Question 1

Find a Cartesian equation for the line whose polar equation is given.
-$r \cos \left( \theta + \frac { 4 \pi } { 3 } \right) = 2 \sqrt { 3 }$

Accepted Answer

A)  $y = \frac { \sqrt { 3 } } { 3 } x - 2 \sqrt { 3 }$ 
B)  $y = \sqrt { 3 } x + 4$ 
C)  $y = \frac { \sqrt { 3 } } { 3 } x + 4$ 
D)  $y = \sqrt { 3 } x + 2$ 
A)  $y = \frac { \sqrt { 3 } } { 3 } x - 2 \sqrt { 3 }$ 
B)  $y = \sqrt { 3 } x + 4$ 
C)  $y = \frac { \sqrt { 3 } } { 3 } x + 4$ 
D)  $y = \sqrt { 3 } x + 2$

Question 2

Graph.
-$x^{2}+25 y^{2}=25$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 3

Find the directrices of the ellipse.
-$16 x ^ { 2 } + 81 y ^ { 2 } = 1296$

Accepted Answer

A)  $x = \pm \frac { 81 } { \sqrt { 97 } }$ 
B)  $y = \pm \frac { 9 } { \sqrt { 65 } }$ 
C)  $y = \pm \frac { 81 } { \sqrt { 65 } }$ 
D)  $x = \pm \frac { 81 } { \sqrt { 65 } }$ 
A)  $x = \pm \frac { 81 } { \sqrt { 97 } }$ 
B)  $y = \pm \frac { 9 } { \sqrt { 65 } }$ 
C)  $y = \pm \frac { 81 } { \sqrt { 65 } }$ 
D)  $x = \pm \frac { 81 } { \sqrt { 65 } }$

Question 4

The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section.
-e = 2, y = -7

Accepted Answer

A)  $r = \frac { 14 } { 1 - 2 \cos \theta }$ 
B)  $r = \frac { 14 } { 1 + 2 \sin \theta }$ 
C)  $r = \frac { 14 } { 1 + 7 \cos \theta }$ 
D)  $r = \frac { 14 } { 1 - 2 \sin \theta }$ 
A)  $r = \frac { 14 } { 1 - 2 \cos \theta }$ 
B)  $r = \frac { 14 } { 1 + 2 \sin \theta }$ 
C)  $r = \frac { 14 } { 1 + 7 \cos \theta }$ 
D)  $r = \frac { 14 } { 1 - 2 \sin \theta }$

Question 5

Find the foci of the ellipse.
-$$81 x ^ { 2 } + 36 y ^ { 2 } = 2916$$

Accepted Answer

A)  $( 0 , \pm 9 )$ 
B)  $( 0 , \pm 6 )$ 
C)  $( \pm 3 \sqrt { 5 } , 0 )$ 
D)  $( 0 , \pm 3 \sqrt { 5 } )$ 
A)  $( 0 , \pm 9 )$ 
B)  $( 0 , \pm 6 )$ 
C)  $( \pm 3 \sqrt { 5 } , 0 )$ 
D)  $( 0 , \pm 3 \sqrt { 5 } )$

Question 6

Find the area of the specified region.
-Inside the circle $r = \sqrt { 3 } \sin \theta$ and outside the cardioid $r = 1 + \cos \theta$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 4 }$ 
B)  $\frac { 3 \sqrt { 3 } } { 8 }$ 
C)  $\frac { 3 \sqrt { 3 } } { 2 }$ 
D)  $\frac { 3 \sqrt { 3 } } { 4 }$ 
A)  $\frac { \sqrt { 3 } } { 4 }$ 
B)  $\frac { 3 \sqrt { 3 } } { 8 }$ 
C)  $\frac { 3 \sqrt { 3 } } { 2 }$ 
D)  $\frac { 3 \sqrt { 3 } } { 4 }$

Question 7

Find the length of the curve.
-The parabolic segment $r = \frac { 3 } { 1 + \cos \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }$

Accepted Answer

A)  $\frac { 3 } { 2 } ( \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) )$ 
B)  $6 \pi$ 
C)  3 
D)  $\frac { 3 } { 2 } ( \sqrt { 2 } - \ln ( \sqrt { 2 } - 1 ) )$ 
A)  $\frac { 3 } { 2 } ( \sqrt { 2 } + \ln ( \sqrt { 2 } + 1 ) )$ 
B)  $6 \pi$ 
C)  3 
D)  $\frac { 3 } { 2 } ( \sqrt { 2 } - \ln ( \sqrt { 2 } - 1 ) )$

Question 8

Graph.
-$36 x^{2}+12 y^{2}=192$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 9

Find an equation for the line tangent to the curve at the point defined by the given value of t.
-$$\mathrm { x } = \mathrm { t } , \mathrm { y } = \sqrt { 2 \mathrm { t } } , \mathrm { t } = 18$$

Accepted Answer

A)  $y = \frac { 1 } { 6 } x$ 
B)  $y = - \frac { 1 } { 6 } x - 3$ 
C)  $y = \frac { 1 } { 6 } x + 3$ 
D)  $y = - \frac { 1 } { 6 } x$ 
A)  $y = \frac { 1 } { 6 } x$ 
B)  $y = - \frac { 1 } { 6 } x - 3$ 
C)  $y = \frac { 1 } { 6 } x + 3$ 
D)  $y = - \frac { 1 } { 6 } x$

Question 10

Find the directrices of the hyperbola.
-$$y ^ { 2 } - 36 x ^ { 2 } = 36$$

Accepted Answer

A)  $y = \pm \frac { 36 } { \sqrt { 35 } }$ 
B)  $x = \pm \frac { 36 } { \sqrt { 37 } }$ 
C)  $y = \pm \frac { 36 } { \sqrt { 37 } }$ 
D)  $y = \pm \frac { 6 } { \sqrt { 35 } }$ 
A)  $y = \pm \frac { 36 } { \sqrt { 35 } }$ 
B)  $x = \pm \frac { 36 } { \sqrt { 37 } }$ 
C)  $y = \pm \frac { 36 } { \sqrt { 37 } }$ 
D)  $y = \pm \frac { 6 } { \sqrt { 35 } }$

Question 11

Graph.
-$16 x^{2}-9 y^{2}=144$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 12

Determine the symmetries of the curve.
-$$r = 5 \theta$$

Accepted Answer

A)  x-axis, y-axis, origin 
B)  No symmetry 
C)  y-axis only 
D)  x-axis only 
A)  x-axis, y-axis, origin 
B)  No symmetry 
C)  y-axis only 
D)  x-axis only

Question 13

Find the standard-form equation for a hyperbola which satisfies the given conditions.
-A hyperbola centered at the origin having focus at $( 9,0 )$ and eccentricity equal to $\frac { 3 } { 2 }$

Accepted Answer

A)  $\frac { y ^ { 2 } } { 36 } + \frac { x ^ { 2 } } { 45 } = 1$ 
B)  $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 45 } = 1$ 
C)  $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 45 } = 1$ 
D)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 45 } = 1$ 
A)  $\frac { y ^ { 2 } } { 36 } + \frac { x ^ { 2 } } { 45 } = 1$ 
B)  $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 45 } = 1$ 
C)  $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 45 } = 1$ 
D)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 45 } = 1$

Question 14

Find the foci of the hyperbola.
-$4 y ^ { 2 } - 25 x ^ { 2 } = 100$

Accepted Answer

A)  $( 0 , \sqrt { 29 } ) , ( 0 , - \sqrt { 29 } )$ 
B)  $( \sqrt { 21 } , 0 ) , ( - \sqrt { 21 } , 0 )$ 
C)  $( 0 , \sqrt { 21 } ) , ( 0 , - \sqrt { 21 } )$ 
D)  $( \sqrt { 29 } , 0 ) , ( - \sqrt { 29 } , 0 )$ 
A)  $( 0 , \sqrt { 29 } ) , ( 0 , - \sqrt { 29 } )$ 
B)  $( \sqrt { 21 } , 0 ) , ( - \sqrt { 21 } , 0 )$ 
C)  $( 0 , \sqrt { 21 } ) , ( 0 , - \sqrt { 21 } )$ 
D)  $( \sqrt { 29 } , 0 ) , ( - \sqrt { 29 } , 0 )$

Question 15

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 16

Find the directrices of the hyperbola.
-$$x ^ { 2 } - y ^ { 2 } = 25$$

Accepted Answer

A)  $y = \pm \frac { 5 } { \sqrt { 2 } }$ 
B)  $x = \pm \frac { 5 } { \sqrt { 2 } }$ 
C)  $y = \pm 5 \sqrt { 2 }$ 
D)  $x = \pm \frac { 25 } { \sqrt { 2 } }$ 
A)  $y = \pm \frac { 5 } { \sqrt { 2 } }$ 
B)  $x = \pm \frac { 5 } { \sqrt { 2 } }$ 
C)  $y = \pm 5 \sqrt { 2 }$ 
D)  $x = \pm \frac { 25 } { \sqrt { 2 } }$

Question 17

Find the vertices and foci of the ellipse.
-$$\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 144 } = 1$$

Accepted Answer

A)  Vertices: $( \pm 20,0 )$; Foci: $( \pm 12,0 )$ 
B)  Vertices: $( \pm 20,0 )$; Foci: $( \pm 16,0 )$ 
C)  Vertices: $( 0 , \pm 20 )$; Foci: $( 0 , \pm 16 )$ 
D)  Vertices: $( 0 , \pm 20 )$; Foci: $( 0 , \pm 12 )$ 
A)  Vertices: $( \pm 20,0 )$; Foci: $( \pm 12,0 )$ 
B)  Vertices: $( \pm 20,0 )$; Foci: $( \pm 16,0 )$ 
C)  Vertices: $( 0 , \pm 20 )$; Foci: $( 0 , \pm 16 )$ 
D)  Vertices: $( 0 , \pm 20 )$; Foci: $( 0 , \pm 12 )$

Question 18

Graph.
-$9 x^{2}-4 y^{2}=36$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 19

Replace the Cartesian equation with an equivalent polar equation.
-$$x ^ { 2 } + ( y - 25 ) ^ { 2 } = 625$$

Accepted Answer

A)  $r = 50 \sin \theta$ 
B)  $\mathrm { r } ^ { 2 } = 50 \cos \theta$ 
C)  $r = 25 \sin \theta$ 
D)  $r = 50 \cos \theta$ 
A)  $r = 50 \sin \theta$ 
B)  $\mathrm { r } ^ { 2 } = 50 \cos \theta$ 
C)  $r = 25 \sin \theta$ 
D)  $r = 50 \cos \theta$

Question 20

Solve the problem.
-Find the foci and asymptotes of the following hyperbola: $x ^ { 2 } - y ^ { 2 } = 32$

Accepted Answer

A)  Foci: (4, 0), (-4, 0); Asymptotes: y = x, y = - x 
B)  Foci: (8, 0), (-8, 0); Asymptotes: y =2x, y = - 2x 
C)  Foci: (0, 8), (0, -8); Asymptotes: y = x, y = - x 
D)  Foci: (8, 0), (-8, 0); Asymptotes: y = x, y = - x 
A)  Foci: (4, 0), (-4, 0); Asymptotes: y = x, y = - x 
B)  Foci: (8, 0), (-8, 0); Asymptotes: y =2x, y = - 2x 
C)  Foci: (0, 8), (0, -8); Asymptotes: y = x, y = - x 
D)  Foci: (8, 0), (-8, 0); Asymptotes: y = x, y = - x

Find a Cartesian equation for the line whose polar equation is given. - $r \cos \left( \theta + \frac { 4 \pi } { 3 } \right) = 2 \sqrt { 3 }$

Graph. - $x^{2}+25 y^{2}=25$ $Graph. - x^{2}+25 y^{2}=25$

Find the directrices of the ellipse. - $16 x ^ { 2 } + 81 y ^ { 2 } = 1296$

The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. -e = 2, y = -7

Find the foci of the ellipse. - $81 x ^ { 2 } + 36 y ^ { 2 } = 2916$

Find the area of the specified region. -Inside the circle $r = \sqrt { 3 } \sin \theta$ and outside the cardioid $r = 1 + \cos \theta$

Find the length of the curve. -The parabolic segment $r = \frac { 3 } { 1 + \cos \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }$

Graph. - $36 x^{2}+12 y^{2}=192$ $Graph. - 36 x^{2}+12 y^{2}=192$

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $\mathrm { x } = \mathrm { t } , \mathrm { y } = \sqrt { 2 \mathrm { t } } , \mathrm { t } = 18$

Find the directrices of the hyperbola. - $y ^ { 2 } - 36 x ^ { 2 } = 36$

Graph. - $16 x^{2}-9 y^{2}=144$ $Graph. - 16 x^{2}-9 y^{2}=144$

Determine the symmetries of the curve. - $r = 5 \theta$

Find the standard-form equation for a hyperbola which satisfies the given conditions. -A hyperbola centered at the origin having focus at $( 9,0 )$ and eccentricity equal to $\frac { 3 } { 2 }$

Find the foci of the hyperbola. - $4 y ^ { 2 } - 25 x ^ { 2 } = 100$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi$

Find the directrices of the hyperbola. - $x ^ { 2 } - y ^ { 2 } = 25$

Find the vertices and foci of the ellipse. - $\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 144 } = 1$

Graph. - $9 x^{2}-4 y^{2}=36$ $Graph. - 9 x^{2}-4 y^{2}=36$

Replace the Cartesian equation with an equivalent polar equation. - $x ^ { 2 } + ( y - 25 ) ^ { 2 } = 625$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $x ^ { 2 } - y ^ { 2 } = 32$

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

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Exam 12: Parametric Equations and Polar Coordinates

Find a Cartesian equation for the line whose polar equation is given. - rcos⁡(θ+4π3)=23r \cos \left( \theta + \frac { 4 \pi } { 3 } \right) = 2 \sqrt { 3 }rcos(θ+34π​)=23​

Graph. - x2+25y2=25x^{2}+25 y^{2}=25x2+25y2=25

Find the directrices of the ellipse. - 16x2+81y2=129616 x ^ { 2 } + 81 y ^ { 2 } = 129616x2+81y2=1296

The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. -e = 2, y = -7

Find the foci of the ellipse. - 81x2+36y2=291681 x ^ { 2 } + 36 y ^ { 2 } = 291681x2+36y2=2916

Find the area of the specified region. -Inside the circle r=3sin⁡θr = \sqrt { 3 } \sin \thetar=3​sinθ and outside the cardioid r=1+cos⁡θr = 1 + \cos \thetar=1+cosθ

Find the length of the curve. -The parabolic segment r=31+cos⁡θ,0≤θ≤π2r = \frac { 3 } { 1 + \cos \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }r=1+cosθ3​,0≤θ≤2π​

Graph. - 36x2+12y2=19236 x^{2}+12 y^{2}=19236x2+12y2=192

Find an equation for the line tangent to the curve at the point defined by the given value of t. - x=t,y=2t,t=18\mathrm { x } = \mathrm { t } , \mathrm { y } = \sqrt { 2 \mathrm { t } } , \mathrm { t } = 18x=t,y=2t​,t=18

Find the directrices of the hyperbola. - y2−36x2=36y ^ { 2 } - 36 x ^ { 2 } = 36y2−36x2=36

Graph. - 16x2−9y2=14416 x^{2}-9 y^{2}=14416x2−9y2=144

Determine the symmetries of the curve. - r=5θr = 5 \thetar=5θ

Find the standard-form equation for a hyperbola which satisfies the given conditions. -A hyperbola centered at the origin having focus at (9,0)( 9,0 )(9,0) and eccentricity equal to 32\frac { 3 } { 2 }23​

Find the foci of the hyperbola. - 4y2−25x2=1004 y ^ { 2 } - 25 x ^ { 2 } = 1004y2−25x2=100

Graph the pair of parametric equations with the aid of a graphing calculator. - x=7t−7sin⁡t,y=7−7cos⁡t,−2π≤t≤2πx=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pix=7t−7sint,y=7−7cost,−2π≤t≤2π

Find the directrices of the hyperbola. - x2−y2=25x ^ { 2 } - y ^ { 2 } = 25x2−y2=25

Find the vertices and foci of the ellipse. - x2400+y2144=1\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 144 } = 1400x2​+144y2​=1

Graph. - 9x2−4y2=369 x^{2}-4 y^{2}=369x2−4y2=36

Replace the Cartesian equation with an equivalent polar equation. - x2+(y−25)2=625x ^ { 2 } + ( y - 25 ) ^ { 2 } = 625x2+(y−25)2=625

Solve the problem. -Find the foci and asymptotes of the following hyperbola: x2−y2=32x ^ { 2 } - y ^ { 2 } = 32x2−y2=32

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Find a Cartesian equation for the line whose polar equation is given. - $r \cos \left( \theta + \frac { 4 \pi } { 3 } \right) = 2 \sqrt { 3 }$

Graph. - $x^{2}+25 y^{2}=25$ $Graph. - x^{2}+25 y^{2}=25$

Find the directrices of the ellipse. - $16 x ^ { 2 } + 81 y ^ { 2 } = 1296$

Find the foci of the ellipse. - $81 x ^ { 2 } + 36 y ^ { 2 } = 2916$

Find the area of the specified region. -Inside the circle $r = \sqrt { 3 } \sin \theta$ and outside the cardioid $r = 1 + \cos \theta$

Find the length of the curve. -The parabolic segment $r = \frac { 3 } { 1 + \cos \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }$

Graph. - $36 x^{2}+12 y^{2}=192$ $Graph. - 36 x^{2}+12 y^{2}=192$

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $\mathrm { x } = \mathrm { t } , \mathrm { y } = \sqrt { 2 \mathrm { t } } , \mathrm { t } = 18$

Find the directrices of the hyperbola. - $y ^ { 2 } - 36 x ^ { 2 } = 36$

Graph. - $16 x^{2}-9 y^{2}=144$ $Graph. - 16 x^{2}-9 y^{2}=144$

Determine the symmetries of the curve. - $r = 5 \theta$

Find the standard-form equation for a hyperbola which satisfies the given conditions. -A hyperbola centered at the origin having focus at $( 9,0 )$ and eccentricity equal to $\frac { 3 } { 2 }$

Find the foci of the hyperbola. - $4 y ^ { 2 } - 25 x ^ { 2 } = 100$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=7 t-7 \sin t, y=7-7 \cos t,-2 \pi \leq t \leq 2 \pi$

Find the directrices of the hyperbola. - $x ^ { 2 } - y ^ { 2 } = 25$

Find the vertices and foci of the ellipse. - $\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 144 } = 1$

Graph. - $9 x^{2}-4 y^{2}=36$ $Graph. - 9 x^{2}-4 y^{2}=36$

Replace the Cartesian equation with an equivalent polar equation. - $x ^ { 2 } + ( y - 25 ) ^ { 2 } = 625$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $x ^ { 2 } - y ^ { 2 } = 32$