along the ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 25 } = 1$ starting and ending at $( 0,5 )$. 
C) 
Moves once counterclockwise along the ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 8 } = 1$ starting and ending at $( - 5,0 )$ 
D) 
Moves once clockwise along the ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 25 } = 1$ starting and ending at $( 0,5 )$. 
E)  Moves once clockwise along the circle $( 8 x ) ^ { 2 } + ( 5 y ) ^ { 2 } = 1$ starting and ending at $( 0,5 )$.

Question 12

A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is $18 \mathrm {~cm}$. Find an equation of the parabola. Let $V$ be the origin. Find the diameter of the opening $| C D | , 19 \mathrm {~cm}$ from the vertex.

Accepted Answer

A)  $| C D | = 6 \sqrt { 38 }$ 
B)  $| C D | = 18$ 
C)  The equation is $y = 18 x ^ { 2 }$ 
D)  $| C D | = 4 \sqrt { 414 }$ 
E)  The equation is $y ^ { 2 } = 18 x$
F)
The equation is $y ^ { 2 } - \frac { x ^ { 2 } } { 18 } = 1$ 
A)  $| C D | = 6 \sqrt { 38 }$ 
B)  $| C D | = 18$ 
C)  The equation is $y = 18 x ^ { 2 }$ 
D)  $| C D | = 4 \sqrt { 414 }$ 
E)  The equation is $y ^ { 2 } = 18 x$
F)
The equation is $y ^ { 2 } - \frac { x ^ { 2 } } { 18 } = 1$

Question 13

The orbit of Hale-Bopp comet, discovered in 1995 , is an ellipse with eccentricity $0.995$ and one focus at the Sun. The length of its major axis is $366.5 \mathrm { AU }$. [An astronomical unit (AU) is the mean distance between Earth and the Sun, about 93 million miles.] Find the maximum distance from the comet to the Sun. (The perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$.) Find the answer in $\mathrm { AU }$ and round to the nearest hundredth.

Accepted Answer

The answer of The orbit of Hale-Bopp comet, discovered in...

Question 14

Find an equation of the hyperbola centered at the origin that satisfies the given condition.
Vertices: $( \pm 4,0 )$, asymptotes: $y = \pm \frac { 7 } { 4 } x$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 49 } = 1$ 
B)  $\frac { y ^ { 2 } } { 49 } - \frac { x ^ { 2 } } { 16 } = 1$ 
C) 
$\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 49 } = 1$ 
D) 
$\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 16 } = 1$ 
A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 49 } = 1$ 
B)  $\frac { y ^ { 2 } } { 49 } - \frac { x ^ { 2 } } { 16 } = 1$ 
C) 
$\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 49 } = 1$ 
D) 
$\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 16 } = 1$

Question 15

Find the area of the region that lies inside the first curve and outside the second curve.
$$r = 3 \cos \theta , \quad r = 1 + \cos \theta$$

Accepted Answer

The answer of Find the area of the region that...

Question 16

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.
$$x = \cos \theta , y = 5 \sec \theta , 0 \leq \theta < \frac { \pi } { 2 }$$

Accepted Answer

The answer of Sketch the parametric curve and eliminate the...

Question 17

Suppose a planet is discovered that revolves around its sun in an elliptical orbit with the sun at one focus. Its perihelion distance (minimum distance from the planet to the sun) is approximately $2.3 \times 10 ^ { 7 } \mathrm {~km}$, and its aphelion distance (maximum distance from the planet to the sun) is approximately $2.7 \times 10 ^ { 7 } \mathrm {~km}$. Approximate the eccentricity of the planet's orbit. Round to three decimal places.

Accepted Answer

The answer of Suppose a planet is discovered that revolves...

Question 18

Write a polar equation in $\mathrm { r }$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$.

Accepted Answer

The answer of Write a polar equation in \(\mathrm {...

Question 19

Find the vertices, foci, and asymptotes of the hyperbola.
$$y ^ { 2 } - 5 x ^ { 2 } = 25$$

Accepted Answer

Vertices:

Question 20

Find the polar equation for the curve represented by the given Cartesian equation. Select the correct answer.
$x + y = 2$

Accepted Answer

A)  $r = 2 ( \cos \theta + \sin \theta )$ 
B) 
$r = \frac { 1 } { \cos \theta - \sin \theta }$ 
C) 
$r = \frac { 2 } { \cos \theta + \sin \theta }$ 
D) 
$r = \frac { 2 } { \cos \theta - \sin \theta }$ 
E)  $r = 1 ( \cos \theta + \sin \theta )$ 
A)  $r = 2 ( \cos \theta + \sin \theta )$ 
B) 
$r = \frac { 1 } { \cos \theta - \sin \theta }$ 
C) 
$r = \frac { 2 } { \cos \theta + \sin \theta }$ 
D) 
$r = \frac { 2 } { \cos \theta - \sin \theta }$ 
E)  $r = 1 ( \cos \theta + \sin \theta )$

Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $\frac { 6 } { 7 }$ and directrix $x = - 13$ .

Write a polar equation in $r$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$ .

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = \cos \theta + \sin 2 \theta + 8 , y = \sin \theta + \cos 2 \theta + 8 , \theta = \pi$

$\text { Find the surface area generated by rotating the lemniscate } r ^ { 2 } = 10 \cos 2 \theta \text { about the line } \theta = \pi \text {. }$

The curve $x = 5 - 10 \cos ^ { 2 } t , y = \tan t \left( 1 - 2 \cos ^ { 2 } t \right)$ cross itself at some point $\left( x _ { 0 } , y _ { 0 } \right)$ . Find the equations of both tangent lines at that point.

Find the eccentricity of the conic. Select the correct answer. $r = \frac { 5 } { 8 - 5 \sin \theta }$

Find an equation of the conic satisfying the given conditions. Hyperbola, foci $( 5,6 )$ and $( 5 , - 4 )$ , asymptotes $x = 2 y + 3$ and $x = - 2 y + 7$

Find the length of the curve. Select the correct answer. $x = 3 t ^ { 2 } + 8 , y = 2 t ^ { 3 } + 8,0 \leq t \leq 1$

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: $( \pm 4,0 )$ , asymptotes: $y = \pm \frac { 7 } { 4 } x$

Describe the motion of a particle with position $( x , y )$ as $t$ varies in the given interval $0 \leq t \leq 2 \pi$ . $x = 8 \sin t , y = 5 \cos t$

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: $( \pm 4,0 )$ , asymptotes: $y = \pm \frac { 7 } { 4 } x$

Find the area of the region that lies inside the first curve and outside the second curve. $r = 3 \cos \theta , \quad r = 1 + \cos \theta$

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $x = \cos \theta , y = 5 \sec \theta , 0 \leq \theta < \frac { \pi } { 2 }$

Write a polar equation in $\mathrm { r }$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$ .

Find the vertices, foci, and asymptotes of the hyperbola. $y ^ { 2 } - 5 x ^ { 2 } = 25$

Find the polar equation for the curve represented by the given Cartesian equation. Select the correct answer. $x + y = 2$

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Exam 10: Parametric Equations and Polar Coordinates

Write a polar equation in rrr and θ\thetaθ of an ellipse with the focus at the origin, with the eccentricity 67\frac { 6 } { 7 }76​ and directrix x=−13x = - 13x=−13 .

Write a polar equation in rrr and θ\thetaθ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix r=−12csc⁡θr = - 12 \csc \thetar=−12cscθ .

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x=cos⁡θ+sin⁡2θ+8,y=sin⁡θ+cos⁡2θ+8,θ=πx = \cos \theta + \sin 2 \theta + 8 , y = \sin \theta + \cos 2 \theta + 8 , \theta = \pix=cosθ+sin2θ+8,y=sinθ+cos2θ+8,θ=π

The curve x=5−10cos⁡2t,y=tan⁡t(1−2cos⁡2t)x = 5 - 10 \cos ^ { 2 } t , y = \tan t \left( 1 - 2 \cos ^ { 2 } t \right)x=5−10cos2t,y=tant(1−2cos2t) cross itself at some point (x0,y0)\left( x _ { 0 } , y _ { 0 } \right)(x0​,y0​) . Find the equations of both tangent lines at that point.

Find the eccentricity of the conic. Select the correct answer. r=58−5sin⁡θr = \frac { 5 } { 8 - 5 \sin \theta }r=8−5sinθ5​

Find an equation of the conic satisfying the given conditions. Hyperbola, foci (5,6)( 5,6 )(5,6) and (5,−4)( 5 , - 4 )(5,−4) , asymptotes x=2y+3x = 2 y + 3x=2y+3 and x=−2y+7x = - 2 y + 7x=−2y+7

Find the length of the curve. Select the correct answer. x=3t2+8,y=2t3+8,0≤t≤1x = 3 t ^ { 2 } + 8 , y = 2 t ^ { 3 } + 8,0 \leq t \leq 1x=3t2+8,y=2t3+8,0≤t≤1

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (±4,0)( \pm 4,0 )(±4,0) , asymptotes: y=±74xy = \pm \frac { 7 } { 4 } xy=±47​x

Describe the motion of a particle with position (x,y)( x , y )(x,y) as ttt varies in the given interval 0≤t≤2π0 \leq t \leq 2 \pi0≤t≤2π . x=8sin⁡t,y=5cos⁡tx = 8 \sin t , y = 5 \cos tx=8sint,y=5cost

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (±4,0)( \pm 4,0 )(±4,0) , asymptotes: y=±74xy = \pm \frac { 7 } { 4 } xy=±47​x

Find the area of the region that lies inside the first curve and outside the second curve. r=3cos⁡θ,r=1+cos⁡θr = 3 \cos \theta , \quad r = 1 + \cos \thetar=3cosθ,r=1+cosθ

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. x=cos⁡θ,y=5sec⁡θ,0≤θ<π2x = \cos \theta , y = 5 \sec \theta , 0 \leq \theta < \frac { \pi } { 2 }x=cosθ,y=5secθ,0≤θ<2π​

Write a polar equation in r\mathrm { r }r and θ\thetaθ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix r=−12csc⁡θr = - 12 \csc \thetar=−12cscθ .

Find the vertices, foci, and asymptotes of the hyperbola. y2−5x2=25y ^ { 2 } - 5 x ^ { 2 } = 25y2−5x2=25

Find the polar equation for the curve represented by the given Cartesian equation. Select the correct answer. x+y=2x + y = 2x+y=2

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $\frac { 6 } { 7 }$ and directrix $x = - 13$ .

Write a polar equation in $r$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$ .

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = \cos \theta + \sin 2 \theta + 8 , y = \sin \theta + \cos 2 \theta + 8 , \theta = \pi$

The curve $x = 5 - 10 \cos ^ { 2 } t , y = \tan t \left( 1 - 2 \cos ^ { 2 } t \right)$ cross itself at some point $\left( x _ { 0 } , y _ { 0 } \right)$ . Find the equations of both tangent lines at that point.

Find the eccentricity of the conic. Select the correct answer. $r = \frac { 5 } { 8 - 5 \sin \theta }$

Find an equation of the conic satisfying the given conditions. Hyperbola, foci $( 5,6 )$ and $( 5 , - 4 )$ , asymptotes $x = 2 y + 3$ and $x = - 2 y + 7$

Find the length of the curve. Select the correct answer. $x = 3 t ^ { 2 } + 8 , y = 2 t ^ { 3 } + 8,0 \leq t \leq 1$

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: $( \pm 4,0 )$ , asymptotes: $y = \pm \frac { 7 } { 4 } x$

Describe the motion of a particle with position $( x , y )$ as $t$ varies in the given interval $0 \leq t \leq 2 \pi$ . $x = 8 \sin t , y = 5 \cos t$

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: $( \pm 4,0 )$ , asymptotes: $y = \pm \frac { 7 } { 4 } x$

Find the area of the region that lies inside the first curve and outside the second curve. $r = 3 \cos \theta , \quad r = 1 + \cos \theta$

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $x = \cos \theta , y = 5 \sec \theta , 0 \leq \theta < \frac { \pi } { 2 }$

Write a polar equation in $\mathrm { r }$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$ .

Find the vertices, foci, and asymptotes of the hyperbola. $y ^ { 2 } - 5 x ^ { 2 } = 25$

Find the polar equation for the curve represented by the given Cartesian equation. Select the correct answer. $x + y = 2$